799 results

equal - being the same in quantity, size, degree, or value

Formula: =

-2 <= x +4 < 9

-2 <= x +4 < 9
Subtract 4 from each piece:
-2 - 4 <= x < 5
Simplify:
[B]-6 <= x < 5
[/B]
To find the interval notation, we set up our notation:
[LIST]
[*]The left side has a solid bracket, since we have an equal sign:
[*]The right side has an open parentheses, since we have no equal sign
[*][B][-6, 5)[/B]
[/LIST]

-28 is the solution to the sum of a number p and 21

-28 is the solution to the sum of a number p and 21
The sum of a number p and 21:
p + 21
The phrase [I]is the solution to[/I] means an equation, so we set p + 21 equal to -28:
[B]p + 21 = -28
[/B]
If the problem asks you to solve for p, then we [URL='https://www.mathcelebrity.com/1unk.php?num=p%2B21%3D-28&pl=Solve']type this into our search engine[/URL] and we get:
p = [B]-49[/B]

-3x<= -9 or 5+x<6

-3x<= -9 or 5+x<6
Take each piece:
-3x<= -9
Divide each side by -3:
x>=3
Now take 5 + x < 6
5 + x < 6
Subtract 5 from each side:
x < 1
Joining together the two inequalities, we have:
x<1 or x>=3
Use our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3C1orx%3E%3D3&pl=Show+Interval+Notation']interval notion calculator[/URL] to find the interval notation of this compound inequality

-65 times the difference between a number and 79 is equal to the number plus 98

-65 times the difference between a number and 79 is equal to the number plus 98
The phrase [I]a number[/I] means an arbitrary variable. Let's call it x.
The first expression, [I]the difference between a number and 79[/I] means we subtract 79 from our arbitrary variable of x:
x - 79
Next, -65 times the difference between a number and 79 means we multiply our result above by -65:
-65(x - 79)
The phrase [I]the number[/I] refers to the arbitrary variable x earlier. The number plus 98 means we add 98 to x:
x + 98
Now, let's bring it all together. The phrase [I]is equal to[/I] means an equation. So we set -65(x - 79) equal to x + [B]98:
-65(x - 79) = x + 98[/B] <-- This is our algebraic expression
If the problem asks you to take it a step further and solve for x, then you [URL='https://www.mathcelebrity.com/1unk.php?num=-65%28x-79%29%3Dx%2B98&pl=Solve']type this equation into our search engine[/URL], and you get:
x = [B]76.31818[/B]

1 apartment equals 10 window each window cost $79.30 there are 30 apartments what is the total for r

1 apartment equals 10 window each window cost $79.30 there are 30 apartments what is the total for replacing all the windows.
30 apartments * 10 windows per apartment * $79.30 per window = [B]$23,790[/B]

1 Die Roll

Free 1 Die Roll Calculator - Calculates the probability for the following events in the roll of one fair dice (1 dice roll calculator or 1 die roll calculator):

* Probability of any total from (1-6)

* Probability of the total being less than, less than or equal to, greater than, or greater than or equal to (1-6)

* The total being even

* The total being odd

* The total being a prime number

* The total being a non-prime number

* Rolling a list of numbers i.e. (2,5,6)

* Simulate (n) Monte Carlo die simulations.

1 die calculator

* Probability of any total from (1-6)

* Probability of the total being less than, less than or equal to, greater than, or greater than or equal to (1-6)

* The total being even

* The total being odd

* The total being a prime number

* The total being a non-prime number

* Rolling a list of numbers i.e. (2,5,6)

* Simulate (n) Monte Carlo die simulations.

1 die calculator

1/2 of x and 10 is 30. Find the x.

1/2 of x and 10 is 30. Find the x.
x and 10 means we add:
x + 10
1/2 of this:
1/2(x + 10)
The phrase is means equal to, so we set 1/2(x + 10) equal to 30 for our algebraic expression
[B]1/2(x + 10) = 30[/B]

1/3 of the sum of a number and 2 plus 5 is -20

1/3 of the sum of a number and 2 plus 5 is -20
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
the sum of a number and 2:
x + 2
1/3 of the sum of a number and 2
1/3(x + 2)
1/3 of the sum of a number and 2 plus 5
1/3(x + 2) + 5
The phrase [I]is[/I] means equal to, so we set 1/3(x + 2) + 5 equal to -20:
[B]1/3(x + 2) + 5 = -20[/B]

1/3 times q plus 5 equal q minus 4

1/3 times q plus 5 equal q minus 4
1/3 times q plus 5:
(q + 5)/3
q minus 4:
q - 4
The word [I]equal[/I] means we set (q + 5)/3 equal to q - 4:
[B](q + 5)/3 = q - 4[/B]

1/4 of the difference of 6 and a number is 200

1/4 of the difference of 6 and a number is 200
Take this [B]algebraic expression[/B] in 4 parts:
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x
[*]The difference of 6 and a number means we subtract x from 6: 6 - x
[*]1/4 of the difference means we divide 6 - x by 4: (6 - x)/4
[*]Finally, the phrase [I]is[/I] means an equation, so we set (6 - x)/4 equal to 200
[/LIST]
[B](6 - x)/4 = 200[/B]

10 is twice the sum of x and 5

10 is twice the sum of x and 5
The sum of x and 5 means we add:
x + 5
Twice the sum means we multiply by 2:
2(x + 5)
The word [I]is[/I] means an equation, so we set 2(x + 5) equal to 10
[B]2(x + 5) = 10[/B]

10 times a number is 420

10 times a number is 420
A number denotes an arbitrary variable, let's call it x.
10 times a number:
10x
The phrase is means equal to, so we set 10x equal to 420
[B]10x = 420 <-- This is our algebraic expression
[/B]
If you want to solve for x, use our [URL='http://www.mathcelebrity.com/1unk.php?num=10x%3D420&pl=Solve']equation calculator[/URL]
We get x = 42

104 subtracted from the quantity 6 times r is the same as r

104 subtracted from the quantity 6 times r is the same as r
The quantity 6 times r means we multiply 6 by r:
6r
104 subtracted from 6r is written as:
6r - 104
[B]The phrase [I]is the same as[/I] means we have an equation. So we set 6r - 104 equal to r
6r - 104 = r[/B]

108 times a, reduced by 147 is k subtracted from 56

108 times a, reduced by 147 is k subtracted from 56
Take this algebraic expression in pieces:
Step 1: 108 times a:
108a
Step 2: Reduced by means subtract, so we subtract 47 from 108a:
108a - 47
Step 3: ksubtracted from 56:
56 - k
Step 4: The phrase [I]is[/I] means equal to, so we set 108a - 47 equal to 56 - k
[B]108a - 47 = 56 - k
[MEDIA=youtube]KrY6uzKeeB0[/MEDIA][/B]

110 subtracted from the product of 244 and w is the product of r and 177 increased by 266

110 subtracted from the product of 244 and w is the product of r and 177 increased by 266
The product of 244 and w:
244w
110 subtracted from the product of 244 and w
244w - 110
the product of r and 177
177r
the product of r and 177 increased by 266
177r + 266
The word [I]is[/I] means equal to, so we set 244w - 110 equal to 177r + 266
[B]244w - 110 = 177r + 266[/B]

12 is multiplied by some number, that product is reduced by 9, and the total is equal to 37

12 is multiplied by some number, that product is reduced by 9, and the total is equal to 37
The phrase [I]some number[/I] means an arbitrary variable, let's call it x.
12 multiplied by this number:
12x
The product of 12x is reduced by 9
12x - 9
The phrase [I]the total is equal to[/I] means an equation, so we set 12x - 9 equal to 37:
[B]12x - 9 = 37[/B]

12 plus 6 times a number is 9 times the number

12 plus 6 times a number is 9 times the number
The phrase [I]a number [/I]means an arbitrary variable. Let's call it x.
6 times a number is written as:
6x
12 plus 6 times the number means we add 6x to 12:
12 + 6x
9 times a number is written as:
9x
The phrase [I]is[/I] means an equation, so we set 12 + 6x equal to 9x
[B]12 + 6x = 9x <-- This is our algebraic expression[/B]
[B][/B]
If the problem asks you to solve for x, then you [URL='https://www.mathcelebrity.com/1unk.php?num=12%2B6x%3D9x&pl=Solve']type this expression into our search engine[/URL] and you get:
x = [B]4[/B]

12 plus the product of 4 and a number is greater than 72

A number means an arbitrary variable, let's call it x.
The product of 4 and a number is 4x.
12 plus that product is 4x + 12
Is greater than means an inequality, so we set the entire expression greater than 72
4x + 12 > 72

13 is the product of 5p and 5

13 is the product of 5p and 5
the product of 5p and 5 means we multiply 5p by 5:
5p * 5
25p
The word [I]is[/I] means equal to, so we set 25p equal to 13
[B]13 = 25p
25p = 13[/B]

13 more than x is greater than 14

13 more than x means we add:
x + 13
This expression is greater than 14, so we write an inequality:
x + 13 > 14

132 is 393 multiplied by y

132 is 393 multiplied by y
393 multiplied by y
393y
The word [I]is[/I] means equal to, so we set 393y equal to 132 as our algebraic expression
[B]393y = 132
[/B]
If you need to solve for y, use our [URL='http://www.mathcelebrity.com/1unk.php?num=393y%3D132&pl=Solve']equation calculator[/URL]

15 added to a number is 16 times the number

15 added to a number is 16 times the number
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
[*]15 added to a number: 15 + x
[*]16 times the number: 16x
[*]The phrase [I]is[/I] means equal to. So we set 15 + x equal to 16x
[/LIST]
[B]15 + x = 16x[/B]

15 added to the quotient of 8 and a number is 7.

15 added to the quotient of 8 and a number is 7.
Take this algebraic expression in pieces:
[LIST]
[*]The phrase [I]a number[/I] means an arbitrary variable. Let's call it x.
[*]The quotient of 8 and a number: 8/x
[*]15 added to this quotient: 8/x + 15
[*]The word [I]is[/I] means an equation, so we set 8/x + 15 equal to 7
[/LIST]
[B]8/x + 15 = 7[/B]

15 minus twice a equals b

15 minus twice a equals b
Twice a means we multiply a by 2:
2a
15 minus 2a:
15 - 2a
Set this equalto b:
[B]15 - 2a = b[/B]

17 decreased by three times d equals c

17 decreased by three times d equals c
three times d means we multiply d by 3:
3d
17 decreased by three times d means we subtract 3d from 17
17 - 3d
The word [I]equals[/I] means an equation, so we set 17 - 3d equal to c:
[B]17 - 3d = c[/B]

175 students separated into n classes is 25

175 students separated into n classes is 25
[U]Divide 175 by n[/U]
175/n
[U]The word [I]is[/I] means equal to, so set this expression equal to 25[/U]
175/n = 25
[U]Cross multiply[/U]
25n = 175
[U]Divide each side by 25[/U]
[B]n = 7[/B]

2 cards have different expressions written on them.: 5y - 2 and 3y + 10. for what value of y do the

2 cards have different expressions written on them.: 5y - 2 and 3y + 10. for what value of y do the 2 cards represent the same number?
If they have the same number, we set them equal to each other and solve for y:
5y - 2 = 3y + 10
To solve for y, we [URL='http://5y - 2 = 3y + 10']type this expression in our search engine [/URL]and we get:
y = [B]6[/B]

2 dice roll

Free 2 dice roll Calculator - Calculates the probability for the following events in a pair of fair dice rolls:

* Probability of any sum from (2-12)

* Probability of the sum being less than, less than or equal to, greater than, or greater than or equal to (2-12)

* The sum being even

* The sum being odd

* The sum being a prime number

* The sum being a non-prime number

* Rolling a list of numbers i.e. (2,5,6,12)

* Simulate (n) Monte Carlo two die simulations. 2 dice calculator

* Probability of any sum from (2-12)

* Probability of the sum being less than, less than or equal to, greater than, or greater than or equal to (2-12)

* The sum being even

* The sum being odd

* The sum being a prime number

* The sum being a non-prime number

* Rolling a list of numbers i.e. (2,5,6,12)

* Simulate (n) Monte Carlo two die simulations. 2 dice calculator

2 is greater than or equal to w and -7 is less than or equal to w

2 is greater than or equal to w and -7 is less than or equal to w
Written as an inequality, we have:
-7 <= w <= 2

2 less than 3 times n is 4 more than n

2 less than 3 times n is 4 more than n
3 times n:
3n
2 less than 3 times n
3n - 2
4 more than n:
n + 4
The word [I]is[/I] means equal to, so we set 3n - 2 equal to n + 4:
[B]3n - 2 = n + 4[/B]

2 numbers that are equal have a sum of 60

2 numbers that are equal have a sum of 60
Let's choose 2 arbitrary variables for the 2 numbers
x, y
Were given 2 equations:
[LIST=1]
[*]x = y <-- Because we have the phrase [I]that are equal[/I]
[*]x + y = 60
[/LIST]
Because x = y in equation (1), we can substitute equation (1) into equation (2) for x:
y + y = 60
Add like terms to get:
2y = 60
Divide each side by 2:
2y/2 = 60/2
Cancel the 2's and we get:
y = [B]30
[/B]
Since x = y, x = y = 30
x = [B]30[/B]

2 times a number equals that number plus 5

2 times a number equals that number plus 5
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
2 times a number means we multiply 2 by x:
2x
That number plus 5 means we add 5 to the number x
x + 5
The phrase [I]equals[/I] means we set both expressions equal to each other
[B]2x = x + 5[/B] <-- This is our algebraic expression
If you want to take this further and solve this equation for x, [URL='https://www.mathcelebrity.com/1unk.php?num=2x%3Dx%2B5&pl=Solve']type this expression in the search engine[/URL] and we get:
[B]x = 5[/B]

2 times the quantity x minus 1 is 12

2 times the quantity x minus 1 is 12
The quantity x minus 1 is written as:
x - 1
2 times this quantity:
2(x - 1)
The word [I]is[/I] means an equation, so we set 2(x - 1) equal to 12:
[B]2(x - 1) = 12[/B]

2 times the sum of 1 and some number is 30. What is the number?

2 times the sum of 1 and some number is 30. What is the number?
We let the phrase "some number" equal the variable x.
The sum of 1 and some number is:
x + 1
2 times the sum:
2(x + 1)
The word "is" means equal to, so we set [B]2(x + 1) = 30[/B]

2 times the sum of a number and 3 is equal to 3x plus 4

2 times the sum of a number and 3 is equal to 3x plus 4
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The sum of a number and 3 means we add 3 to x:
x + 3
2 times this sum means we multiply the quantity x + 3 by 2
2(x + 3)
3x plus 4 means 3x + 4 since the word plus means we use a (+) sign
3x + 4
The phrase [I]is equal to[/I] means an equation, where we set 2(x + 3) equal to 3x + 4
[B]2(x + 3) = 3x + 4[/B]

2 tons and 500 pounds is equivalent to

2 tons and 500 pounds is equivalent to
1 ton equals 2000 pounds.
So 2 tons equal 2(2000) = 4000 pounds.
Add this to 500 pounds, we get:
4000 + 500 = [B]4,500 pounds[/B].

2/3 of a number 17 is at least 29

2/3 of a number 17 is at least 29
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
2/3 of a number means we multiply x by 2/3:
2x/3
The phrase [I]is at least[/I] also means greater than or equal to, so we set up the inequality:
[B]2x/3 >= 29[/B]

20 teachers made a bulk purchase of some textbooks. The teachers received a 24% discount for the bul

20 teachers made a bulk purchase of some textbooks. The teachers received a 24% discount for the bulk purchase, which originally cost $5230. Assuming the cost was divided equally among the teachers, how much did each teacher pay?
[U]Calculate Discount Percent:[/U]
If the teachers got a 24% discount, that means they paid:
100% - 24% = 76%
[URL='https://www.mathcelebrity.com/perc.php?num=+5&den=+8&num1=+16&pct1=+80&pct2=+90&den1=+80&pct=76&pcheck=4&decimal=+65.236&astart=+12&aend=+20&wp1=20&wp2=30&pl=Calculate']76% as a decimal[/URL] = 0.76 (Discount Percent)
[U]Calculate discount price:[/U]
Discount Price = Full Price * (Discount Percent)
Discount Price = 5230 * 0.76
Discount Price = 3974.80
Price per teacher = Discount Price / Number of Teachers
Price per teacher = 3974.80 / 20
Price per teacher = [B]$198.74[/B]

217 times u, reduced by 180 is the same as q

217 times u, reduced by 180 is the same as q.
Take this algebraic expression pieces:
Step 1: 217 times u
We multiply the variable u by 217
217u
Step 2: reduced by 180
Subtract 180 from 217u
217u - 180
The phrase [I]is the same as[/I] means an equation, so we set 217u - 180 equal to q
[B]217u - 180 = q[/B]

223 subtracted from the quantity 350 times a is equal to b

223 subtracted from the quantity 350 times a is equal to b
Take this algebraic expression in parts:
[LIST]
[*]the quantity 350 times a: 350a
[*]223 subtracted from the quantity: 350a - 223
[*]The phrase [I]is equal to[/I] means an equation, so we set 350a - 223 equal to b
[/LIST]
[B]350a - 223 = b[/B]

23 decreased by thrice of y is not equal to 15

Thrice of y means multiply y by 3
3y
23 decreased by 3y means we subtract
23 - 3y
Is not equal to means we set up an equation with not equal sign
23 - 3y <> 15

231 is 248 subtracted from the quantity h times 128

231 is 248 subtracted from the quantity h times 128
Let's take this algebraic expression in parts:
[LIST=1]
[*]h times 128: 128h
[*]24 subtracted from this: 128h - 248
[*]The word [I]is[/I] means an equation, so we set 128h - 248 equal to 231
[/LIST]
[B]128h - 248 = 231[/B] <-- This is our algebraic expression
If the problem asks you to solve for h, then you [URL='https://www.mathcelebrity.com/1unk.php?num=128h-248%3D231&pl=Solve']type in this equation into our search engine[/URL] and get:
h = [B]3.742[/B]

249 equals 191 times c, decreased by 199

249 equals 191 times c, decreased by 199
[U]Take this in pieces:[/U]
191 times c: 191c
The phrase [I]decreased by[/I] means we subtract 199 from 191c: 191c - 199
We set this expression equal to 249:
[B]191c - 199 = 249[/B] <-- This is our algebraic expression
If you want to solve for c, type this equation into the search engine and we get:
[B]c = 2.346[/B]

298 is the same as c and 230 more

[I]Is the same as[/I] means equal to. 230 more means we add 230.
Set up this equation:
c + 230 = 298
To solve for c if needed, visit our [URL='http://www.mathcelebrity.com/1unk.php?num=c%2B230%3D298&pl=Solve']calculator[/URL].
c = 68

2x decreased by 15 is equal to -27

2x decreased by 15 is equal to -27
The phrase [I]decreased by[/I] 15 means we subtract 15 from 2x:
2x - 15
The phrase [I]is equal to[/I] means an equation, so we set 2x - 15 equal to -27
[B]2x - 15 = -27 [/B] <-- This is our algebraic expression
To solve this, [URL='https://www.mathcelebrity.com/1unk.php?num=2x-15%3D-27&pl=Solve']type 2x - 15 = -27 into the search engine[/URL].

2x plus 4 increased by 15 is 57

2x plus 4 increased by 15 is 57
Take this algebraic expression in parts:
[LIST]
[*]2x plus 4: 2x + 4
[*][I]Increased by[/I] means we add 15 to 2x + 4: 2x + 4 + 15 = 2x + 19
[*]The word [I]is[/I] means an equation, so we set 2x + 19 equal to 57:
[/LIST]
Our final algebraic expression is:
[B]2x + 19 = 57
[/B]
To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B19%3D57&pl=Solve']type this equation into our search engine [/URL]and we get
x = [B]19[/B]

3 times a number increased by 1 is between -8 and 13

3 times a number increased by 1 is between -8 and 13.
Let's take this algebraic expression in [U]4 parts[/U]:
Part 1 - The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Part 2 - 3 times this number means we multiply x by 3:
3x
Part 3 - Increased by 1 means we add 1 to 3x:
3x + 1
The phrase [I]between[/I] means we have an inequality:
[B]-8 <= 3x + 1 <=13[/B]

3 times a number is 3 more a number

3 times a number is 3 more a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
3 times a number:
3x
3 more than a number means we add 3 to x:
x + 3
The word [I]is[/I] means and equation, so we set 3x equal to x + 3
[B]3x = x + 3[/B]

3 times the difference of a and b is equal to 4 times c

3 times the difference of a and b is equal to 4 times c
[U]The difference of a and b:[/U]
a - b
[U]3 times the difference of a and b:[/U]
3(a - b)
[U]4 times c:[/U]
4c
The phrase [I]is equal to[/I] means an equation. So we set 3(a - b) equal to 4c:
[B]3(a - b) = 4c[/B]

3 times the difference of x and 5 is 15

The difference of x and 5 means we subtract:
x - 5
3 times the difference means we multiply (x - 5) by 3
3(x - 5)
Is, means equal to, so we set our expression equal to 15
[B]3(x - 5) = 15
[/B]
If you need to take it one step further to solve for x, use our [URL='http://www.mathcelebrity.com/1unk.php?num=3%28x-5%29%3D15&pl=Solve']equation calculator[/URL]

3 times the quantity 2 decreased by x is 9

3 times the quantity 2 decreased by x is 9
The quantity 2 decreased by x. The phrase [I]decreased by[/I] means we subtract:
2 - x
3 times the quantity:
3(2 - x)
The word [I]is[/I] means equal to, so we set 3(2 - x) equal to 9:
[B]3(2 - x) = 9
[MEDIA=youtube]Hzyt_GajZA4[/MEDIA][/B]

3 times the sum of 2 decreased by x is 9

3 times the sum of 2 decreased by x is 9
2 decreased by x:
2 - x
3 times the sum means we multiply 2 - x by 3:
3(2 - x)
The phrase [I]is 9[/I] means equal to, so we set 3(2 - x) equal to 9:
[B]3(2 - x) = 9[/B]

3 times x minus y is 5 times the sum of y and 2 times x

3 times x minus y is 5 times the sum of y and 2 times x
Take this algebraic expression in pieces:
3 times x:
3x
Minus y means we subtract y from 3x
3x - y
The sum of y and 2 times x mean we add y to 2 times x
y + 2x
5 times the sum of y and 2 times x:
5(y + 2x)
The word [I]is[/I] means an equation, so we set 3x - y equal to 5(y + 2x)
[B]3x - y = 5(y + 2x)[/B]

3 to the power of 2 times 3 to the power of x equals 3 to the power of 7

3 to the power of 2 times 3 to the power of x equals 3 to the power of 7.
Write this out:
3^2 * 3^x = 3^7
When we multiply matching coefficients, we add exponents, so we have:
3^(2 + x) = 3^7
Therefore, 2 + x = 7. To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=2%2Bx%3D7&pl=Solve']type it into our search engine[/URL] and we get:
x = [B]5[/B]

3, 8, 13, 18, .... , 5008 What term is the number 5008?

3, 8, 13, 18, .... , 5008 What term is the number 5008?
For term n, we have a pattern:
f(n) = 5(n - 1) + 3
Set this equal to 5008
5(n - 1) + 3 = 5008
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=5%28n-1%29%2B3%3D5008&pl=Solve']equation solver,[/URL] we get:
n = [B]1002[/B]

3/10 of a circle equal how many degrees

3/10 of a circle equal how many degrees
A circle is 365 degrees. So [URL='https://www.mathcelebrity.com/fraction.php?frac1=365&frac2=3%2F10&pl=Multiply']we multiply 365 * 3/10 in our search engine[/URL] and get:
219/2
219/2 = [B]109.5 degrees[/B]

3/5 of workers at a company have enrolled in the 403(b) program. If 24 workers have enrolled in the

3/5 of workers at a company have enrolled in the 403(b) program. If 24 workers have enrolled in the program, how many workers are employed at this company?
We read this as 3/5 of the total workers employed at the company equals 24. Let w be the number of workers. We have the following equation:
3/5w = 24
Run [URL='http://www.mathcelebrity.com/1unk.php?num=3%2F5w%3D24&pl=Solve']3/5w = 24[/URL] through the search engine, we get [B]w = 40[/B].

3/8 of income is rent. 360 is rent. How much is annual income

3/8 of income is rent. 360 is rent. How much is annual income
Let i equal the annual income.
We're told the following:
3i/8 = 360
To solve for i, we [URL='https://www.mathcelebrity.com/prop.php?num1=3i&num2=360&den1=8&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']type this in our math engine[/URL] and we get:
i = [B]960
[MEDIA=youtube]8gue05tlEGQ[/MEDIA][/B]

30 increased by 3 times the square of a number

Let "a number" equal the arbitrary variable x.
The square of that is x^2.
3 times the square of that is 3x^2.
Now, 30 increased by means we add 3x^2 to 30
30 + 3x^2

30 is equal to thrice y decreased by z

30 is equal to thrice y decreased by z
Thrice y means we multiply y by 3:
3y
Decreased by z means we subtract z from 3y
3y - z
The phrase [I]is[/I] means an equal to, so we set up an equation where 3y - z is equal to 30
[B]3y - z = 30[/B]

300 reduced by 5 times my age is 60

300 reduced by 5 times my age is 60
Let my age be a. We have:
5 times my age = 5a
300 reduced by 5 times my age means we subtract 5a from 300:
300 - 5a
The word [I]is[/I] means an equation, so we set 300 - 5a equal to 60 to get our final algebraic expression:
[B]300 - 5a = 60
[/B]
If you have to solve for a, you [URL='https://www.mathcelebrity.com/1unk.php?num=300-5a%3D60&pl=Solve']type this equation into our search engine[/URL] and you get:
a = [B]48[/B]

309 is the same as 93 subtracted from the quantity f times 123

309 is the same as 93 subtracted from the quantity f times 123.
The quantity f times 123:
123f
Subtract 93:
123f - 93
The phrase [I]is the same as[/I] means an equation, so we set 123f - 93 equal to 309
[B]123f - 93 = 309[/B] <-- This is our algebraic expression
If you wish to solve for f, [URL='https://www.mathcelebrity.com/1unk.php?num=123f-93%3D309&pl=Solve']type this equation into the search engine[/URL], and we get f = 3.2683.

32 girls and 52 boys were on an overseas learning trip . They were divided into as many groups as po

32 girls and 52 boys were on an overseas learning trip . They were divided into as many groups as possible where the number of groups of girls and the number of groups of boys is the same .how many boys and how many girls were in each group
We want a number such that our total members divided by this number equals our group size.
We take the greatest common factor (32,52) = 4
Therefore, we have:
[LIST]
[*][B]32/4 = 8 girls in each group[/B]
[*][B]52/4 = 13 boys in each group[/B]
[/LIST]

324 times z, reduced by 12 is z

324 times z, reduced by 12 is z.
Take this algebraic expression in pieces:
324 [I]times[/I] z means we multiply 324 by the variable z.
324z
[I]Reduced by[/I] 12 means we subtract 12 from 324z
324z - 12
The word [I]is[/I] means we have an equation, so we set 324z - 12 equal to z
[B]324z - 12 = z [/B] <-- This is our algebraic expression

339 equals 303 times w, minus 293

339 equals 303 times w, minus 293
Take this algebraic expression in pieces:
303 times w:
303w
Minus 293:
303w - 293
The phrase [I]equals[/I] means we have an equation. We set 303w - 293 = 339
[B]303w - 293 = 339[/B] <-- This is our algebraic expression
To solve for w, [URL='https://www.mathcelebrity.com/1unk.php?num=303w-293%3D339&pl=Solve']we type this equation into our search engine[/URL] to get:
[B]w = 2.086[/B]

346 times w, decreased by 79 equals w

346 times w, decreased by 79 equals w
346 times w
346w
Decreased by 79
346w - 79
Equals w
[B]346w - 79 = w[/B]

35 added to n is greater than or equal to the sum of k and 21

35 added to n is greater than or equal to the sum of k and 21
Take this algebraic expression in 3 parts:
[LIST=1]
[*]35 added to n means we have a sum: n + 35
[*]The sum of k and 21 means we add 21 to k: k +21
[*]The phrase [I]greater than or equal to[/I] means an inequality using this sign (>=), so we write this as follows:
[/LIST]
[B]n + 35 >= k + 21[/B]

365 subtracted from the quantity q times 146 is the same as w

[U]q times 146:[/U]
146q
[U]365 subtracted from that:[/U]
146q - 365
[U]Is the same as means equal to, so we have:[/U]
[B]146q - 365 = w[/B]

3x less than 2 times the sum of 2x and 1 is equal to the sum of 2 and 5

3x less than 2 times the sum of 2x and 1 is equal to the sum of 2 and 5
This is an algebraic expression. Let's take this algebraic expression in 5 parts:
[LIST=1]
[*]The sum of 2x and 1 means we add 1 to 2x: 2x + 1
[*]2 times the sum of 2x and 1: 2(2x + 1)
[*]3x less than the sum of 2x and 1 means we subtract 3x from 2(2x + 1): 2(2x + 1) - 3x
[*]The sum of 2 and 5 means we add 5 to 2: 2 + 5
[*]Finally, the phrase [I]equal[/I] means an equation, so we set #3 equal to #4
[/LIST]
Our algebraic expression is:
[B]2(2x + 1) - 3x = 2 + 5[/B]
Now, some problems may ask you to simplify. In this case, we multiply through and group like terms:
4x + 2 - 3x = 7
[B]x + 2 = 7 <-- This is our simplified algebraic expression
[/B]
Now, what if the problem asks you to solve for x, [URL='https://www.mathcelebrity.com/1unk.php?num=x%2B2%3D7&pl=Solve']you type this into our search engine[/URL] and get:
x =[B] 5
[MEDIA=youtube]3hzyc2NPCGI[/MEDIA][/B]

3x over 27 equals 2x minus 2 over 15

3x over 27 equals 2x minus 2 over 15
3x over 27:
3x/27
2x minus 2 over 15:
(2x - 2)/15
Set them equal to each other:
3x/27 = (2x - 2)/15

4 consecutive integers such that the sum of the first 3 integers is equal to the 4th

4 consecutive integers such that the sum of the first 3 integers is equal to the 4th
Let n be our first consecutive integer.
[LIST=1]
[*]n
[*]n + 1
[*]n + 2
[*]n + 3
[/LIST]
The sum of the first 3 integers is equal to the 4th:
n + n + 1 + n + 2 = n + 3
Simplify by grouping like terms:
(n + n + n) + (1 + 2) = n + 3
3n + 3 = n + 3
3n = n
n = 0
n = 0
n + 1 = 1
n + 2 = 2
n + 3 = 3
Check our work:
0 + 1 +2 ? 3
3 = 3
Our final answer is [B](0, 1, 2, 3}[/B]

4 minus 3p equals 36

4 minus 3p equals 36
4 minus 3p:
4 - 3p
The phrase [I]equals[/I] means an equation, so we set 4 - 3p equal to 36:
[B]4 - 3p = 36[/B]

4 times a number added to 8 times a number equals 36

4 times a number added to 8 times a number equals 36
Let [I]a number[/I] be an arbitrary variable, let us call it x.
4 times a number:
4x
8 times a number:
8x
We add these together:
4x + 8x = 12x
We set 12x equal to 36 to get our final algebraic expression of:
[B]12x = 36
[/B]
If the problem asks you to solve for x, you [URL='https://www.mathcelebrity.com/1unk.php?num=12x%3D36&pl=Solve']type this algebraic expression into our search engine[/URL] and get:
x = [B]3[/B]

4 times a number is the same as the number increased by 78

4 times a number is the same as the number increased by 78.
Let's take this algebraic expression in parts:
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
[*]4 times a number is written as 4x
[*]The number increased by 78 means we add 78 to x: x + 78
[*]The phrase [I]the same as[/I] mean an equation, so we set #2 equal to #3
[/LIST]
[B]4x = x + 78[/B] <-- This is our algebraic expression
If the problem asks you to take it a step further, then [URL='https://www.mathcelebrity.com/1unk.php?num=4x%3Dx%2B78&pl=Solve']we type this equation into our search engine [/URL]and get:
x = 26

4 times x plus 2 is at most 10

4 times x plus 2 is at most 10
4 times x
4x
Plus 2
4x + 2
At most means less than or equal to, so we have:
[B]4x + 2 <= 10[/B]

400 reduced by 3 times my age is 214

400 reduced by 3 times my age is 214
Let my age be a. We have:
3 times my age:
3a
400 reduced by 3 times my age:
400 - 3a
The word [I]is[/I] means an equation. So we set 400 - 3a equal to 214
400 - 3a = 214
Now if you want to solve this equation for a, we [URL='https://www.mathcelebrity.com/1unk.php?num=400-3a%3D214&pl=Solve']type it in the search engin[/URL]e and we get;
a = [B]62[/B]

48 is the difference of Chrissys height and 13 .

48 is the difference of Chrissys height and 13 .
Let Chrissy's height = h.
The difference of the height and 13 is h - 13.
We set this expression equal to 48:
[B]h - 13 = 48
[/B]
Note: To solve this, [URL='http://www.mathcelebrity.com/1unk.php?num=h-13%3D48&pl=Solve']paste this problem into the search engine[/URL].

4subtractedfrom6timesanumberis32

4 subtracted from 6 times a number is 32.
Take this algebraic expression in pieces.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
6 times this number means we multiply by x by 6
6x
4 subtracted from this expression means we subtract 4
6x - 4
The phrase [I]is[/I] means an equation, so we set 6x - 4 equal to 32
[B]6x - 4 = 32
[/B]
If you need to solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=6x-4%3D32&pl=Solve']type it in the search engine here[/URL].

5 added to xis 11

5 added to x means we use the plus sign for a sum.
x + 5
"is" means equals, so we set that equal to 11.
x + 5 = 11 <-- This is our algebraic expression.

5 is one-fourth of a number c

5 is one-fourth of a number c
[LIST]
[*]A number c is just written as c
[*]one-fourth of c means we multiply c by 1/4: c/4
[*]The phrase [I]is[/I] means equal to, so we set c/4 equal to 5
[/LIST]
[B]c/4 = 5[/B]

5 less than x is y

5 less than x means we subtract 5 from x.
x - 5
Is, means equal to, so we set x - 5 equal to y
x - 5 = y

5 subtracted from 3 times a number is 44

5 subtracted from 3 times a number is 44.
The problem asks for an algebraic expression.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
3 times this number is 3x.
5 subtracted from this is written as 3x - 5.
The phrase [I]is[/I] means an equation, so we set 3x - 5 equal to 44
[B]3x - 5 = 44[/B]

5 times a number is 4 more than twice a number

5 times a number is 4 more than twice a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
5 times a number:
5x
Twice a number means we multiply x by 2:
2x
4 more than twice a number
2x + 4
The word [I]is[/I] means equal to, so we set 5x equal to 2x + 4
[B]5x = 2x + 4[/B]

5 times a number is that number minus 3

5 times a number is that number minus 3
The phrase [I]a number[/I] means an arbitrary variable. Let's call it x.
[LIST]
[*]5 times a number: 5x
[*]That number means we use the same number from above which is x
[*]That number minus 3: x - 3
[*]The phrase [I]is[/I] means an equation, so we set 5x equal to x - 3
[/LIST]
[B]5x = x - 3[/B]

5/8 Of a class are boys. what fraction of the class are girls

5/8 Of a class are boys. what fraction of the class are girls?
The total class equals 1. Since 5/8 are boys, we subtract 5/8 from 1:
1 - 5/8
But we can write 1 as 8/8. So we have
8/8 - 5/8
[URL='https://www.mathcelebrity.com/fraction.php?frac1=8%2F8&frac2=5%2F8&pl=Subtract']Type this fraction operation into our search engine[/URL] and we get:
[B]3/8[/B] are girls

50 is more than the product of 4 and w

50 is more than the product of 4 and w
Take this algebraic expression in pieces:
The product of 4 and w mean we multiply the variable w by 4:
4w
The phrase [I]is more than[/I] means an inequality using the (>) sign, where 50 is greater than 4w:
[B]50 > 4w[/B]

54 is the sum of 15 and Vidyas score

54 is the sum of 15 and Vidyas score.
Let Vida's score be s.
The sum of 15 and s:
s + 15
When they say "is", they mean equal to, so we set s + 15 equal to 54. Our algebraic expression is below:
[B]s + 15 = 54
[/B]
To solve this equation for s, use our [URL='http://www.mathcelebrity.com/1unk.php?num=s%2B15%3D54&pl=Solve']equation calculator[/URL]

54 is the sum of 24 and Julies score. Use the variable J to represent Julies score.

54 is the sum of 24 and Julies score. Use the variable J to represent Julies score.
Sum of 24 and Julie's score:
24 + J
The phrase [I]is[/I] means an equation, so we set 24 + J equal to 54 to get an algebraic expression:
[B]24 + J = 54[/B]

56 is the sum of 20 and Donnie's savings. Use the variable d to represent Donnie's savings

56 is the sum of 20 and Donnie's savings. Use the variable d to represent Donnie's savings
The sum of 20 and Donnie's savings using [I]d[/I] to represent Donnie's savings:
20 + d
The word [I]is[/I] means equal to, so we set 20 + d equal to 56:
[B]20 + d = 56[/B]

59 is the difference of vanessas height and 20

59 is the difference of vanessas height and 20.
Let h be Vanessa's height. We have the difference of h and 20:
h - 20
The phrase [I]is[/I] means equal to, so we set h - 20 equal to 59
[B]h - 20 = 59[/B]

59 is the sum of 16 and Donnie's saving. Use the variable d to represent Donnie's saving.

59 is the sum of 16 and Donnie's saving. Use the variable d to represent Donnie's saving.
The phrase [I]the sum of[/I] means we add Donnie's savings of d to 16:
d + 16
The phrase [I]is[/I] means an equation, so we set d + 16 equal to 59
d + 16 = 59 <-- [B]This is our algebraic expression[/B]
Now, if the problem asks you to solve for d, then you[URL='https://www.mathcelebrity.com/1unk.php?num=d%2B16%3D59&pl=Solve'] type the algebraic expression into our search engine to get[/URL]:
d = [B]43[/B]

6 diminished by twice x is at most 8

6 diminished by twice x is at most 8
Twice x means we multiply x by 2:
2x
6 diminished by twice x means we subtract 2x from 6:
6 - 2x
The phrase [I]is at most[/I] is an inequality using the sign <=, so we have:
[B]6 - 2x <= 8[/B]

6 is one third of a number s

6 is one third of a number s
A number s is written as s:
s
One third of a number s means we multiply s by 1/3
s/3
The word [I]is[/I] means equal to, so we set s/3 equal to 6
[B]s/3 = 6[/B]

6 subtracted from the product of 5 and a number is 68

6 subtracted from the product of 5 and a number is 68
Take this algebraic expression in parts.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The product of 5 and this number is:
5x
We subtract 6 from 5x:
5x - 6
The phrase [I]is[/I] means an equation, so we set 5x - 6 equal to 68
[B]5x - 6 = 68[/B]

6 times a number, x, is at least 22.

6 times a number, x, is at least 22.
6 times a number x:
6x
The phrase [I]is at least[/I] means greater than or equal to. So we have an inequality:
[B]6x >= 22[/B] <-- This is our algebraic expression
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=6x%3E%3D22&pl=Show+Interval+Notation']To solve this for x, paste this into the search engine[/URL] and we get:
[B]x >= 3.666667[/B]

6 times the reciprocal of a number equals 2 times the reciprocal of 7. What is the number

6 times the reciprocal of a number equals 2 times the reciprocal of 7. What is the number
We've got two algebraic expressions here. Let's take it in parts:
Term 1:
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The reciprocal is 1/x
Multiply this by 6: 6/x
Term 2:
Reciprocal of 7: 1/7
2 times this: 2/7
We set these terms equal to each other:
6/x = 2/7
[URL='https://www.mathcelebrity.com/prop.php?num1=6&num2=2&den1=x&den2=7&propsign=%3D&pl=Calculate+missing+proportion+value']Type this proportion into the search engine[/URL], and we get:
[B]x = 21[/B]

6 times the reciprocal of a number equals 3 times the reciprocal of 7 .

6 times the reciprocal of a number equals 3 times the reciprocal of 7 .
This is an algebraic expression. Let's take it in parts:
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The reciprocal of a number x means we divide 1 over x:
1/x
6 times the reciprocal means we multiply 6 by 1/x:
6/x
The reciprocal of 7 means we divide 1/7
1/7
3 times the reciprocal means we multiply 1/7 by 3:
3/7
Now, the phrase [I]equals[/I] mean an equation, so we set 6/x = 3/7
[B]6/x = 3/7[/B] <-- This is our algebraic expression
If the problem asks you to solve for x, then [URL='https://www.mathcelebrity.com/prop.php?num1=6&num2=3&den1=x&den2=7&propsign=%3D&pl=Calculate+missing+proportion+value']we type this proportion in our search engine[/URL] and get:
x = 14

6 times the sum of a number and 3 is equal to 42. What is this number?

6 times the sum of a number and 3 is equal to 42. What is this number?
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The sum of a number and 3 means we add 3 to x:
x + 3
6 times the sum:
6(x + 3)
The word [I]is[/I] means an equation, so we set 6(x + 3) equal to 42 to get our [I]algebraic expression[/I] of:
[B]6(x + 3) = 42[/B]
[B][/B]
If the problem asks you to solve for x, then [URL='https://www.mathcelebrity.com/1unk.php?num=6%28x%2B3%29%3D42&pl=Solve']you type this equation into our search engine[/URL] and you get:
x = [B]4[/B]

6 times the sum of a number and 5 is 16

6 times the sum of a number and 5 is 16
A number represents an arbitrary variable, let's call it x
x
The sum of x and 5
x + 5
6 times the sum of x and 5
6(x + 5)
Is means equal to, so set 6(x + 5) equal to 16
[B]6(x + 5) = 16 <-- This is our algebraic expression
Solve for x[/B]
Multiply through:
6x + 30 = 16
Subtract 30 from each side:
6x - 30 + 30 = 16 - 30
6x = -14
Divide each side by 6
6x/6 = -14/6
Simplify this fraction by dividing top and bottom by 2:
x = [B]-7/3
[MEDIA=youtube]oEx5dsYK7DY[/MEDIA][/B]

60 is the sum of 22 and Helenas height. Use the variable h to represent Helenas height.

60 is the sum of 22 and Helenas height. Use the variable h to represent Helenas height.
If height is represented by h, we have: 22 and h
22 + h
When they say "is the sum of", we set 22 + h equal to 60
[B]22 + h = 60[/B]

60 percent of a number minus 17 is -65

60 percent of a number minus 17 is -65
Using our [URL='https://www.mathcelebrity.com/perc.php?num=+5&den=+8&num1=+16&pct1=+80&pct2=+90&den1=+80&pct=60&pcheck=4&decimal=+65.236&astart=+12&aend=+20&wp1=20&wp2=30&pl=Calculate']percent to decimal calculator[/URL], we see that 60% is 0.6, so we have:
0.6
The phrase [I]a number[/I] means an arbitrary variable, let's call it x. So 60% of a number is:
0.6x
Minus 17:
0.6x - 17
The word [I]is[/I] means an equation, so we set 0.6x - 17 equal to -65 to get our algebraic expression of:
[B]0.6x - 17 = -65[/B]
[B][/B]
If you want to solve for x in this equation, you [URL='https://www.mathcelebrity.com/1unk.php?num=0.6x-17%3D-65&pl=Solve']type it in our search engine and you get[/URL]:
[B]x = -80[/B]

64 is 4 times the difference between Sarah's age, a, and 44. Assume Sarah is older than 44.

64 is 4 times the difference between Sarah's age, a, and 44. Assume Sarah is older than 44.
The phrase [I]difference between[/I] means we subtract 44 from a:
a - 44
The phrase [I]64 is[/I] means an equation, so we set a - 44 equal to 64
[B]a - 44 = 64 <-- This is our algebraic expression
[/B]
If you want to solve for a, then we [URL='https://www.mathcelebrity.com/1unk.php?num=a-44%3D64&pl=Solve']type this expression into our search engine[/URL] and get:
[B]a = 108[/B]

64 is 4 times the difference between Sarah’s age a, and 44.Assume Sarah is older than 44

64 is 4 times the difference between Sarah’s age a, and 44.Assume Sarah is older than 44
Difference between Sarah's age (a) and 44 (Assuming Sarah is older than 44):
a - 44
4 times the difference:
4(a - 44)
The word [I]is[/I] means equal to, so we set 4(a - 44) equal to 64 to get our algebraic expression:
[B]4(a - 44) = 64[/B]
If the problem asks you to solve for a, we [URL='https://www.mathcelebrity.com/1unk.php?num=4%28a-44%29%3D64&pl=Solve']type this equation into our search engine[/URL] and we get:
a = [B]60[/B]

66 decreased by Janelle's savings is 15

66 decreased by Janelle's savings is 15
Let Janelle's savings be s.
66 decreased by s is:
66 - s
The word [I]is[/I] means equal so we set 66 - s equal to 15
[B]66 - s = 15[/B]

7 and 105 are successive terms in a geometric sequence. what is the term following 105?

7 and 105 are successive terms in a geometric sequence. what is the term following 105?
Geometric sequences are set up such that the next term in the sequence equals the prior term multiplied by a constant. Therefore, we express the relationship in the following equation:
7k = 105 where k is the constant
[URL='https://www.mathcelebrity.com/1unk.php?num=7k%3D105&pl=Solve']Type this equation into our search engine[/URL] and we get:
k = 15
The next term in the geometric sequence after 105 is found as follows:
105*15 = [B]1,575[/B]

7 is 1/4 of some number

7 is 1/4 of some number
The phrase [I]some number[/I] means an arbitrary variable, let's call it x.
1/4 of this is written as:
x/4
The word [I]is[/I] means an equation, so we set x/4 equal to 7:
[B]x/4 = 7[/B]

7 less than -2 times a number x is greater than or equal to 41

-2 times a number x is written as -2x.
Less means subtract, so we have 7 less than this is -2x - 7.
Finally, greater than or equal to is >=, so our expression becomes:
-2x - 7 >= 41

7 less than -2 times a number x is greater than or equal to 41

-2 times a number x is denoted as -2x.
7 less than that means we subtract 7:
-2x - 7
Finally, that entire expression is greater than or equal to 41
-2x - 7 >= 41

7 less than -2 times a number x is greater than or equal to 41

-2 times a number x is denoted as -2x.
7 less means we subtract, so 7 less than that is -2x - 7.
Finally, that entire expression is greater than or equal to 41
-2x - 7 >= 41

7 less than -2 times a number x is greater than or equal to 41

7 less than -2 times a number x is greater than or equal to 41
-2 times a number x
-2x
7 less than this
-2x - 7
Now we set this expressions greater than or equal to 41
[B]-2x - 7 >= 41[/B]

7 plus the quotient of 12 and x is 2

7 plus the quotient of 12 and x is 2
The quotient of 12 and x:
12/x
7 plus the quotient of 12 and x:
7 + 12/x
The word [I]is[/I] means equal to, so we set 7 + 12/x equal to 2:
[B]7 + 12/x = 2[/B]

7 times a number and 2 is equal to 4 times a number decreased by 8

7 times a number and 2 is equal to 4 times a number decreased by 8
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
7 times a number:
7x
and 2 means we add 2:
7x + 2
4 times a number
4x
decreased by 8 means we subtract 8:
4x - 8
The phrase [I]is equal to[/I] means an equation, so we set 7x + 2 equal to 4x - 8:
[B]7x + 2 = 4x - 8[/B]

7 times a number is the same as 12 more than 3 times a number

7 times a number is the same as 12 more than 3 times a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
[B][U]Algebraic Expression 1:[/U][/B]
7 times a number means we multiply 7 by x:
7x
[B][U]Algebraic Expression 2:[/U][/B]
3 times a number means we multiply 3 by x:
3x
12 more than 3 times a number means we add 12 to 3x:
3x + 12
The phrase [I]is the same as[/I] means an equation, so we set 7x equal to 3x + 12
[B]7x = 3x + 12[/B] <-- Algebraic Expression

7 times a positive number n is decreased by 3, it is less than 25

7 times a positive number n is decreased by 3, it is less than 25
7 times a positive number n:
7n
Decreased by 3:
7n - 3
The phrase [I]it is less than [/I]means an inequality. So we relate 7n - 3 less than 25 using the < sign to get our algebraic expression of:
[B]7n - 3 < 25[/B]

76 subtracted from p is equal to the total of g and 227

76 subtracted from p is equal to the total of g and 227
We've got two algebraic expressions. Take them in pieces:
Part 1:
76 subtracted from p
We subtract 76 from the variable p
p - 76
Part 2:
The total of g and 227
The total means a sum, so we add 227 to g
g + 227
Now the last piece, the phrase [I]is equal to[/I] means an equation. So we set both algebraic expressions equal to each other:
[B]p - 76 = g + 227[/B]

8 increased by the product of a number and 7 is greater than or equal to -18

Take this in parts:
First, the phrase, "a number" means we pick an arbitrary variable, let's call it x.
The product of a number and 7 is 7x.
8 increased by the product of 7x means we add them together.
7x + 8
Finally that entire expression is greater than [U]or equal to[/U] -18
[B]7x + 8 >=-18[/B]

8 less than x is 31

8 less than X means X - 8.
The word is means equal to. So we have:
X - 8 = 31

8 more than the product of x and 2 equals 4

8 more than the product of x and 2 equals 4
The product of x and 2:
2x
8 more than this, means we add 8:
2x + 8
Set this equal to 4:
[URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B8%3D4&pl=Solve']2x + 8 = 4[/URL] <-- Algebraic expression
to solve for x, type this into the search engine and we get [B]x = -2[/B].

8 more than twice a number is less than 6 more than the number

8 more than twice a number is less than 6 more than the number.
This is an algebraic expression, let's take it in pieces...
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
8 more than twice a number:
Twice a number means multiply x by 2: 2x
Then add 8: 2x + 8
6 more than the number, means we add 6 to x
x + 6
The phrase [I]is less than[/I] means an inequality, where we set 2x + 8 less than x + 6
[B]2x + 8 < x + 6[/B]

8 times the difference of a number and 2 is the same as 3 times the sum of the number and 3. What is

8 times the difference of a number and 2 is the same as 3 times the sum of the number and 3. What is the number?
Let the number be n. We're given two expressions:
[LIST=1]
[*]8(n - 2) [I]difference means we subtract[/I]
[*]3(n + 3) [I]sum means we add[/I]
[/LIST]
The phrase [I]is the same as[/I] mean an equation. So we set the first expression equal to the second expression:
8(n - 2) = 3(n + 3)
To solve this equation for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=8%28n-2%29%3D3%28n%2B3%29&pl=Solve']type it in our search engine[/URL] and we see that:
n =[B] 5[/B]

9 divided by the sum of x and 4 is equal to 6 divided by x minus 4

9 divided by the sum of x and 4 is equal to 6 divided by x minus 4.
Build our two algebraic expressions first:
9 divided by the sum of x and 4
9/(x + 4)
6 divided by x minus 4
6/(x - 4)
The phrase [I]is equal to[/I] means and equation, so we set the algebraic expressions equal to each other:
[B]9/(x + 4) = 6/(x - 4) <-- This is our algebraic expression[/B]
[B][/B]
If the problem asks you to solve for x, we cross multiply:
9(x - 4) = 6(x + 4)
To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=9%28x-4%29%3D6%28x%2B4%29&pl=Solve']type this equation into our search engine[/URL] and we get:
x = [B]20[/B]

9 is greater than x and x is greater than 3

9 is greater than x and x is greater than 3
This is a compound inequality. We write this as:
[B]3 < x < 9[/B]

9 is one-third of a number x

9 is one-third of a number x
A number x can be written as x
x
one-third of a number x means we multiply x by 1/3:
x/3
The phrase [I]is[/I] means an equation, so we set 9 equal to x/3 to get our final algebraic expression of:
[B]x/3 = 9[/B]
If the problem asks you to solve for x, you [URL='https://www.mathcelebrity.com/prop.php?num1=x&num2=9&den1=3&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']type this algebraic expression into our search engine[/URL] and you get:
[B]x = 27[/B]

9 is the sum of 7 and twice a number

9 is the sum of 7 and twice a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
Twice a number means we multiply x by 2:
2x
The sum of 7 and twice a number
7 + 2x
The word [I]is[/I] mean equal to, so we set 7 + 2x equal to 9:
[B]7 + 2x = 9[/B]

9 is the sum of thrice x and y

9 is the sum of thrice x and y
Thrice x means multiply x by 3:
3x
Sum of this and y:
3x + y
Now we set this expression equal to 9:
[B]3x + y = 9[/B]

9 less than 5 times a number is 3 more than 2x

9 less than 5 times a number is 3 more than 2x
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
5 times a number means we multiply x by 5:
5x
9 less than 5x means we subtract 9 from 5x:
5x - 9
3 more than 2x means we add 3 to 2x:
2x + 3
The word [I]is[/I] means an equation, so we set 5x - 9 equal to 2x + 3:
[B]5x - 9 = 2x + 3 <-- This is our algebraic expression[/B]
[B][/B]
If you want to solve for x, [URL='https://www.mathcelebrity.com/1unk.php?num=5x-9%3D2x%2B3&pl=Solve']type this equation into the search engine[/URL], and we get:
x = [B]4[/B]

9 less than twice x is twice y

9 less than twice x is twice y
Twice x means we multiply x by 2:
2x
9 less than Twice x means we subtract 9 from 2x
2x - 9
Twice y means we multiply y by 2:
2y
The word [I]is[/I] means equal to, so we set 2x - 9 equal to 2y:
[B]2x - 9 = 2y[/B]

9 subtracted from the product of 3 and a number is greater than or equal to 16

9 subtracted from the product of 3 and a number is greater than or equal to 16
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x
[*]The product of 3 and a number means we multiply 3 times x: 3x
[*]9 subtracted from the product: 3x - 9
[*]The phrase is greater than or equal to means an inequality. So we set up an inequality with >= for the greater than or equal to sign in relation to 3x - 9 and 16
[/LIST]
Our algebraic expression (inequality) becomes:
[B]3x - 19 >= 16[/B]

9 times a number is that number minus 10

9 times a number is that number minus 10
The phrase [I]a number[/I] means we define a random/arbitrary variable, let's call it x:
x
9 times a number means we multiply x by 9:
9x
The phrase [I]that number[/I] refers back to the original arbitrary variable we defined above, which is x:
x
That number minus 10 means we subtract 10 from x:
x - 10
The word [I]is[/I] means equal to, so we set 9x equal to x - 10
[B]9x = x - 10[/B]

9 times a number is that number minus 3

9 times a number is that number minus 3
Let [I]a number[/I] be an arbitrary variable, let's call it x. We're given:
9 times a number is 9x
The number minus 3 is x - 3
The word [I]is[/I] means an equation, so we set 9x equal to x - 3 to get our [I]algebraic expression[/I]:
[B]9x = x - 3[/B]
To solve for x, we type this equation into our search engine and we get:
x = [B]-0.375 or -3/8[/B]

9 times x is twice the sum of x and 5

9 times x is twice the sum of x and 5
9 times x:
9x
the sum of x and 5
x + 5
twice the sum of x and 5
2(x + 5)
The phrase [I]is[/I] means equal to, so we set 9x equal to 2(x + 5)
[B]9x = 2(x + 5)[/B]

993 cold drinks bottles are to be placed in crates. Each crate can hold 9 bottles. How many crates w

993 cold drinks bottles are to be placed in crates. Each crate can hold 9 bottles. How many crates would be needed and how many bottles will remain?
Let c equal the number of crates
9 bottles per crate * c = 993
9c = 993
Solve for [I]c[/I] in the equation 9c = 993
[SIZE=5][B]Step 1: Divide each side of the equation by 9[/B][/SIZE]
9c /9 = 993/9
c = 110.33333333333
Since we can't have fractional crates, we round up 1 to the next full crate
c = [B]111[/B]

a 12 sided die is rolled find the probability of rolling a number greater than 7

a 12 sided die is rolled find the probability of rolling a number greater than 7
We assume this is a fair die, not loaded.
This means each side 1-12 has an equal probability of 1/12 of being rolled.
The problem asks, P(Roll > 7)
Greater than 7 means our sample space is {8, 9, 10, 11, 12}
If each of these 5 faces have an equal probability of being rolled, then we have:
P(Roll > 7) = P(Roll = 8) + P(Roll = 9) + P(Roll = 10) + P(Roll = 11) + P(Roll = 12)
P(Roll > 7) = 1/12 + 1/12 + 1/12 + 1/12 + 1/12
P(Roll > 7) =[B] 5/12[/B]

A 12-sided die is rolled. The set of equally likely outcomes is {1,2,3,4,5,6,7,,8,9,10,11,12}. Find

A 12-sided die is rolled. The set of equally likely outcomes is {1,2,3,4,5,6,7,,8,9,10,11,12}. Find the probability of rolling a number less than 6.
We want a {1, 2, 3, 4, 5}
P(X < 6) =[B] 5/12[/B]

A 12-sided die is rolled. The set of equally likely outcomes is {1,2,3,4,5,6,7,8,9,10,11,12}. Find t

A 12-sided die is rolled. The set of equally likely outcomes is {1,2,3,4,5,6,7,8,9,10,11,12}. Find the probability of rolling a number less than 6.
We have 12 outcomes.
Less than 6 means 1, 2, 3, 4, 5.
Our probability P(x < 6) is:
P(x < 6) = [B]5/12[/B]

A 3-gallon bucket of paint costs $87.12. What is the price per quart?

A 3-gallon bucket of paint costs $87.12. What is the price per quart?
3 gallons equals 12 quarts with our [URL='https://www.mathcelebrity.com/liqm.php?quant=3&pl=Calculate&type=gallon#quart']conversion calculator[/URL]. We divide 87.12 for 12 quarts by 12:
[URL='https://www.mathcelebrity.com/perc.php?num=87.12&den=12&pcheck=1&num1=16&pct1=80&pct2=70&den1=80&idpct1=10&hltype=1&idpct2=90&pct=82&decimal=+65.236&astart=12&aend=20&wp1=20&wp2=30&pl=Calculate']87.12 / 12[/URL] = [B]$7.26 per quart[/B]

A bakery has a fixed cost of $119.75 per a day plus $2.25 for each pastry. The bakery would like to

A bakery has a fixed cost of $119.75 per a day plus $2.25 for each pastry. The bakery would like to keep its daily costs at or below $500 per day. Which inequality shows the maximum number of pastries, p, that can be baked each day.
Set up the cost function C(p), where p is the number of pastries:
C(p) = Variable Cost + Fixed Cost
C(p) = 2.25p + 119.75
The problem asks for C(p) at or below $500 per day. The phrase [I]at or below[/I] means less than or equal to (<=).
[B]2.25p + 119.75 <= 500[/B]

A bank charges a service fee of $7.50 per month for a checking account. A bank account has $85.00. I

A bank charges a service fee of $7.50 per month for a checking account. A bank account has $85.00. If no money is deposited or withdrawn except the service charge, how many months until the account balance is negative?
Let m be the number of months. Our balance is denoted by B(m):
B(m) = 85 - 7.5m
The question asks when B(m) is less than 0. So we set up an inequality:
85 - 7.5m < 0
To solve this inequality for m, [URL='https://www.mathcelebrity.com/1unk.php?num=85-7.5m%3C0&pl=Solve']we type it in our search engine[/URL] and we get:
m > 11.3333
We round up to the next whole integer and get [B]m = 12[/B]

a baseball park charges $4.50 per admission ticket. the park has already sold 42 tickets. how many m

a baseball park charges $4.50 per admission ticket. the park has already sold 42 tickets. how many more tickets would they need to sell to earn at least $441?
Let the number of tickets above 42 be t.
A few things to note on this question:
[LIST]
[*]The phrase [I]at least[/I] means greater than or equal to, so we have an inequality.
[*]Earnings = Price * Quantity
[/LIST]
We're given:
Earnings = 4.50 * 42 + 4.5t >= 441
Earnings = 189 + 4.5t >= 441
We want to solve this inequality for t:
Solve for [I]t[/I] in the inequality 189 + 4.5t ? 441
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 189 and 441. To do that, we subtract 189 from both sides
4.5t + 189 - 189 ? 441 - 189
[SIZE=5][B]Step 2: Cancel 189 on the left side:[/B][/SIZE]
4.5t ? 252
[SIZE=5][B]Step 3: Divide each side of the inequality by 4.5[/B][/SIZE]
4.5t/4.5 ? 252.4.5
[B]t ? 56[/B]

A bicycle store costs $1500 per month to operate. The store pays an average of $60 per bike. The ave

A bicycle store costs $1500 per month to operate. The store pays an average of $60 per bike. The average selling price of each bicycle is $80. How many bicycles must the store sell each month to break even?
Profit = Revenue - Cost
Let the number of bikes be b.
Revenue = 80b
Cost = 60b + 1500
Break even is when profit equals 0, which means revenue equals cost. Set them equal to each other:
60b + 1500 = 80b
We [URL='https://www.mathcelebrity.com/1unk.php?num=60b%2B1500%3D80b&pl=Solve']type this equation into our search engine[/URL] and we get:
b = [B]75[/B]

A bicycle store costs $2750 per month to operate. The store pays an average of $45 per bike. The a

A bicycle store costs $2750 per month to operate. The store pays an average of $45 per bike. The average selling price of each bicycle is $95. How many bicycles must the store sell each month to break even?
Let the number of bikes be b.
Set up our cost function, where it costs $45 per bike to produce
C(b) = 45b
Set up our revenue function, where we earn $95 per sale for each bike:
R(b) = 95b
Set up our profit function, which is how much we keep after a sale:
P(b) = R(b) - C(b)
P(b) = 95b - 45b
P(b) = 50b
The problem wants to know how many bikes we need to sell to break-even. Note: break-even means profit equals operating cost, which in this case, is $2,750. So we set our profit function of 50b equal to $2,750
50b = 2750
[URL='https://www.mathcelebrity.com/1unk.php?num=50b%3D2750&pl=Solve']We type this equation into our search engine[/URL], and we get:
b = [B]55[/B]

a bicycle store costs $3600 per month to operate. The store pays an average of $60 per bike. the ave

a bicycle store costs $3600 per month to operate. The store pays an average of $60 per bike. the average selling price of each bicycle is $100. how many bicycles must the store sell each month to break even?
Cost function C(b) where b is the number of bikes:
C(b) = Variable Cost + Fixed Cost
C(b) = Cost per bike * b + operating cost
C(b) = 60b + 3600
Revenue function R(b) where b is the number of bikes:
R(b) = Sale price * b
R(b) = 100b
Break Even is when Cost equals Revenue, so we set C(b) = R(b):
60b + 3600 = 100b
To solve this equation for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=60b%2B3600%3D100b&pl=Solve']type it in our math engine[/URL] and we get:
b = [B]90[/B]

A board must be cut into three pieces that are the same length. If it takes five minutes for each cu

A board must be cut into three pieces that are the same length. If it takes five minutes for each cut, how long will it take to saw the board into three pieces that are the same size?
Three equal pieces means only 2 cuts on the board:
2 cuts * 5 minutes per cut = [B]10 minutes[/B]

A boat is marked up 1/5 of the original price. The original price was $50. What is the new price of

A boat is marked up 1/5 of the original price. The original price was $50. What is the new price of the boat
1/5 of 50 equals 10. So we add the markup of 10 to the original price of 50:
50 + 10 = [B]$60[/B]

A book publishing company has fixed costs of $180,000 and a variable cost of $25 per book. The books

A book publishing company has fixed costs of $180,000 and a variable cost of $25 per book. The books they make sell for $40 each.
[B][U]Set up Cost Function C(b) where b is the number of books:[/U][/B]
C(b) = Fixed Cost + Variable Cost x Number of Units
C(b) = 180,000 + 25(b)
[B]Set up Revenue Function R(b):[/B]
R(b) = 40b
Set them equal to each other
180,000 + 25b = 40b
Subtract 25b from each side:
15b = 180,000
Divide each side by 15
[B]b = 12,000 for break even[/B]

A box has y oranges. 5 oranges are rotten and removed. The remaining oranges are placed equally into

A box has y oranges. 5 oranges are rotten and removed. The remaining oranges are placed equally into 2 containers . Each container has ______oranges
Remove 5 rotten oranges means we subtract 5 from y:
y - 5
If each of the two remaining boxes contains an equal amount of the remaining oranges, we have:
[B](y - 5)/2[/B] oranges in each box

A bus ride cost 1.50. A 30 day pass cost $24. Write an inequallity to show that the 30 day pass is t

A bus ride cost 1.50. A 30 day pass cost $24. Write an inequallity to show that the 30 day pass is the better deal
Let the number of days be d. We have the inequality below where we show when the day to day cost is greater than the monthly pass:
1.5d > 24
To solve this inequality for d, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=1.5d%3E24&pl=Show+Interval+Notation']type it in our search engine[/URL] and we get:
[B]d > 16[/B]

A cable company charges $75 for installation plus $20 per month. Another cable company offers free i

A cable company charges $75 for installation plus $20 per month. Another cable company offers free installation but charges $35 per month. For how many months of cable service would the total cost from either company be the same
[U]Set ups the cost function for the first cable company C(m) where m is the number of months:[/U]
C(m) = cost per month * m + installation fee
C(m) = 20m + 75
[U]Set ups the cost function for the second cable company C(m) where m is the number of months:[/U]
C(m) = cost per month * m + installation fee
C(m) = 35m
The problem asks for m when both C(m) functions are equal. So we set both C(m) functions equal and solve for m:
20m + 75 = 35m
To solve for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=20m%2B75%3D35m&pl=Solve']type this equation into our search engine[/URL] and we get:
m = [B]5[/B]

A cake is to be divided into 8 equal parts. After division, each equal portion is again divided into

A cake is to be divided into 8 equal parts. After division, each equal portion is again divided into 2 equal individual parts. How big is each of the new equal parts?
1 cake * 8 parts * 2 parts = 16 parts.
So each slice is 1/16 of a cake.

A cake that weighs 2 pounds was split equally among 5 children. How much cake did each child get?

A cake that weighs 2 pounds was split equally among 5 children. How much cake did each child get?
2 pounds / 5 children = [B]0.4 pounds per child[/B]

A candlestick burns at a rate of 0.2 inches per hour. After eight straight hours of burning, the can

A candlestick burns at a rate of 0.2 inches per hour. After eight straight hours of burning, the candlestick is 13.4 inches tall. Write and solve a linear equation to find the original height of the candle.
Let h equal the number of hours the candlestick burns. We have a candlestick height equation of C.
C = 13.4 + 0.2(8) <-- We need to add back the 8 hours of candlestick burning
C = 13.4 + 1.6
C = [B]15 inches[/B]

A car salesman earns $800 per month plus a 10% commission on the value of sales he makes for the mon

A car salesman earns $800 per month plus a 10% commission on the value of sales he makes for the month. If he is aiming to earn a minimum of $3200 a month, what is the possible value of sales that will enable this?
to start, we have:
[LIST]
[*]Let the salesman's monthly sales be s.
[*]With a 10% discount as a decimal of 0.1
[*]The phrase [I]a minimum[/I] also means [I]at least[/I] or [I]greater than or equal to[/I]. This tells us we want an inequality
[*]We want 10% times s + 800 per month is greater than or equal to 3200
[/LIST]
We want the inequality:
0.1s + 800 >= 3200
To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.1s%2B800%3E%3D3200&pl=Solve']type this inequality into our search engine[/URL] and we get:
[B]s >= 24000[/B]

A car who’s original value was $25600 decreases in value by $90 per month. How Long will it take bef

A car who’s original value was $25600 decreases in value by $90 per month. How Long will it take before the cars value falls below $15000
Let m be the number of months.We have our Book Value B(m) given by:
B(m) = 25600 - 90m
We want to know when the Book value is less than 15,000. So we have an inequality:
25600 - 90m < 15000
Typing [URL='https://www.mathcelebrity.com/1unk.php?num=25600-90m%3C15000&pl=Solve']this inequality into our search engine and solving for m[/URL], we get:
[B]m > 117.78 or m 118 months[/B]

a carnival charges $6 admission and $2.50 per ride. You have $50 to spend at the carnival. Which of

a carnival charges $6 admission and $2.50 per ride. You have $50 to spend at the carnival. Which of the following inequalities represents the situation if r is the number of rides?
We set up our inequality using less than or equal to, since our cash is capped at $50. We use S for our :
Cost per ride * r + Admission <= 50
Plugging in our numbers, we get:
2.50r + 6 <= 50
[B][/B]
Now, if the problem asks you to put this in terms of r, then [URL='https://www.mathcelebrity.com/1unk.php?num=2.50r%2B6%3C%3D50&pl=Solve']we plug this inequality into our search engine[/URL] and we get:
r <= 17.6
Since we cannot do fractional rides, we round down to 17:
[B]r <= 17[/B]

A carnival charges a $15 admission price. Each game at the carnival costs $4. How many games would a

A carnival charges a $15 admission price. Each game at the carnival costs $4. How many games would a person have to play to spend at least $40?
Let g be the number of games. The Spend function S(g) is:
S(g) = Cost per game * number of games + admission price
S(g) = 4g + 15
The problem asks for g when S(g) is at least 40. At least is an inequality using the >= sign:
4g + 15 >= 40
To solve this inequality for g, we type it in our search engine and we get:
g >= 6.25
Since you can't play a partial game, we round up and get:
[B]g >= 7[/B]

A cell phone costs $20 for 400 minutes and $2 for each extra minute. Gina uses 408 minutes. How m

A cell phone costs $20 for 400 minutes and $2 for each extra minute. Gina uses 408 minutes. How much will it cost?
Set up the cost function for minutes (m) if m is greater than or equal to 400
C(m) = 20 + 2(m - 400)
For m = 408, we have:
C(408) = 20 + 2(408 - 400)
C(408) = 20 + 2(8)
C(408) = [B]36[/B]

A cell phone provider is offering an unlimited data plan for $70 per month or a 5 GB plan for $55 pe

A cell phone provider is offering an unlimited data plan for $70 per month or a 5 GB plan for $55 per month. However, if you go over your 5 GB of data in a month, you have to pay an extra $10 for each GB. How many GB would be used to make both plans cost the same?
Let g be the number of GB.
The limited plan has a cost as follows:
C = 10(g - 5) + 55
C = 10g - 50 + 55
C = 10g + 5
We want to set the limited plan equal to the unlimited plan and solve for g:
10g + 5 = 70
Solve for [I]g[/I] in the equation 10g + 5 = 70
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 5 and 70. To do that, we subtract 5 from both sides
10g + 5 - 5 = 70 - 5
[SIZE=5][B]Step 2: Cancel 5 on the left side:[/B][/SIZE]
10g = 65
[SIZE=5][B]Step 3: Divide each side of the equation by 10[/B][/SIZE]
10g/10 = 65/10
g = [B]6.5[/B]
Check our work for g = 6.5:
10(6.5) + 5
65 + 5
70

A certain species of fish costs $3.19 each. You can spend at most $35. How many of this type of f

A certain species of fish costs $3.19 each. You can spend at most $35. How many of this type of fish can you buy for your aquarium?
Let the number of fish be f. We have the following inequality where "at most" means less than or equal to:
3.19f <= 35
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=3.19f%3C%3D35&pl=Show+Interval+Notation']Typing this inequality into our search engine[/URL], we get:
f <= 10.917
Since we need a whole number of fish, we can buy a maximum of [B]10 fish[/B].

A classroom had x students. Then 9 of them went home. There are now 27 students in the classroom.

A classroom had x students. Then 9 of them went home. There are now 27 students in the classroom.
Take this one piece at a time:
[LIST]
[*]We start with x students
[*]9 of them went home. This means we have 9 less students. So we subtract 9 from x: x - 9
[*]The phrase [I]there are now[/I] means an equation, so we set x - 9 equal to 27
[/LIST]
x - 9 = 27
To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=x-9%3D27&pl=Solve']type it in our search engine[/URL] and we get:
x = [B]36[/B]

a company has revenue given by R(x)=500x dollars and total cost given by C(x)=48,000 100x dollars, w

a company has revenue given by R(x)=500x dollars and total cost given by C(x)=48,000 + 100x dollars, where x is the number of units produced and sold. How many units will give a profit
Profit P(x) is given by:
R(x) - C(x)
So we have:
P(x) = 500x - (100x + 48,000)
P(x) = 500x - 100x - 48,000
P(x) = 400x - 48,000
A profit is found when P(x) > 0, so we have:
400x - 48000 > 0
To solve this inequality, [URL='https://www.mathcelebrity.com/1unk.php?num=400x-48000%3E0&pl=Solve']we type it into our search engine [/URL]and we get:
[B]x > 120[/B]

A company makes toy boats. Their monthly fixed costs are $1500. The variable costs are $50 per boat.

A company makes toy boats. Their monthly fixed costs are $1500. The variable costs are $50 per boat. They sell boats for $75 a piece. How many boats must be sold each month to break even?
[U]Set up Cost function C(b) where t is the number of tapestries:[/U]
C(b) = Cost per boat * number of boats + Fixed Cost
C(b) = 50b + 1500
[U]Set up Revenue function R(b) where t is the number of tapestries:[/U]
R(b) = Sale Price * number of boats
R(b) = 75b
[U]Break even is where Revenue equals Cost, or Revenue minus Cost is 0, so we have:[/U]
R(b) - C(b) = 0
75b - (50b + 1500) = 0
75b - 50b - 1500 = 0
25b - 1500 = 0
To solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=25b-1500%3D0&pl=Solve']type this equation in our math engine[/URL] and we get:
b = [B]60[/B]

A company now has 4900 employees nationwide. It wishes to reduce the number of employees by 300 per

A company now has 4900 employees nationwide. It wishes to reduce the number of employees by 300 per year through retirements, until its total employment is 2560. How long will this take?
Figure out how many reductions are needed
4900 - 2560 = 2340
We want 300 per year for retirements, so let x equal how many years we need to get 2340 reductions.
300x = 2340
Divide each side by 300
x = 7.8 years.
If we want full years, we would do 8 full years

A company specializes in personalized team uniforms. It costs the company $15 to make each uniform a

A company specializes in personalized team uniforms. It costs the company $15 to make each uniform along with their fixed costs at $640. The company plans to sell each uniform for $55.
[U]The cost function for "u" uniforms C(u) is given by:[/U]
C(u) = Cost per uniform * u + Fixed Costs
[B]C(u) = 15u + 640[/B]
Build the revenue function R(u) where u is the number of uniforms:
R(u) = Sale Price per uniform * u
[B]R(u) = 55u[/B]
Calculate break even function:
Break even is where Revenue equals cost
C(u) = R(u)
15u + 640 = 55u
To solve for u, we [URL='https://www.mathcelebrity.com/1unk.php?num=15u%2B640%3D55u&pl=Solve']type this equation into our search engine[/URL] and we get:
u = [B]16
So we break even selling 16 uniforms[/B]

A cup of coffee costs $1.75. A monthly unlimited coffee card costs $25.00. Which inequality represe

A cup of coffee costs $1.75. A monthly unlimited coffee card costs $25.00. Which inequality represents the number x of cups of coffee you must purchase for the monthly card to be a better deal?
Let c be the number of cups. We want to know how many cups (x) where:
1.75x > 25
Using our [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=1.75x%3E25&pl=Show+Interval+Notation']inequality solver[/URL], we see:
[B]x > 14.28[/B]

A cup that is filled with equal parts red, green, and blue dye spills half of its contents. Enough g

A cup that is filled with equal parts red, green, and blue dye spills half of its contents. Enough green dye is then poured into the cup to fill it again. What is the ratio of red to green to blue dye now?
Original Cup:
[LIST=1]
[*]Blue
[*]Green
[*]Red
[/LIST]
Spilled Cup
[LIST=1]
[*]Empty
[*]Blue
[*]Green
[*]Red
[/LIST]
Refilled Cup
[LIST=1]
[*]Green
[*]Blue
[*]Green
[*]Red
[/LIST]
[B]1:4:1[/B]

A dad gave his 3 sons each the same amount of money in an envelope. He took $20 from one son for get

A dad gave his 3 sons each the same amount of money in an envelope. He took $20 from one son for getting a D on a math test and he gave another son an extra $35 for doing extra chores. Combined, the sons had $81. Figure out how much each son had.
Let x, y, and z be the money each son received.
To begin, x = y = z.
But then, Dad took 20 from son X and gave it to son Y.
So now, x = y - 20
Next, he gave Son Z an extra $35 for chores
So z is now y + 35 since y and z used to be equal
Combined, they all have 81.
x + y + z = 181
But with the changes, it is:
(y - 20) + y + (y + 35)
Combine like terms:
3y - 20 + 35 = 81
3y + 15 = 81
Subtract 15 from each side:
3y = 66
Divide each side by 3 to isolate y
y = 22
Since x = y - 20, x = 2
Since z = y + 35, we have z = 57
Checking our work, 2 + 22 + 57 = 81.

A daily pass costs $62. A season ski pass costs $450. The skier would have to rent skis with eithe

A daily pass costs $62. A season ski pass costs $450. The skier would have to rent skis with either pass for $30 per day. How many days would the skier have to go skiing in order to make the season pass less expensive than the daily passes?
Let d be the number of days the skier attends.
Calculate the daily cost:
Daily Total Cost = Daily Cost + Rental Cost
Daily Total Cost = 62d + 30d
Daily Total Cost = 92d
Calculate Season Cost:
Season Total Cost = Season Fee + Rental Cost
Season Total Cost = 450 + 30d
Set the daily total cost and season cost equal to each other:
450 + 30d = 92d
[URL='https://www.mathcelebrity.com/1unk.php?num=450%2B30d%3D92d&pl=Solve']Typing this equation into the search engine[/URL], we get d = 7.258.
We round up to the next full day of [B]8[/B].
Now check our work:
Daily Total Cost for 8 days = 92(8) = 736
Season Cost for 8 days = 30(8) + 450 = 240 + 450 = 710.
Therefore, the skier needs to go at least [B]8 days[/B] to make the season cost less than the daily cot.

A fair die is rolled. What is the probability of rolling a 3 or a 6?

A fair die is rolled. What is the probability of rolling a 3 or a 6?
P(3 or 6) can be written as:
P(3) + P(6)
A fair die means all faces have an equal probability of 1/6
P(3) = 1/6
P(6) = 1/6
P(3 or 6) = P(3) + P(6)
P(3 or 6) = 1/6 + 1/6
P(3 or 6) = 2/6
[URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F6&frac2=3%2F8&pl=Simplify']Using our fractions simplifier for 2/6[/URL], we get:
P(3 or 6) = [B]1/3[/B]

A family is taking a cross-country trip of 3000 miles by car. They are bringing two spare tires with

A family is taking a cross-country trip of 3000 miles by car. They are bringing two spare tires with them and want all six tires to go an equal distance. How many miles will each tire go?
3000 * 4 tires = 12,000 miles traveled
12,000 / 6 tires = [B]2,000 miles[/B]

A farmer bought a number of pigs for $232. However, 5 of them died before he could sell the rest at

A farmer bought a number of pigs for $232. However, 5 of them died before he could sell the rest at a profit of 4 per pig. His total profit was $56. How many pigs did he originally buy?
Let p be the purchase price of pigs. We're given:
[LIST]
[*]Farmer originally bought [I]p [/I]pigs for 232 which is our cost C.
[*]5 of them died, so he has p - 5 left
[*]He sells 4(p - 5) pigs for a revenue amount R
[*]Since profit is Revenue - Cost, which equals 56, we have:
[/LIST]
Calculate Profit
P = R - C
Plug in our numbers:
4(p - 5) - 232 = 56
4p - 20 - 232 = 56
To solve for p, [URL='https://www.mathcelebrity.com/1unk.php?num=4p-20-232%3D56&pl=Solve']we type this equation into our search engine[/URL] and we get:
p = [B]77[/B]

a fever is generally considered to be a body temperature greater than 100 F. You friend has a temper

a fever is generally considered to be a body temperature greater than 100 F. You friend has a temperature of 37 C. Does you friend have a fever?
37 Celsius equals 98.6 F. Since this is less than 100F, your friend does not have a fever.

a firefighter must cut a 2,700 foot rope into 75 equal pieces how long will each be

a firefighter must cut a 2,700 foot rope into 75 equal pieces how long will each be
2700/75 = [B]36 feet per piece[/B]

A first number plus twice a second number is 10. Twice the first number plus the second totals 35. F

A first number plus twice a second number is 10. Twice the first number plus the second totals 35. Find the numbers.
[U]The phrase [I]a number[/I] means an arbitrary variable[/U]
A first number is written as x
A second number is written as y
[U]Twice a second number means we multiply y by 2:[/U]
2y
[U]A first number plus twice a second number:[/U]
x + 2y
[U]A first number plus twice a second number is 10 means we set x + 2y equal to 10:[/U]
x + 2y = 10
[U]Twice the first number means we multiply x by 2:[/U]
2x
[U]Twice the first number plus the second:[/U]
2x + y
[U]Twice the first number plus the second totals 35 means we set 2x + y equal to 35:[/U]
2x + y = 35
Therefore, we have a system of two equations:
[LIST=1]
[*]x + 2y = 10
[*]2x + y = 35
[/LIST]
Since we have an easy multiple of 2 for the x variable, we can solve this by multiply the first equation by -2:
[LIST=1]
[*]-2x - 4y = -20
[*]2x + y = 35
[/LIST]
Because the x variables are opposites, we can add both equations together:
(-2 + 2)x + (-4 + 1)y = -20 + 35
The x terms cancel, so we have:
-3y = 15
To solve this equation for y, we [URL='https://www.mathcelebrity.com/1unk.php?num=-3y%3D15&pl=Solve']type it in our search engine[/URL] and we get:
y = [B]-5
[/B]
Now we substitute this y = -5 into equation 2:
2x - 5 = 35
To solve this equation for x, we[URL='https://www.mathcelebrity.com/1unk.php?num=2x-5%3D35&pl=Solve'] type it in our search engine[/URL] and we get:
x = [B]20[/B]

A first number plus twice a second number is 14. Twice the first number plus the second totals 40. F

A first number plus twice a second number is 14. Twice the first number plus the second totals 40. Find the numbers.
[B][U]Givens and assumptions:[/U][/B]
[LIST]
[*]Let the first number be x.
[*]Let the second number be y.
[*]Twice means multiply by 2
[*]The phrases [I]is[/I] and [I]totals[/I] mean equal to
[/LIST]
We're given two equations:
[LIST=1]
[*]x + 2y = 14
[*]2x + y = 40
[/LIST]
To solve this system, we can take a shortcut, and multiply the top equation by -2 to get our new system:
[LIST=1]
[*]-2x - 4y = -28
[*]2x + y = 40
[/LIST]
Now add both equations together
(-2 _ 2)x (-4 + 1)y = -28 + 40
-3y = 12
To solve this equation, we [URL='https://www.mathcelebrity.com/1unk.php?num=-3y%3D12&pl=Solve']type it in our search engine[/URL] and we get:
y = [B]-4
[/B]
We substitute this back into equation 1 for y = -4:
x + 2(-4) = 14
x - 8 = 14
To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=x-8%3D14&pl=Solve']type it in our search engine[/URL] and we get:
x = [B]22[/B]

A first number plus twice a second number is 7

A first number plus twice a second number is 7
Let the first number be x. Let the second number be y. We're given:
[LIST]
[*]A first number is x
[*]A second number is y
[*]Twice the second number means we multiply y by 2: 2y
[*][I]Plus [/I]means we add x to 2y: x + 2y
[*]The phrase [I]is[/I] means an equation, so we set x + 2y equal to 7
[/LIST]
[B]x + 2y = 7[/B]

A football team lost 7 yards each play for four consecutive plays. Represent the team’s total change

A football team lost 7 yards each play for four consecutive plays. Represent the team’s total change in position for the four plays as an integer.
A net loss in yardage for 7 yards is written as -7
4 plays * -7 yards equals [B]-28[/B]

A fraction has a value of 3/4. If 7 is added to the numerator, the resulting fraction is equal to th

A fraction has a value of 3/4. If 7 is added to the numerator, the resulting fraction is equal to the reciprocal of the original fraction. Find the original fraction.
Let the fraction be x/y. We're given two equations:
[LIST=1]
[*]x/y = 3/4
[*](x + 7)/y = 4/3. [I](The reciprocal of 3/4 is found by 1/(3/4)[/I]
[/LIST]
Cross multiply equation 1 and equation 2:
[LIST=1]
[*]4x = 3y
[*]3(x + 7) = 4y
[/LIST]
Simplifying, we get:
[LIST=1]
[*]4x = 3y
[*]3x + 21 = 4y
[/LIST]
If we divide equation 1 by 4, we get:
[LIST=1]
[*]x = 3y/4
[*]3x + 21 = 4y
[/LIST]
Substitute equation (1) into equation (2) for x:
3(3y/4) + 21 = 4y
9y/4 + 21 = 4y
Multiply the equation by 4 on both sides to eliminate the denominator:
9y + 84 = 16y
To solve this equation for y, we type it in our math engine and we get:
y = [B]12
[/B]
We then substitute y = 12 into equation 1 above:
x = 3 * 12/4
x = 36/4
x = [B]9
[/B]
So our original fraction x/y = [B]9/12[/B]

A furniture company plans to have 25 employees at its corporate headquarters and 25 employees at eac

A furniture company plans to have 25 employees at its corporate headquarters and 25 employees at each store it opens. Let s represent the number of stores and m represent the total number of employees.
There is only one corporate headquarters. So we have the number of employees (m) as:
m = Store Employees + Corporate Employees
Each store has 25 employees. Total store employees equal 25 per store times the number of stores (s).
[B]m = 25s + 25[/B]

A grocer is selling oranges at 3 for $2. How much would it cost to buy a dozen oranges?

A grocer is selling oranges at 3 for $2. How much would it cost to buy a dozen oranges?
Set up a proportion of oranges per cost where c is the cost of a dozen oranges:
3/2 = 12/c <-- A dozen equals 12
[URL='https://www.mathcelebrity.com/prop.php?num1=3&num2=12&den1=2&den2=c&propsign=%3D&pl=Calculate+missing+proportion+value']Typing this proportion into our search engine[/URL], we get:
c = [B]8[/B]

A group of scientists studied the effect of a chemical on various strains of bacteria. Strain A star

A group of scientists studied the effect of a chemical on various strains of bacteria. Strain A started with 6000 cells and decreased at a constant rate of 2000 cells per hour after the chemical was applied. Strain B started with 2000 cells and decreased at a constant rate of 1000 cells per hour after the chemical was applied. When will the strains have the same number of cells? Explain.
Set up strain equations where h is the number of hours since time 0:
[LIST]
[*]Strain A: 6000 - 2000h
[*]Strain B: 2000 - 1000h
[/LIST]
Set them equal to each other
6000 - 2000h = 2000 - 1000h
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=6000-2000h%3D2000-1000h&pl=Solve']equation solver[/URL], we see that [B]h = 4[/B]

A hexagon has a total 126 dots and a equal number of dots on each side. how many dots on each side?

A hexagon has a total 126 dots and a equal number of dots on each side. how many dots on each side?
Since it has an equal number of dots on each side, each side has:
Number of dots on each side = 126 dots / 6 sides
Number of dots on each side = [B]21 dots per side[/B]

A house painting company charges $376 plus $12 per hour. Another painting company charges $280 plus

A house painting company charges $376 plus $12 per hour. Another painting company charges $280 plus $15 per hour. How long is a job for which companies will charge the same amount?
Set up the cost function C(h) where h is the number of hours.
Company 1:
C(h) = 12h + 376
Company 2:
C(h) = 15h + 280
To see when the companies charge the same amount, set both C(h) functions equal to each other.
12h + 376 = 15h + 280
To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=12h%2B376%3D15h%2B280&pl=Solve']type this equation into our search engine[/URL] and we get:
h = [B]32[/B]

A house painting company charges $376 plus $12 per hour. Another painting company charges $280 plus

A house painting company charges $376 plus $12 per hour. Another painting company charges $280 plus $15 per hour. How long is a job for which both companies will charge the same amount?
[U]Set up the cost function for the first company C(h) where h is the number of hours:[/U]
C(h) = Hourly Rate * h + flat rate
C(h) = 12h + 376
[U]Set up the cost function for the first company C(h) where h is the number of hours:[/U]
C(h) = Hourly Rate * h + flat rate
C(h) = 15h + 280
The problem asks how many hours will it take for both companies to charge the same. So we set the cost functions equal to each other:
12h + 376 = 15h + 280
Plugging this equation [URL='https://www.mathcelebrity.com/1unk.php?num=12h%2B376%3D15h%2B280&pl=Solve']into our search engine and solving for h[/URL], we get:
h = [B]32[/B]

A is the set of integers greater than or equal to -5 and less than or equal to -2

A is the set of integers greater than or equal to -5 and less than or equal to -2
[B]-5 <= A <= -2[/B]

A jug holds 1.2 litres of orange juice. All of the juice is poured equally into six glasses. How muc

A jug holds 1.2 litres of orange juice. All of the juice is poured equally into six glasses. How much orange is in each glass?
1.2 litres / 6 glasses = [B]0.2 litres[/B] in each glass.

A light bulb consumes 17100 watt-hours in 4 days and 18 hours. How many watt-hours does it consume a

A light bulb consumes 17100 watt-hours in 4 days and 18 hours. How many watt-hours does it consume a day?
Since one day equals 24 hours, we have:
4 days and 18 hours equals:
4(24) + 18 hours
96 + 18 hours
114 hours
Therefore, we have a proportion, where w is the number of watt-hours in a 24-hour period.
17,100 watt-hours/114 hours = w/24
[URL='https://www.mathcelebrity.com/prop.php?num1=17100&num2=w&den1=114&den2=24&propsign=%3D&pl=Calculate+missing+proportion+value']Typing 1711/114 = w/24 into our calculator[/URL], we get:
[B]w = 3,600[/B]

a lighthouse blinks every 12 minutes. A second lighthouse blinks every 10 minutes if they both blink

a lighthouse blinks every 12 minutes. A second lighthouse blinks every 10 minutes if they both blink at 8:10 P.M at what time will they next blink together
We want the least common multiple of (10, 12). This will be the next time each number times a multiple equals the same number.
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=10&num2=12&num3=&pl=GCF+and+LCM']Typing in LCM 10,12 into our search engine[/URL], we get:
60
So if we start at 8:10, and 60 minutes later is when both lighthouses blink. 60 minutes equals 1 hour.
So we add 1 hour to 8:10, we have [B]9:10[/B]

A local bank charges 19 per month plus 3 cents per check. The credit union charges7 per month plus

A local bank charges 19 per month plus 3 cents per check. The credit union charges7 per month plus 7 cents per check. How many checks should be written each month to make the credit union a better deal?
Set up the cost function B(c) for the local bank where c is the number of checks:
B(c) = 0.03c + 19
Set up the cost function B(c) for the credit union where c is the number of checks:
B(c) = 0.07c + 7
We want to find out when:
0.07c + 7 < 0.03c + 19
[URL='https://www.mathcelebrity.com/1unk.php?num=0.07c%2B7%3C0.03c%2B19&pl=Solve']Typing this inequality into our search engine[/URL], we get:
c < 300

A local sports centre charges $8 per visit. For an annual membership fee of$45, the cost per visit i

A local sports centre charges $8 per visit. For an annual membership fee of$45, the cost per visit is only $5.50. What is the least number of visits needed in a year in order for the membership to be a better deal?
Set up the cost for the visitors plan C(v) where v is the number of visits:
C(v) = 8v
Set up the cost for the membership plan C(v) where v is the number of visits:
C(v) = 5v + 45
The problem asks for v where:
5v + 45 < 8v
[URL='https://www.mathcelebrity.com/1unk.php?num=5v%2B45%3C8v&pl=Solve']Type this inequality into our search engine[/URL] and get:
v > 15
This means, the least number of visits is 1 more which is [B]16[/B]

A man is 5 years older than his wife, and the daughter age is half of the mother, and if you add the

A man is 5 years older than his wife, and the daughter age is half of the mother, and if you add their ages is equal 100
Let the man's age be m. Let the wife's age be w. Let the daughter's age be d. We're given:
[LIST=1]
[*]m = w + 5
[*]d = 0.5m
[*]d + m + w = 100
[/LIST]
Rearrange equation 1 in terms of w my subtracting 5 from each side:
[LIST=1]
[*]w = m - 5
[*]d = 0.5m
[*]d + m + w = 100
[/LIST]
Substitute equation (1) and equation (2) into equation (3)
0.5m + m + m - 5 = 100
We [URL='https://www.mathcelebrity.com/1unk.php?num=0.5m%2Bm%2Bm-5%3D100&pl=Solve']type this equation into our search engine[/URL] to solve for m and we get:
m = [B]42
[/B]
Now, substitute m = 42 into equation 2 to solve for d:
d = 0.5(42)
d = [B]21
[/B]
Now substitute m = 42 into equation 1 to solve for w:
w = 42 - 5
w = [B]37
[/B]
To summarize our ages:
[LIST]
[*]Man (m) = 42 years old
[*]Daughter (d) = 21 years old
[*]Wife (w) = 37 years old
[/LIST]

A man's age (a) 10 years ago is 43

A man's age (a) 10 years ago is 43
[U]10 years ago means we subtract 10 from a:[/U]
a - 10
[U]The word [I]is[/I] means an equation. So we set a - 10 equal to 43 to get our algebraic expression[/U]
[B]a - 10 = 43[/B]
If the problem asks you to solve for a, [URL='https://www.mathcelebrity.com/1unk.php?num=a-10%3D43&pl=Solve']we type this equation into our search engine[/URL] and we get:
a = 53

A manufacturer has a monthly fixed cost of $25,500 and a production cost of $7 for each unit produce

A manufacturer has a monthly fixed cost of $25,500 and a production cost of $7 for each unit produced. The product sells for $10/unit.
Set up cost function where u equals each unit produced:
C(u) = 7u + 25,500
Set up revenue function
R(u) = 10u
Break Even is where Cost equals Revenue
7u + 25,500 = 10u
Plug this into our [URL='http://www.mathcelebrity.com/1unk.php?num=7u%2B25500%3D10u&pl=Solve']equation calculator[/URL] to get [B]u = 8,500[/B]

A math teacher bought 40 calculators at $8.20 each and a number of other calculators costing$2.95 ea

A math teacher bought 40 calculators at $8.20 each and a number of other calculators costing$2.95 each. In all she spent $387. How many of the cheaper calculators did she buy
Let the number of cheaters calculators be c. Since amount equals price * quantity, we're given the following equation:
8.20 * 40 + 2.95c = 387
To solve this equation for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=8.20%2A40%2B2.95c%3D387&pl=Solve']type it in our search engine [/URL]and we get:
c = [B]20[/B]

A mechanic will charge a new customer $45.00 for an initial diagnosis plus $20 an hour of labor. How

A mechanic will charge a new customer $45.00 for an initial diagnosis plus $20 an hour of labor. How long did the mechanic work on a car if he charged the customer $165?
We set up a cost function C(h) where h is the number of hours of labor:
C(h) = Hourly Labor Rate * h + Initial Diagnosis
C(h) = 20h + 45
The problem asks for the number of hours if C(h) = 165. So we set our cost function C(h) above equal to 165:
20h + 45 = 165
To solve for h, [URL='https://www.mathcelebrity.com/1unk.php?num=20h%2B45%3D165&pl=Solve']we plug this equation into our search engine[/URL] and we get:
h = [B]6[/B]

A Middleweight UFC fighter weighs between 170 lbs and 185 lbs.

A Middleweight UFC fighter weighs between 170 lbs and 185 lbs.
Let w be the UFC fighter's weight:
We have a compound inequality.
Right side includes 185 lbs. because between means includes 185lbs.
Left side includes 170 lbs. because between means includes 17lb0s
[B]170 <= w <= 185[/B]

a more than b is greater than 6

a more than b is greater than 6
a more than b:
b + a
Is greater than 6 means an inequality using the > sign:
[B]b + a > 6[/B]

A motorist pays $4.75 per day in tolls to travel to work. He also has the option to buy a monthly pa

A motorist pays $4.75 per day in tolls to travel to work. He also has the option to buy a monthly pass for $80. How many days must he work (i.e. pass through the toll) in order to break even?
Let the number of days be d. Break even means both costs are equal. We want to find when:
4.75d = 80
To solve for d, we [URL='https://www.mathcelebrity.com/1unk.php?num=4.75d%3D80&pl=Solve']type this equation into our search engine[/URL] and we get:
d = 16.84 days
We round up to an even [B]17 days[/B].

a number is twice another number

a number is twice another number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x
The phrase [I]another number [/I]means another arbitrary variable, let's call it y
Twice means we multiply y by 2:
2y
The phrase [I]is [/I]means an equation, so we set x equal to 2y:
[B]x = 2y[/B]

A number m is no less than -8 and fewer than 9.

A number m is no less than -8 and fewer than 9.
No less than means greater than or equal to:
m >= -8
Fewer than 9 means less than 9:
m < 9
Combine these two inequalities to get
[B]-8 <= m < 9[/B]

A number multiplied by 6 and divided by 5 give four more than a number?

A number multiplied by 6 and divided by 5 give four more than a number?
A number is represented by an arbitrary variable, let's call it x.
Multiply by 6:
6x
Divide by 5
6x/5
The word "gives" means equals, so we set this equal to 4 more than a number, which is x + 4.
6x/5 = x + 4
Now, multiply each side of the equation by 5, to eliminate the fraction on the left hand side:
6x(5)/5 = 5(x + 4)
The 5's cancel on the left side, giving us:
6x = 5x + 20
Subtract 5x from each side
[B]x = 20[/B]
Check our work from our original equation:
6x/5 = x + 4
6(20)/5 ? 20 + 4
120/5 ?24
24 = 24 <-- Yes, we verified our answer

A number n diminished by 8 gives 12

A number n diminished by 8 gives 12
A number n can be written as n:
n
Diminished by means we subtract, so we subtract 8 from n:
n - 8
The word [I]gives[/I] means an equation, so we set n - 8 equal to 12:
[B]n - 8 = 12[/B]

A number n is no less than 2 and no more than 49.

A number n is no less than 2 and no more than 49.
This is a compound inequality. Let's break it into parts.
Step 1: No more than 49 means 49 or less. Or, less than or equal to 49
<= 49
Step 2: no less than 2 means 2 or greater. Or, greater than or equal to 2
>=2
Writing this in terms of the number n, we have:
[B]2 <= n <= 49[/B]

A number of dogs are to equally share a bag of dog food. If there are n dogs in the group and one do

A number of dogs are to equally share a bag of dog food. If there are [I]n[/I] dogs in the group and one dog eats its share, what percent of the bag is left?
Fraction of the bag left is:
(n - 1)/n
Multiply by 100 to get a percentage:
[B]100(n - 1)/n[/B]

a number of pennies splits into 4 equal groups

a number of pennies splits into 4 equal groups
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
We take x and divide it by 4 to get 4 equal groups:
[B]x/4[/B]

A number p subtracted by its double is 10

A number p subtracted by its double is 10
The double of a number means we multiply p by 2:
2p
A number p is subtracted by its double
p - 2p
The phrase [I]is[/I] means equal to, so we set p - 2p equal to 10:
[B]p - 2p = 10[/B]

A number t is no less than 30 and fewer than 40.

A number t is no less than 30 and fewer than 40.
This is a compound inequality. Take it in 3 parts:
Step 1: fewer than 40 means less than (does not include 40)
t < 40
Step 2: no less than 30 means greater than or equal to
t >= 30
Step 3: Combine these 2 statements into one compound inequality:
[B]30 <= t < 40[/B]

A pair of dice is rolled. Find the probability of rolling a sum of not less than 5

A pair of dice is rolled. Find the probability of rolling a sum of not less than 5.
The phrase [I]not less than[/I] also means greater than or equal to. So we [URL='https://www.mathcelebrity.com/2dice.php?gl=3&pl=5&opdice=1&rolist=+&dby=&ndby=&montect=+']use our 2 dice calculator for a sum roll of 5 or greater[/URL] and we get:
[B]5/6[/B]

A parabola has a Vertex at (4,-2) and a Focus at (6,-2). Find the equation of the parabola

A parabola has a Vertex at (4,-2) and a Focus at (6,-2). Find the equation of the parabola and the lotus rectum.
Equation of a parabola given the vertex and focus is:
([I]x[/I] – [I]h[/I])^2 = 4[I]p[/I]([I]y[/I] – [I]k[/I])
The vertex (h, k) is 4, -2
The distance is p, and since the y coordinates of -2 are equal, the distance is 6 - 4 = 2.
So p = 2
Our parabola equation becomes:
(x - 4)^2 = 4(2)(y - -2)
[B](x - 4)^2 = 8(y + 2)[/B]
Latus rectum of a parabola is 4p, where p is the distance between the vertex and the focus
LR = 4p
LR = 4(2)
[B]LR = 8[/B]

A parallelogram has a perimeter of 48 millimeters. Two of the sides are each 20 millimeters long. Wh

A parallelogram has a perimeter of 48 millimeters. Two of the sides are each 20 millimeters long. What is the length of each of the other two sides?
2 sides * 20 mm each is 40 mm
subtract this from the perimeter of 48:
48 - 40 = 8
Since the remaining two sides equal each other, their length is:
8/2 = [B]4mm[/B]

A parking lot has seventy-one parking spaces numbered from 1 to 71. There are no cars in the parking

A parking lot has seventy-one parking spaces numbered from 1 to 71. There are no cars in the parking lot when Jillian pulls in and randomly parks. What is the probability that the number on the parking space where she parks is greater than or equal to 31?
Greater than or equal to means including 31 all the way through 71
31-71 is 40 spaces
P(s>=31) = [B]40/71[/B]

A parking lot has sixty-eight parking spaces numbered from 1 to 68. There are no cars in the parking

A parking lot has sixty-eight parking spaces numbered from 1 to 68. There are no cars in the parking lot when Jillian pulls in and randomly parks. What is the probability that the number on the parking space where she parks is greater than or equal to 21?
We want P(X>=21). This is also found by taking 1 - P(X <= 20).
P(X<=20) = 20/68. Reduced using a [URL='http://www.mathcelebrity.com/gcflcm.php?num1=20&num2=68&num3=&pl=GCF']GCF of 4[/URL], we get 5/17.
P(X >=21) = 1 - 5/17 = [B]12/17[/B]

A pet store has 10 rabbits. What’s the probability that they have at least 1 female rabbit.

A pet store has 10 rabbits. What’s the probability that they have at least 1 female rabbit.
At least 1 female rabbit means we [U]must[/U] have a female rabbit
First, we calculate the probability of 0 females
A rabbit can be either male or female with equal probabilities of 1/2 or 0.5.
Since each birth is independent, we can multiply to get the probability of all males:
P(MMMMMMMMMM) = 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2
P(MMMMMMMMMM) = 1/1024
Then, we subtract this probability from 1 to get the probability of [B]at least[/B] one female:
P(At least one F) = 1 - 1/1024
Since 1 = 1024/1024, we have:
P(At least one F) = (1024 - 1)/1024
P(At least one F) = [B]1023/1024[/B]

A phone company offers two monthly charge plans. In Plan A, there is no monthly fee, but the custome

A phone company offers two monthly charge plans. In Plan A, there is no monthly fee, but the customer pays 8 cents per minute of use. In Plan B, the customer pays a monthly fee of $1.50 and then an additional 7 cents per minute of use.
For what amounts of monthly phone use will Plan A cost more than Plan B?
Set up the cost equations for each plan. The cost equation for the phone plans is as follows:
Cost = Cost Per Minute * Minutes + Monthly Fee
Calculate the cost of Plan A:
Cost for A = 0.08m + 0. <-- Since there's no monthly fee
Calculate the cost of Plan B:
Cost for B = 0.07m + 1.50
The problem asks for what amounts of monthly phone use will Plan A be more than Plan B. So we set up an inequality:
0.08m > 0.07m + 1.50
[URL='https://www.mathcelebrity.com/1unk.php?num=0.08m%3E0.07m%2B1.50&pl=Solve']Typing this inequality into our search engine[/URL], we get:
[B]m > 150
This means Plan A costs more when you use more than 150 minutes per month.[/B]

a piece of ribbon is y cm long. Ashley cut it into 4 equal pieces to do her 4 sisters hair. write an

a piece of ribbon is y cm long. Ashley cut it into 4 equal pieces to do her 4 sisters hair. write an expression for the amount of ribbon used for each sister
We take y cm and divide it equal among 4 sisters:
[B]y/4[/B]

A plant is 15 cm high and grows 4.5 cm every month. How many months will it take until the plant is

A plant is 15 cm high and grows 4.5 cm every month. How many months will it take until the plant is 27.5 cm
We set up the height function H(m) where m is the number of months since now. We have:
H(m) = 4.5m + 15
We want to know when H(m) = 27.5, so we set our H(m) function equal to 27.5:
4.5m + 15 = 27.5
To solve for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=4.5m%2B15%3D27.5&pl=Solve']type this equation into our search engine[/URL] and we get:
m = 2.78
So we round up to [B]3 whole months[/B]

A police officer is trying to catch a fleeing criminal. The criminal is 20 feet away from the cop, r

A police officer is trying to catch a fleeing criminal. The criminal is 20 feet away from the cop, running at a rate of 5 feet per second. The cop is running at a rate of 6.5 feet per second. How many seconds will it take for the police officer to catch the criminal?
Distance = Rate * Time
[U]Criminal:[/U]
5t + 20
[U]Cop[/U]:
6.5t
We want to know when their distances are the same (cop catches criminal). So we set the equations equal to each other:
5t + 20 = 6.5t
To solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=5t%2B20%3D6.5t&pl=Solve']we type it in our search engine[/URL] and we get:
t = 13.333 seconds

A quantity x is at least 10 and at most 20

A quantity x is at least 10 and at most 20
The phrase [I]at most[/I] means less than or equal to
The phrase [I]at least[/I] means greater than or equal to.
So we have the following inequality
[B]10 <= x <= 20[/B]

A quarter of a number is greater than or equal to 38

A quarter of a number is greater than or equal to 38.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
A quarter of a number means 1/4, so we have:
x/4
The phrase [I]is greater than or equal to[/I] means an inequality, so we use the >= sign in relation to 38:
[B]x/4 >= 38 <-- This is our algebraic expression
[/B]
If you want to solve this inequality, [URL='https://www.mathcelebrity.com/prop.php?num1=x&num2=38&propsign=%3E%3D&den1=4&den2=1&pl=Calculate+missing+proportion+value']we type it in the search engine[/URL] to get:
x >= [B]152[/B]

A rectangular football pitch has its length equal to twice its width and a perimeter of 360m. Find i

A rectangular football pitch has its length equal to twice its width and a perimeter of 360m. Find its length and width.
The area of a rectangle (A) is:
A = lw --> where l is the length and w is the width
We're given l = 2w, so we substitute this into the Area equation:
A = (2w)w
A = 2w^2
We're given the area of the pitch is 360, so we set:
2w^2 = 360
We [URL='https://www.mathcelebrity.com/1unk.php?num=2w%5E2%3D360&pl=Solve']type this equation into our search engine[/URL], follow the links, and get:
w = [B]6*sqrt(5)
[/B]
Now we take this, and substitute it into this equation:
6*sqrt(5)l = 360
Dividing each side by 6*sqrt(5), we get:
l = [B]60/sqrt(5)[/B]

A rental truck costs $49.95+$0.59 per mile and another costs $39.95 plus $0.99, set up an equation t

A rental truck costs $49.95+$0.59 per mile and another costs $39.95 plus $0.99, set up an equation to determine the break even point?
Set up the cost functions for Rental Truck 1 (R1) and Rental Truck 2 (R2) where m is the number of miles
R1(m) = 0.59m + 49.95
R2(m) = 0.99m + 39.95
Break even is when we set the cost functions equal to one another:
0.59m + 49.95 = 0.99m + 39.95
[URL='https://www.mathcelebrity.com/1unk.php?num=0.59m%2B49.95%3D0.99m%2B39.95&pl=Solve']Typing this equation into the search engine[/URL], we get [B]m = 25[/B].

A researcher believed that there was a difference in the amount of time boys and girls at 7th grade

A researcher believed that there was a difference in the amount of time boys and girls at 7th grade studied by using a two-tailed t test. Which of the following is the null hypothesis?
a. Mean of hours that boys studied per day was equal to mean of hours that girls studied per day
b. Mean of hours that boys studied per day was greater than mean of hours that girls studied per day
c. Mean of hours that boys studied per day was smaller than mean of hours that girls studied per day
d. Mean of hours that boys studied per day was smaller than or equal to mean of hours that girls studied per day
[B]a. Mean of hours that boys studied per day was equal to mean of hours that girls studied per day[/B]
Reason is that in hypothesis testing, you take a position other than what is assumed or what is being tested as the null hypothesis

A revenue function is R(x) = 22x and a cost function is C(x) = -9x + 341. The break-even point is

A revenue function is R(x) = 22x and a cost function is C(x) = -9x + 341. The break-even point is
Break even is when C(x) = R(x). So we set them equal and solve for x:
-9x + 341 = 22x
Typing[URL='https://www.mathcelebrity.com/1unk.php?num=-9x%2B341%3D22x&pl=Solve'] this equation into our search engine[/URL], we get:
x = [B]11[/B]

A river is rising at a rate of 3 inches per hour. If the river rises more than 2 feet, it will excee

A river is rising at a rate of 3 inches per hour. If the river rises more than 2 feet, it will exceed flood stage. How long can the river rise at this rate without exceeding flood stage?
Let the number of inches be i. Remember 12 inches to a foot, so we have 2 feet = 12*2 = 24 inches.
[LIST]
[*]Inequality: 3i <= 24. (since more than means the river can go [U]up to[/U] 2 feet or 24 inches
[/LIST]
To solve the inequality for I, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=3i%3C%3D24&pl=Show+Interval+Notation']type it in our search engine[/URL] and we get:
[B]I <= 8
This means after 8 hours, the river will flood[/B]

A Sales Manager buys antacid in bottles by the gross. If he goes through 3 bottles of antacid everyd

A Sales Manager buys antacid in bottles by the gross. If he goes through 3 bottles of antacid everyday, how long will the gross last?
[U][B]Calculate a gross[/B][/U]
[URL='https://en.wikipedia.org/wiki/Gross_(unit)#:~:text=A%20gross%20refers%20to%20a,cubic%20dozen%2C%20123).']A gross equals[/URL] 144 because 1 gross = 12 dozen
12 dozen * 12 items/dozen = 144
[B]
[U]Answer[/U][/B]
Days lasted = Total Bottles / Bottles per day
Days lasted = 144 bottles / 3 days
[B]Days lasted = 48 days[/B]

A salesperson works 40 hours per week at a job where he has two options for being paid. Option A is

A salesperson works 40 hours per week at a job where he has two options for being paid. Option A is an hourly wage of $24. Option B is a commission rate of 4% on weekly sales. How much does he need to sell this week to earn the same amount with the two options?
Option A payment function:
24h
With a 40 hour week, we have:
24 * 40 = 960
Option B payment function with sales amount (s):
0.04s
We want to know the amount of sales (s) where Option A at 40 hours = Option B. So we set both equal to each other:
0.04s = 960
To solve this equation for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.04s%3D960&pl=Solve']type it in our math engine[/URL] and we get:
s = [B]24,000[/B]

A school spent $150 on advertising for a breakfast fundraiser. Each plate of food was sold for $8.00

A school spent $150 on advertising for a breakfast fundraiser. Each plate of food was sold for $8.00 but cost the school $2.00 to prepare. After all expenses were paid, the school raised $2,400 at the fundraiser. Which equation can be used to find x, the number of plates that were sold?
Set up the cost equation C(x) where x is the number of plates sold:
C(x) = Cost per plate * x plates
C(x) = 2x
Set up the revenue equation R(x) where x is the number of plates sold:
R(x) = Sales price per plate * x plates
C(x) = 8x
Set up the profit equation P(x) where x is the number of plates sold:
P(x) = R(x) - C(x)
P(x) = 8x - 2x
P(x) = 6x
We're told the profits P(x) for the fundraiser were $2,400, so we set 6x equal to 2400 and solve for x:
6x = 2400
To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=6x%3D2400&pl=Solve']type it in our math engine[/URL] and we get:
x =[B]400 plates[/B]

A school theater group is selling candy to raise funds in order to put on their next performance. Th

A school theater group is selling candy to raise funds in order to put on their next performance. The candy cost the group $0.20 per piece. Plus, there was a $9 shipping and handling fee. The group is going to sell the candy for $0.50 per piece. How many pieces of candy must the group sell in order to break even?
[U]Set up the cost function C(p) where p is the number of pieces of candy.[/U]
C(p) = Cost per piece * p + shipping and handling fee
C(p) = 0.2p + 9
[U]Set up the Revenue function R(p) where p is the number of pieces of candy.[/U]
R(p) = Sale price * p
R(p) = 0.5p
Break-even means zero profit or loss, so we set the Cost Function equal to the Revenue Function
0.2p + 9 = 0.5p
To solve this equation for p, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.2p%2B9%3D0.5p&pl=Solve']type it in our math engine[/URL] and we get:
p = [B]30[/B]

A secret number is added to 6. The total is multiplied by 5 to get 50. What is the secret number?

A secret number is added to 6. The total is multiplied by 5 to get 50. What is the secret number?
Take this algebraic expression in pieces:
[LIST]
[*]Let the secret number be n.
[*]Added to means we add 6 to n: n + 6
[*]The total is multiplied by 5: 5(n + 6)
[*]The phrase [I]to get[/I] means equal to, so we set 5(n + 6) equal to 50
[/LIST]
5(n + 6) = 50
To solve this equation for n, we type it in our search engine and we get:
n = [B]4[/B]

A section of land measuring 3 & 3/6 acres is divided equally among 7 people. How many acres will eac

A section of land measuring 3 & 3/6 acres is divided equally among 7 people. How many acres will each person get?
We want 3&3/6 /7
[URL='https://www.mathcelebrity.com/fraction.php?frac1=3%263%2F6&frac2=7&pl=Divide']Using our fraction calculator[/URL], we get:
[B]1/2 acre per person[/B]

A shipping service charges $0.43 for the first ounce and $0.29 for each additional ounce of package

A shipping service charges $0.43 for the first ounce and $0.29 for each additional ounce of package weight. Write an equation to represent the price P of shipping a package that weighs x ounces, for any whole number of ounces greater than or equal to 1.
Set up the price function P(x)
[B]P(x) = 0.43 + 0.29(x - 1)[/B]

A shirt costs 15 and jeans costs 25. write an expressions for the costs of j jeans and s shirts

A shirt costs 15 and jeans costs 25. write an expressions for the costs of j jeans and s shirts.
Cost equals quantity times price, so we have the total cost C:
[B]C(s, j) = 15s + 25j[/B]

A spinner has 10 equally sized sections, 2 are gray 8 are blue. The spinner is spun twice. What is t

A spinner has 10 equally sized sections, 2 are gray 8 are blue. The spinner is spun twice. What is the probability that the first spin lands on blue and the second lands on gray?
P(blue) = Blue sections / Total Sections
P(blue) = 8/10
[URL='https://www.mathcelebrity.com/fraction.php?frac1=8%2F10&frac2=3%2F8&pl=Simplify']Reducing this using our simplified fraction calculator[/URL], we get:
P(blue) = 4/5
P(gray) = Gray sections / Total Sections
P(blue) = 2/10
[URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F10&frac2=3%2F8&pl=Simplify']Reducing this using our simplified fraction calculator[/URL], we get:
P(gray) = 1/5
We want the probability of blue,gray. Since each spin is independent, we multiply the two probabilities to get our answer:
P(blue, gray) = P(blue) * P(gray)
P(blue, gray) = 4/5 * 1/5
P(blue, gray) = [B]4/25[/B]

A spinner has 3 equal sections labelled A, B, C. A bag contains 3 marbles: 1 grey, 1 black, and 1 w

A spinner has 3 equal sections labelled A, B, C. A bag contains 3 marbles: 1 grey, 1 black, and 1 white. The pointer is spun and a marble is picked at random.
a) Use a tree diagram to list the possible outcomes.
[LIST=1]
[*][B]A, Grey[/B]
[*][B]A, Black[/B]
[*][B]A, White[/B]
[*][B]B, Grey[/B]
[*][B]B, Black[/B]
[*][B]B, White[/B]
[*][B]C, Grey[/B]
[*][B]C, Black[/B]
[*][B]C, White[/B]
[/LIST]
b) What is the probability of:
i) spinning A?
P(A) = Number of A sections on spinner / Total Sections
P(A) = [B]1/3[/B]
---------------------------------
ii) picking a grey marble?
P(A) = Number of grey marbles / Total Marbles
P(A) = [B]1/3[/B]
---------------------------------
iii) spinning A and picking a white marble?
Since they're independent events, we multiply to get:
P(A AND White) = P(A) * P(White)
P(A) was found in i) as 1/3
Find P(White):
P(White) = Number of white marbles / Total Marbles
P(White) = 1/3
[B][/B]
Therefore, we have:
P(A AND White) = 1/3 * 1/3
P(A AND White) = [B]1/9[/B]
---------------------------------
iv) spinning C and picking a pink marble?
Since they're independent events, we multiply to get:
P(C AND Pink) = P(C) * P(Pink)
Find P(C):
P(C) = Number of C sections on spinner / Total Sections
P(C) = 1/3
[B][/B]
Find P(Pink):
P(Pink) = Number of pink marbles / Total Marbles
P(Pink) = 0/3
[B][/B]
Therefore, we have:
P(C AND Pink) = 1/3 * 0
P(C AND Pink) = [B]0[/B]

A spinner has 6 equal sections, of which 2 are green. If you spin the spinner once, what is the prob

A spinner has 6 equal sections, of which 2 are green. If you spin the spinner once, what is the probability that it will land on a green section? Write your answer as a fraction or whole number.
P(green) = Total Green / Total spaces
P(green) = 2/6
We can simplify this fraction. So we [URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F6&frac2=3%2F8&pl=Simplify']type 2/6 into our search engine[/URL], choose Simplify, and we get:
P(green) = [B]1/3[/B]

A spinner is divided into 4 equal sections numbered 1 to 4. The theoretical probability of the spinn

A spinner is divided into 4 equal sections numbered 1 to 4. The theoretical probability of the spinner stopping on 3 is 25%. Which of the following is most likely the number of 3s spun in 10,000 spins?
We want Expected Value of s spins. Set up the expected value formula for any number 1-4
E(s) = 0.25 * n where n is the number of spins.
Using s = 3, n = 10,000, we have:
E(10,000) = 0.25 * 10,000
E(10,000) = [B]2,500[/B]

A store had 600 pounds of feed. After delivering equal amounts to 4 farmers, there are 60 pounds lef

A store had 600 pounds of feed. After delivering equal amounts to 4 farmers, there are 60 pounds left. How many pounds did each farmer receive?
If there were 60 pounds left, then the store had 600 - 60 = 540 pounds delivered.
540 pounds delivered in equal amounts to 4 farmers means each farmer got:
540/4 = [B]135 pounds of feed[/B]

A survey was conducted that asked 1007 people how many books they had read in the past year. Results

A survey was conducted that asked 1007 people how many books they had read in the past year. Results indicated that x overbarequals11.3 books and sequals16.6 books. Construct a 90% confidence interval for the mean number of books people read. Interpret the interval.
x bar = 11.3
s = 16.6
n = 1007
[URL='https://www.mathcelebrity.com/normconf.php?n=1007&xbar=11.3&stdev=16.6&conf=90&rdig=4&pl=Not+Sure']We use our confidence interval calculator[/URL] and get [B]10.4395 < u < 12.1605[/B].
[B][I]We interpret this as:
If we repeated experiments, the proportion of such intervals containing u would be 90%[/I][/B]

A sweater that you love costs $32. You really want the sweater but only have $35. If there’s a sales

A sweater that you love costs $32. You really want the sweater but only have $35. If there’s a sales tax of 4% on the item, do you have enough to buy the sweater?
Calculate after-tax amount:
After tax amount = Sale Price * (1 + sales tax percent)
After tax amount = 32 * (1 + 0.04) <-- Since 4% = 0.04
After tax amount = 32 * (1.04)
After tax amount = $33.28
[B]Yes[/B], since $33.28 is less than or equal to $35, you have enough to buy the sweater.

A taxi cab in Chicago charges $3 per mile and $1 for every person. If the taxi cab ride for two peop

A taxi cab in Chicago charges $3 per mile and $1 for every person. If the taxi cab ride for two people costs $20. How far did the taxi cab travel.
Set up a cost function C(m) where m is the number of miles driven:
C(m) = cost per mile * m + per person fee
[U]Calculate per person fee:[/U]
per person fee = $1 per person * 2 people
per person fee = $2
[U]With a cost per mile of $3 and per person fee of $2, we have:[/U]
C(m) = cost per mile * m + per person fee
C(m) = 3m + 2
The problem asks for m when C(m) = 20, so we set 3m + 2 equal to 20:
3m + 2 = 20
To solve this equation for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=3m%2B2%3D20&pl=Solve']plug it in our search engine[/URL] and we get:
m = [B]6[/B]

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to s

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to spend less than $12 on a ride. Which inequality can be used to find the distance Samantha can travel?
Set up the travel cost equation where m is the number of miles:
C(m) = 0.8m + 1.50
If Samantha wants to spend less than 12 per ride, we have an inequality where C(m) < 12:
[B]0.8m + 1.50 < 12[/B]

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to s

A taxi charges a flat rate of $1.50 with an additional charge of $0.80 per mile. Samantha wants to spend less than $12 on a ride. Which inequality can be used to find the distance Samantha can travel?
[LIST]
[*]Each ride will cost 1.50 + 0.8x where x is the number of miles per trip.
[*]This expression must be less than 12.
[/LIST]
[U]Setup the inequality:[/U]
1.5 + 0.8x < 12
[U]Subtracting 1.5 from each side of the inequality[/U]
0.8x < 10.5
[U]Simplifying even more by dividing each side of the inequality by 0.8, we have:[/U]
[B]x < 13.125[/B]

a total of one and x is less than 5

a total of one and x is less than 5
A total of one and x means we add x to 1:
1 + x
We set up an inequality, where 1 + x is less than (<) 5
[B]1 + x < 5[/B]

A toy company makes "Teddy Bears". The company spends $1500 for factory expenses plus $8 per bear. T

A toy company makes "Teddy Bears". The company spends $1500 for factory expenses plus $8 per bear. The company sells each bear for $12.00 each. How many bears must this company sell in order to break even?
[U]Set up the cost function C(b) where b is the number of bears:[/U]
C(b) = Cost per bear * b + factory expenses
C(b) = 8b + 1500
[U]Set up the revenue function R(b) where b is the number of bears:[/U]
R(b) = Sale Price per bear * b
R(b) = 12b
[U]Break-even is where cost equals revenue, so we set C(b) equal to R(b) and solve for b:[/U]
C(b) = R(b)
8b + 1500 = 12b
To solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=8b%2B1500%3D12b&pl=Solve']type this equation into our search engine[/URL] and we get:
b = [B]375[/B]

A train leaves San Diego at 1:00 PM. A second train leaves the same city in the same direction at 3

A train leaves San Diego at 1:00 PM. A second train leaves the same city in the same direction at 3:00 PM. The second train travels 30mph faster than the first. If the second train overtakes the first at 6:00 PM, what is the speed of each of the two trains?
Distance = Rate x Time
Train 1:
d = rt
t = 1:oo PM to 6:00 PM = 5 hours
So we have d = 5r
Train 2:
d = (r + 30)t
t = 3:oo PM to 6:00 PM = 3 hours
So we have d = 3(r + 30)
Set both distances equal to each other since overtake means Train 2 caught up with Train 1, meaning they both traveled the same distance:
5r = 3(r + 30)
Multiply through:
3r + 90 = 5r
[URL='https://www.mathcelebrity.com/1unk.php?num=3r%2B90%3D5r&pl=Solve']Run this equation through our search engine[/URL], and we get [B]r = 45[/B]. This is Train 1's Speed.
Train 2's speed = 3(r + 30).
Plugging r = 45 into this, we get 3(45 + 30).
3(75)
[B]225[/B]

A vendor sells h hot dogs and s sodas. If a hot dog costs twice as much as a soda, and if the vendor

A vendor sells h hot dogs and s sodas. If a hot dog costs twice as much as a soda, and if the vendor takes in a total of d dollars, how many cents does a soda cost?
Let the cost of the soda be p. So the cost of a hot dog is 2p.
The total cost of hot dogs:
2hp
The total cost of sodas:
ps
The total cost of both equals d. So we set the total cost of hots dogs plus sodas equal to d:
2hp + ps = d
We want to know the cost of a soda (p). So we have a literal equation. We factor out p from the left side:
p(2h + s) = d
Divide each side of the equation by (2h + s)
p(2h + s)/(2h + s) = d/(2h + s)
Cancel the (2h + s) on the left side, we get:
p = [B]d/(2h + s[/B])

A video store charges a monthly membership fee of $7.50, but the charge to rent each movie is only $

A video store charges a monthly membership fee of $7.50, but the charge to rent each movie is only $1.00 per movie. Another store has no membership fee, but it costs $2.50 to rent each movie. How many movies need to be rented each month for the total fees to be the same from either company?
Set up a cost function C(m) where m is the number of movies you rent:
C(m) = Rental cost per movie * m + Membership Fee
[U]Video Store 1 cost function[/U]
C(m) = 1m + 7.5
Video Store 2 cost function:
C(m) = 2.50m
We want to know when the costs are the same. So we set each C(m) equal to each other:
m + 7.5 = 2.50m
To solve this equation for m, [URL='https://www.mathcelebrity.com/1unk.php?num=m%2B7.5%3D2.50m&pl=Solve']we type it in our search engine[/URL] and we get:
m = [B]5[/B]

A water tank holds 236 gallons but is leaking at a rate of 3 gallons per week. A second water tank h

A water tank holds 236 gallons but is leaking at a rate of 3 gallons per week. A second water tank holds 354 gallons but is leaking at a rate if 5 gallons per week. After how many weeks will the amount of water in the two tanks be the same
Let w be the number of weeks of leaking. We're given two Leak equations L(w):
[LIST=1]
[*]L(w) = 236 - 3w
[*]L(w) = 354 - 5w
[/LIST]
When the water in both tanks is the same, we can set both L(w) equations equal to each other:
236 - 3w = 354 - 5w
To solve this equation for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=236-3w%3D354-5w&pl=Solve']type it in our search engine[/URL] and we get:
w = [B]59[/B]

Aaron is staying at a hotel that charges $99.95 per night plus tax for a room. A tax of 8% is applie

Aaron is staying at a hotel that charges $99.95 per night plus tax for a room. A tax of 8% is applied to the room rate, and an additional onetime untaxed fee of $5.00 is charged by the hotel. Which of the following represents Aaron’s total charge, in dollars, for staying [I]x[/I] nights?
[LIST]
[*]The Room cost equals 99.95 times x where x is the number of rooms
[*]We express an 8% tax by multiplying the room cost by 1.08
[*]Finally, we add on $5, which is [I]untaxed[/I]
[/LIST]
[I][/I]
Take this in pieces:
Room Cost: 99.95x
Tax on Room 1.08(99.95x)
Add on $5 which is untaxed: [B]1.08(99.95x) + 5[/B]

absolute value of x is less than or equal to 4

absolute value of x is less than or equal to 4
Absolute value of x:
|x|
Set up an inequality where this is less than or equal to 4:
[B]|x| <= 4 [/B] <-- This is our algebraic expression
To solve this, we have the following compound inequality:
-4 < x < 4

Absolute value of x less than 8

These are now available as shortcuts. You can type any number or variable in the following forms:
[LIST]
[*]Absolute value of x less than 8
[*]Absolute value of x less than or equal to 8
[*]Absolute value of x greater than 8
[*]Absolute value of x greater than or equal to 8
[*]Absolute value of x equal to 8
[/LIST]

Adam has 20 sweets he eats a quarter of them how many does he have left?

Adam has 20 sweets he eats a quarter of them how many does he have left?
A quarter means 1/4. There's 2 ways you can approach this problem.
[B][U]Approach #1:[/U][/B]
Adam eats a quarter, or 1/4 of the sweets. So he eats:
20 * 1/4 = 5
Remaining sweets = Total Apples - Eaten Apples
Remaining sweets = 0 - 5
Remaining sweets= 15
[U][B]Approach #2:[/B][/U]
If Adam eats 1/4 of the sweets, this means he has:
1 - 1/4 sweets remaining.
Since 1 equals 4/4, we have:
4/4 - 1/4 = 3/4
Therefore, he has 20 * 3/4 sweets remaining.
This is 60/4, or [B]15[/B]

Addition Equality Property

Free Addition Equality Property Calculator - Demonstrates the Addition Equality Property
Numerical Properties

Addition Property Of Inequality

Free Addition Property Of Inequality Calculator - Demonstrates the Addition Property Of Inequality.
Numerical Properties

Admir works at a coffee shop and earns $9/hour he also works at a grocery store and earns $15/hour.

Admir works at a coffee shop and earns $9/hour he also works at a grocery store and earns $15/hour. Last week he earned $500 dollars. Write an equation that represents the situation.
[LIST]
[*]Let c be the hours Admir works at the coffee shop.
[*]Let g be the hours Admir works at the grocery store.
[/LIST]
Since earnings equal hourly rate times hours, We have the following equation:
[B]9c + 15g = 500[/B]

Al's Rentals charges $25 per hour to rent a sailboard and a wetsuit. Wendy's Rentals charges $20 per

Al's Rentals charges $25 per hour to rent a sailboard and a wetsuit. Wendy's Rentals charges $20 per hour plus $15 extra for a wetsuit. Find the number of hours for which the total charges for both companies would be the same.
Al's Rentals Cost Equation C(h) where h is the number of hours you rent a sailboard and wetsuit:
C(h) = 25h
Wendy's Rentals Cost Equation C(h) where h is the number of hours you rent a sailboard and wetsuit:
C(h) = 20h + 15
We want to set both cost equation equal to each other, and solve for h:
20h + 15 = 25h
[URL='https://www.mathcelebrity.com/1unk.php?num=20h%2B15%3D25h&pl=Solve']Typing this equation into our search engine[/URL], we get:
h = [B]3[/B]

algexpress: letthefirstnumberequalx.thesecondnumberis3morethantwicethefirstnumber.expressthesecondnu

Let the first number equal x. The second number is 3 more than twice the first number. Express the second number in terms of the first number x.
[LIST]
[*]Let the second number be y.
[*]Twice means multiply by 2
[*]3 more than means we add 3
[/LIST]
So we have the following algebraic expression:
[B]y = 2x + 3[/B]

Ali runs each lap in 6 minutes. He will run at least 11 laps today. What are the possible numbers of

Ali runs each lap in 6 minutes. He will run at least 11 laps today. What are the possible numbers of minutes he will run today?
Let m be the number of minutes. The phrase [I]at least[/I] means an inequality, also known as greater than or equal to. So we have:
m >= 11*6
[B]m >= 66
You can read this as Ali will run 66 or more minutes today. Or at least 66 minutes. Or greater than or equal to 66 minutes[/B]

Alicia deposited $41 into her checking account. She wrote checks for $31 and $13. Now her account ha

Alicia deposited $41 into her checking account. She wrote checks for $31 and $13. Now her account has a balance of $81. How much did she have in her account to start with?
We start with a balance of b.
Depositing 41 means we add to the account balance:
b + 41
Writing checks for 31 and 13 means we subtract from the account balance:
b + 41 - 31 - 13
The final balance is 81, so we set b + 41 - 31 - 13 equal to 81:
b + 41 - 31 - 13 = 81
To solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=b%2B41-31-13%3D81&pl=Solve']type this equation into our math engine[/URL] and we get:
b = [B]84[/B]

All real numbers that are less than equal to -1 or greater than 5

We have two expressions here, so we need a union since we have the word [U]or[/U].
First, All real numbers less than or equal to -1 is x <= -1.
All real numbers greater than 5 is x > 5
So we have x <= -1 U x > 5
[MEDIA=youtube]boOueZTCSuU[/MEDIA]

all real numbers y greater than or equal to 12

all real numbers y greater than or equal to 12
Greater than or equal to means we use the sign >=
[B]y >= 12[/B]

Amanda runs each lap in 4 minutes. She will run less than 44 minutes today. What are the possible nu

Amanda runs each lap in 4 minutes. She will run less than 44 minutes today. What are the possible number of laps she will run today?
Notes for this problem:
[LIST]
[*]Let laps be l.
[*]Lap time = Time per lap * number of laps (l)
[*]Less than means we have an inequality using the < sign
[/LIST]
We have the inequality:
4l < 44
To solve this inequality for l, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=4l%3C44&pl=Show+Interval+Notation']type it in our math engine[/URL] and we get:
[B]l < 11[/B]

Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for ea

Amara currently sells televisions for company A at a salary of $17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of $29,000 plus a $20 commission for each television she sells. How many televisions would Amara need to sell for the options to be equal?
Let the number of tv's be t. Set up the salary function S(t):
S(t) = Commision * tv's sold + Salary
Company A:
S(t) = 100t + 17,000
Company B:
S(t) = 20t + 29,000
The problem asks for how many tv's it takes to make both company salaries equal. So we set the S(t) functions equal to each other:
100t + 17000 = 20t + 29000
[URL='https://www.mathcelebrity.com/1unk.php?num=100t%2B17000%3D20t%2B29000&pl=Solve']Type this equation into our search engine[/URL] and we get:
t = [B]150[/B]

Amy and ryan operate a car dealing and repair service. For a car detailing (full wash outside and in

Amy and ryan operate a car dealing and repair service. For a car detailing (full wash outside and inside. Amy charges 40$ and Ryan charges 50$ . In addition they charge a hourly rate. Amy charges $35/h and ryan charges $30/h. How many hours does amy and ryan have to work to make the same amount of money?
Set up the cost functions C(h) where h is the number of hours.
[U]Amy:[/U]
C(h) = 35h + 40
[U]Ryan:[/U]
C(h) = 30h + 50
To make the same amount of money, we set both C(h) functions equal to each other:
35h + 40 = 30h + 50
To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=35h%2B40%3D30h%2B50&pl=Solve']type this equation into our search engine[/URL] and we get:
h = [B]2[/B]

An airplane is flying at 38,800 feet above sea level. The airplane starts to descend at a rate of 18

An airplane is flying at 38,800 feet above sea level. The airplane starts to descend at a rate of 1800 feet per minute. Let m be the number of minutes. Which of the following expressions describe the height of the airplane after any given number of minutes?
Let m be the number of minutes. Since a descent equals a [U]drop[/U] in altitude, we subtract this in our Altitude function A(m):
[B]A(m) = 38,800 - 1800m[/B]

An Amazon delivery package receives a bonus if he delivers a package to a costumer in 20 minutes plu

An Amazon delivery package receives a bonus if he delivers a package to a costumer in 20 minutes plus or minus 5 minutes. Which inequality or equation represents the drivers allotted time (x) to receive a bonus
20 plus 5 minutes = 25 minutes
20 minus 5 minutes = 15 minutes
So we have the inequality:
[B]15 <= x <= 25[/B]

An avocado is not ripe until 4 days after picking and will go bad after 7 days after picking. Repres

An avocado is not ripe until 4 days after picking and will go bad after 7 days after picking. Represent the days the avocado is ripe
Our sweet spot for ripeness is 4 days or more but not more than 7 days. Using d as our days, we have the following inequality:
[B]4 <= d <= 7[/B]

An elevator can hold less than 2700 pounds of extra weight. If an average person weighs 150 pounds,

An elevator can hold less than 2700 pounds of extra weight. If an average person weighs 150 pounds, what is the maximum number of people (p) that can be on the elevator at one time?
Total weight = average weight per person * Number of people
Total weight = 150p
We know from the problem that:
150p < 2700
We want to solve this inequality for p. Divide each side of the inequality by 150:
150p/150 < 2700/150
Cancel the 150's on the left side and we get:
p < [B]18[/B]

An elevator has a maximum weight of 3000 pounds. How many 150-pound people can safely ride the eleva

An elevator has a maximum weight of 3000 pounds. How many 150-pound people can safely ride the elevator? (Use "p" to represent the number of people)
Maximum means less than or equal to. We have the inequality:
150p <= 3000
To solve this inequality for p, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=150p%3C%3D3000&pl=Show+Interval+Notation']type it in our search engine[/URL] and we get:
[B]p <= 20[/B]

An equilateral triangle has three sides of equal length. What is the equation for the perimeter of a

An equilateral triangle has three sides of equal length. What is the equation for the perimeter of an equilateral triangle if P = perimeter and S = length of a side?
P = s + s + s
[B]P = 3s[/B]

Ana's height is strictly between 63 and 66 inches. Write a symbolic inequality to represent this sce

Ana's height is strictly between 63 and 66 inches. Write a symbolic inequality to represent this scenario. let h be height
[B]63 < h < 66
[/B]
You can also type [I][URL='https://www.mathcelebrity.com/algexpress.php?num=between63and66&pl=Write+Expression']between 63 and 66[/URL][/I] in our search engine.

Angad was thinking of a number. Angad adds 20 to it, then doubles it and gets an answer of 53. What

Angad was thinking of a number. Angad adds 20 to it, then doubles it and gets an answer of 53. What was the original number?
The phrase [I]a number[/I] means an arbitrary variable, let's call it n.
[LIST]
[*]Start with n
[*]Add 20 to it: n + 20
[*]Double it means we multiply the expression by 2: 2(n + 20)
[*]Get an answer of 53: means an equation, so we set 2(n + 20) equal to 53
[/LIST]
2(n + 20) = 53
To solve for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=2%28n%2B20%29%3D53&pl=Solve']type this equation into our search engine[/URL] and we get:
n = [B]6.5[/B]

Angelica’s financial aid stipulates that her tuition cannot exceed $1000. If her local community col

Angelica’s financial aid stipulates that her tuition cannot exceed $1000. If her local community college charges a $35 registration fee plus $375 per course, what is the greatest number of courses for which Angelica can register?
We set up the Tuition function T(c), where c is the number of courses:
T(c) = Cost per course * c + Registration Fee
T(c) = 35c + 375
The problem asks for the number of courses (c) where her tuition [I]cannot exceed[/I] $1000. The phrase [I]cannot exceed[/I] means less than or equal to, or no more than. So we setup the inequality for T(c) <= 1000 below:
35c + 375 <= 1000
To solve this inequality for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=35c%2B375%3C%3D1000&pl=Solve']type it in our search engine and we get[/URL]:
c <= 17.85
Since we cannot have fractional courses, we round down and get:
c[B] <= 17[/B]

As a salesperson you will earn $600 per month plus a commission of 20% of sales. Find the minimum am

As a salesperson you will earn $600 per month plus a commission of 20% of sales. Find the minimum amount of sales you need to make in order to receive a total income of at least $1500 per month.
Let the amount of sales be s. The phrase [I]at least[/I] means greater than or equal to. Since 20% is 0.2, We want to know when:
0.20s + 600 >= 1500
We [URL='https://www.mathcelebrity.com/1unk.php?num=0.20s%2B600%3E%3D1500&pl=Solve']type this inequality into our search engine to solve for s[/URL] and we get:
s >= [B]4500[/B]

As a salesperson, you are paid $50 per week plus $2 per sale. This week you want your pay to be at l

As a salesperson, you are paid $50 per week plus $2 per sale. This week you want your pay to be at least $100. What is the minimum number of sales you must make to earn at least $100?
Set up the inequality where s is the amount of sales you make:
50 + 2s >= 100
We use >= because the phrase [I]at least[/I] 100 means 100 or more
Subtract 50 from each side:
2s >= 50
Divide each side by 2
[B]s >= 25[/B]

ason decided that he will sell his stocks if their values per share (x) goes below $5 or above $15.

Jason decided that he will sell his stocks if their values per share (x) goes below $5 or above $15. Write a compound inequality represents the values at which Jason will sell his stocks?
Below $5 is also known as less than $5:
x < 5
Above $15 is also known as greater than $15
x > 15
We write the compound inequality:
[B]x < 5 U x > 15[/B]

Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a

Assume the speed of vehicles along a stretch of I-10 has an approximately normal distribution with a mean of 71 mph and a standard deviation of 8 mph.
a. The current speed limit is 65 mph. What is the proportion of vehicles less than or equal to the speed limit?
b. What proportion of the vehicles would be going less than 50 mph?
c. A new speed limit will be initiated such that approximately 10% of vehicles will be over the speed limit. What is the new speed limit based on this criterion?
d. In what way do you think the actual distribution of speeds differs from a normal distribution?
a. Using our [URL='http://www.mathcelebrity.com/probnormdist.php?xone=65&mean=71&stdev=8&n=+1&pl=P%28X+%3C+Z%29']z-score calculator[/URL], we see that P(x<65) = [B]22.66%[/B]
b. Using our [URL='http://www.mathcelebrity.com/probnormdist.php?xone=+50&mean=71&stdev=8&n=+1&pl=P%28X+%3C+Z%29']z-score calculator[/URL], we see that P(x<50) = [B]0.4269%[/B]
c. [URL='http://www.mathcelebrity.com/zcritical.php?a=0.9&pl=Calculate+Critical+Z+Value']Inverse of normal for 90% percentile[/URL] = 1.281551566
Plug into z-score formula: (x - 71)/8 = 1.281551566
[B]x = 81.25241252[/B]
d. [B]The shape/ trail differ because the normal distribution is symmetric with relatively more values at the center. Where the actual has a flatter trail and could be expected to occur.[/B]

At a concert, 20% of the audience members were teenagers. If the number of teenagers at the concert

At a concert, 20% of the audience members were teenagers. If the number of teenagers at the concert was 360, what was the total number of audience members?
We're looking for total audience members where [I]20% of what equals 360[/I]?
[URL='https://www.mathcelebrity.com/perc.php?num=+5&den=+8&num1=360&pct1=20&pcheck=2&pct2=+70&den1=+80&pct=+82&decimal=+65.236&astart=+12&aend=+20&wp1=20&wp2=30&pl=Calculate']Type this expression into our search engine[/URL] and we get:
Audience = [B]1,800[/B]

At a local fitness center, members pay a $10 membership fee and $3 for each aerobics class. Nonme

At a local fitness center, members pay a $10 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost for members and nonmembers be the same?
Set up the cost functions where x is the number of aerobics classes:
[LIST]
[*]Members: C(x) = 10 + 3x
[*]Non-members: C(x) = 5x
[/LIST]
Set them equal to each other
10 + 3x = 5x
Subtract 3x from both sides:
2x = 10
Divide each side by 2
[B]x = 5 classes[/B]

At a local fitness center, members pay an $8 membership fee and $3 for each aerobics class. Nonmembe

At a local fitness center, members pay an $8 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost for members be equal to nonmembers?
Set up two cost equations C(x):
[LIST=1]
[*]Members: C(x) = 8 + 3x
[*]Nonmembers: C(x) = 5x
[/LIST]
Set the two cost equations equal to each other:
8 + 3x = 5x
Subtract 3x from each side
2x = 8
Divide each side by 2
[B]x = 4[/B]

At the end of the week, Francesca had a third of her babysitting money left after spending $14.65 on

At the end of the week, Francesca had a third of her babysitting money left after spending $14.65 on a movie and popcorn and another $1.35 on a pen. How much did she earn babysitting?
Let the original amount of money earned for babysitting be b. We're given:
[LIST=1]
[*]Start with b
[*]Spending 14.65 for a movie means we subtract 14.65 from b: b - 14.65
[*]Spending 1.35 on a pen means we subtract another 1.35 from step 2: b - 14.65 - 1.35
[*]Francesca has a third of her money left. So we set step 3 equal to 1/3 of b
[/LIST]
b - 14.65 - 1.35 = b/3
Multiply each side of the equation by 3 to remove the fraction
3(b - 14.65 - 1.35) = 3b/3
3b - 43.95 - 4.05 = b
To solve this equation for b, [URL='https://www.mathcelebrity.com/1unk.php?num=3b-43.95-4.05%3Db&pl=Solve']we type it in our search engine[/URL] and we get:
b =[B] 24[/B]

At the movie theater, Celeste bought 2 large drinks and 2 large popcorns for $8.50. She paid with a

At the movie theater, Celeste bought 2 large drinks and 2 large popcorns for $8.50. She paid with a twenty-dollar bill. What is the fewest number of bills and coins that she could have received as change?r of bills and coins that she could have received as change?
Calculate change:
Change = Amount Paid - Bill
Change = $20.00 - $8.50
Change = $11.50
Largest bill we can start with is a 10 dollar bill:
$11.50 - 10 = $1.50
Next largest bill is a $1 bill
$1.50 - $1 = 0.50
Now we're down to coins. Largest coin(s) we can use are quarters (assuming no half-dollars)
2 quarters equals 0.50
0.50 - 0.50 = 0
[U]Therefore, our answer is:[/U]
[B]Ten dollar Bill, 1 dollar bill, and 2 quarters[/B]

At Zabowood’s Gadget Store, some items are paid on instalment basis through credit cards. Clariza wa

[B]A[/B]t Zabowood’s Gadget Store, some items are paid on instalment basis through credit cards. Clariza was able to sell 10 cellphones costing Php 18,000.00 each. Each transaction is payable in 6 months equally divided into 6 equal instalments without interest. Clariza gets 2% commission on the first month for each of the 10 cellphones. Commission decreases by 0.30% every month thereafter and computed on the outstanding balance for the month. How much commission does Clariza receive on the third month?
Calculate Total Sales Amount:
Calculate Total Sales Amount = 10 cellphones * 18000 per cellphone
Calculate Total Sales Amount = 180000
Calculate monthly sales amount installment:
monthly sales amount installment = Total Sales Amount / 6
monthly sales amount installment = 180000/6
monthly sales amount installment = 30000 per month
Calculate Third Month Commission:
Third month commission = First Month Commission - 0.30% - 0.30%
Third month Commission = 2% - 0.30% - 0.30% = 1.4%
Calculate 3rd month commission amount:
3rd month Commission amount = 1.4% * 30000
3rd month Commission amount = [B]420[/B]

Austin needs $240 to buy a new bike if he can save $16 per week and how many weeks can you purchase

Austin needs $240 to buy a new bike if he can save $16 per week and how many weeks can you purchase the bike?
Set up the equation, where w equals the number of weeks needed. We have:
16w = 240
[URL='https://www.mathcelebrity.com/1unk.php?num=16w%3D240&pl=Solve']Typing this into our search engine[/URL], we get [B]w = 15[/B].

average of 16 and x is three. find x

average of 16 and x is three. find x
Average of 16 and x is written as:
(16 + x)/2
We set this equal to 3:
(16 + x)/2 = 3
Cross multiply;
x + 16 = 6
[URL='https://www.mathcelebrity.com/1unk.php?num=x%2B16%3D6&pl=Solve']Type this equation into our search engine[/URL] and we get:
x = [B]-10[/B]

B is the midpoint of AC and BC=5

B is the midpoint of AC and BC=5
Since the midpoint divides a segment into two equal segments, we know that:
AB = BC
So AB =[B] 5[/B]
And AC = 5 + 5 = [B]10[/B]

B out of 6 is 12

B out of 6 is 12
b out of 6:
b/6
The phrase [I]is[/I] means an equation, so we set b/6 equal to 12:
[B]b/6 = 12[/B]

Barbara bought a piece of rope that was 7 1/3 meters long. She cut the rope into 3 equal pieces. How

Barbara bought a piece of rope that was 7 1/3 meters long. She cut the rope into 3 equal pieces. How long is each piece of rope?
Using our mixed number converter, we see that:
[URL='https://www.mathcelebrity.com/fraction.php?frac1=7%261%2F3&frac2=3%2F8&pl=Simplify']7&1/3[/URL] = 22/3
Split into [URL='https://www.mathcelebrity.com/fraction.php?frac1=22%2F9&frac2=3&pl=Simplify']3 equal pieces[/URL], we have:
22/3 / 3 = 22/9 or 2&4/9

Barbra is buying plants for her garden. She notes that potato plants cost $3 each and corn plants co

Barbra is buying plants for her garden. She notes that potato plants cost $3 each and corn plants cost $4 each. If she plans to spend at least $20 and purchase less than 15 plants in total, create a system of equations or inequalities that model the situation. Define the variables you use.
[U]Define variables[/U]
[LIST]
[*]Let c be the number of corn plants
[*]Let p be the number of potato plants
[/LIST]
Since cost = price * quantity, we're given two inequalities:
[LIST=1]
[*][B]3p + 4c >= 20 (the phrase [I]at least[/I] means greater than or equal to)[/B]
[*][B]c + p < 15[/B]
[/LIST]

Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will

Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will it take for them to have the same amount of money?
Let w be the number of weeks that go by for saving/spending.
Set up Barney's balance equation, B(w). Spending means we [U]subtract[/U]
B(w) = Initial Amount - spend per week * w weeks
B(w) = 450 - 3w
Set up Betty's balance equation, B(w). Saving means we [U]add[/U]
B(w) = Initial Amount + savings per week * w weeks
B(w) = 120 + 8w
The same amount of money means both of their balance equations B(w) are equal. So we set Barney's balance equal to Betty's balance and solve for w:
450 - 3w = 120 + 8w
Add 3w to each side to isolate w:
450 - 3w + 3w = 120 + 8w + 3w
Cancelling the 3w on the left side, we get:
450 = 120 + 11w
Rewrite to have constant on the right side:
11w + 120 = 450
Subtract 120 from each side:
11w + 120 - 120 = 450 - 120
Cancelling the 120's on the left side, we get:
11w = 330
To solve for w, we divide each side by 11
11w/11 = 330/11
Cancelling the 11's on the left side, we get:
w = [B]30
[MEDIA=youtube]ifG_q-utgJI[/MEDIA][/B]

Bawi solves a problem that has an answer of x = -4. He first added 7 to both sides of the equal sign

Bawi solves a problem that has an answer of x = -4. He first added 7 to both sides of the equal sign, then divided by 3. What was the original equation
[LIST=1]
[*]If we added 7 to both sides, that means we had a minus 7 (-7) to start with as a constant. Since subtraction undoes addition.
[*]If we divided by 3, this means we multiplied x by 3 to begin with. Since division undoes multiplication
[/LIST]
So we have the start equation:
3x - 7
If the answer was x = -4, then we plug this in to get our number on the right side of the equation:
3(-4) - 7
-12 - 7
-19
This means our original equation was:
[B]3x - 7 = -19[/B]
And if we want to solve this to prove our answer, we [URL='https://www.mathcelebrity.com/1unk.php?num=3x-7%3D-19&pl=Solve']type the equation into our search engine [/URL]and we get:
x = -4

Before Barry Bonds, Mark McGwire, and Sammy Sosa, Roger Maris held the record for the most home runs

Before Barry Bonds, Mark McGwire, and Sammy Sosa, Roger Maris held the record for the most home runs in one season. Just behind Maris was Babe Ruth. The numbers of home runs hit by these two athletes in their record-breaking seasons form consecutive integers. Combined, the two athletes hit 121 home runs. Determine the number of home runs hit by Maris and Ruth in their record-breaking seasons.
We want [URL='https://www.mathcelebrity.com/consecintwp.php?num=121&pl=Sum']the sum of 2 consecutive integers equals 121[/URL].
[B]We get Maris at 61 and Ruth at 60[/B]

Ben is 4 times as old as Ishaan and is also 6 years older than Ishaan.

Ben is 4 times as old as Ishaan and is also 6 years older than Ishaan.
Let b be Ben's age and i be Ishaan's age. We're given:
[LIST=1]
[*]b = 4i
[*]b = i + 6
[/LIST]
Set (1) and (2) equal to each other:
4i = i + 6
[URL='https://www.mathcelebrity.com/1unk.php?num=4i%3Di%2B6&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]i = 2[/B]
Substitute this into equation (1):
b = 4(2)
[B]b = 8
[/B]
[I]Therefore, Ishaan is 2 years old and Ben is 8 years old.[/I]

Benny bought 8 new baseball trading cards to add to his collection. The next day his dog ate half of

Benny bought 8 new baseball trading cards to add to his collection. The next day his dog ate half of his collection. There are now only 47 cards left. How many cards did Benny start with?
Let b be the number of baseball trading cards Benny started with. We have the following events:
[LIST=1]
[*]Benny buys 8 new cards, so we add 8 to get b + 8
[*]The dog ate half of his cards the next day, so Benny has (b + 8)/2
[*]We're told he has 47 cards left, so we set (b + 8)/2 equal to 47
[/LIST]
(b + 8)/2 = 47
[B][U]Cross multiply:[/U][/B]
b + 8 = 47 * 2
b + 8 = 94
[URL='https://www.mathcelebrity.com/1unk.php?num=b%2B8%3D94&pl=Solve']Type this equation into the search engine[/URL], we get [B]b = 86[/B].

Beth is 5 years younger than celeste. Next year, their ages will have a sum equal to 57. How old is

Beth is 5 years younger than celeste. Next year, their ages will have a sum equal to 57. How old is each now?
Let b = Beth's age
Let c = Celeste's age
We are given:
[LIST=1]
[*]b = c - 5
[*]b + 1 + c + 1 = 57
[/LIST]
Substitute (1) into (2)
(c - 5) + 1 + c + 1 = 57
Group like terms:
2c - 3 = 57
[URL='https://www.mathcelebrity.com/1unk.php?num=2c-3%3D57&pl=Solve']Type 2c - 3 = 57 into our search engine[/URL], we get [B]c = 30[/B]
Substitute c = 30 into Equation (1), we get:
b = 30 - 5
[B]b = 25
[/B]
Therefore, Beth is 25 and Celeste is 30.

Beverly has $50 to spend at an amusement park. She plans to spend $10 for food, and $15 for admissio

Beverly has $50 to spend at an amusement park. She plans to spend $10 for food, and $15 for admission to the park. Each ride costs $1.50 to ride. Write an inequality to represent the possible number of rides she can ride?
First, we subtract the food and admission cost from Beverly's starting balance of $50:
Cost available for rides = Starting Balance - Food - Admission
Cost available for rides = 50 - 10 - 15
Cost available for rides = 25
Now we set up an inequality for the number of rides (r) that Beverly can ride with the remaining balance:
1.50r <= 25
To solve for r, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=1.50r%3C%3D25&pl=Show+Interval+Notation']type this inequality into our search engine[/URL] and we get:
[B]r <=[/B] [B]16.67[/B]

Bob has 4/5 of a pound of fudge. He wants to share it with Sue. How much of a pound of fudge will bo

Bob has 4/5 of a pound of fudge. He wants to share it with Sue. How much of a pound of fudge will both of them get?
If Bob shares the fudge with Sue, we assume they split equal parts. This means:
We take 4/5 total and divide into 2 for 2 people:
4/5/2
This is the same as 4/5 * 1/2
4/10
This fraction is not simplified.
Factor of 4 = {1, [U]2[/U], 4}
Factors of 10 = {1, [U]2[/U], 5, 10}
In both of these lists, we see the greatest common factor is 2.
So we divide top and bottom of 4/10 by 2:
4/2 / 10 / 2
[B]2/5
Bob gets 2/5 of a pound of fudge and Sue gets [B]2/5 of a pound of fudge[/B][/B]

Bob weighs between 125 and 135

Bob weighs between 125 and 135
Let w be Bob's weight. Between means includes, so we have a compound inequality:
[B]125 <= w <= 135[/B]

Brighthouse charges $120 a month for their basic plan, plus $2.99 for each on demand movie you buy.

Brighthouse charges $120 a month for their basic plan, plus $2.99 for each on demand movie you buy. Write and solve and inequality to find how many on demand movies could you buy if you want your bill to be less than $150 for the month.
Let x equal to the number room movie rentals per month. Our inequality is:
120 + 2.99x < 150
To solve for the number of movies, Add 120 to each side
2.99x < 30
Divide each side by 2.99
x < 10.03, which means 10 since you cannot buy a fraction of a movie

Carlos age increased by is 16 is 62

Carlos age increased by is 16 is 62.
Let a be Carlos's age.
Increased by 16 means we add 16
a + 16
Now the phrase [I]is[/I] means equal to, so we set [B]a + 16 = 62[/B]

Carmen has $30 in store bucks and a 25% discount coupon for a local department store. What maximum d

Carmen has $30 in store bucks and a 25% discount coupon for a local department store. What maximum dollar amount can Carmen purchase so that after her store bucks and discount are applied, her total is no more than $60 before sales tax
Let the original price be p.
p
Apply 25% discount first, which is the same as subtracting 0.25:
p(1 - 0.25)
Subtract 30 for in store buck
p(1 - 0.25) - 30
The phrase [I]no more than[/I] means an inequality using less than or equal to:
p(1 - 0.25) - 30 <= 60
To solve this inequality for p, we [URL='https://www.mathcelebrity.com/1unk.php?num=p%281-0.25%29-30%3C%3D60&pl=Solve']type it in our math engine[/URL] and we get:
[B]p <= 120[/B]

Carmen is serving her child french fries and chicken wings for lunch today. Let f be the number of f

Carmen is serving her child french fries and chicken wings for lunch today. Let f be the number of french fries in the lunch, and let c be the number of chicken wings. Each french fry has 25 calories, and each chicken wing has 100 calories. Carmen wants the total calorie count from the french fries and chicken wings to be less than 500 calories. Using the values and variables given, write an inequality describing this.
We have:
25f + 100c < 50
Note: We use < and not <= because it states less than in the problem.

Cathy wants to buy a gym membership. One gym has a $150 joining fee and costs $35 per month. Another

Cathy wants to buy a gym membership. One gym has a $150 joining fee and costs $35 per month. Another gym has no joining fee and costs $60 per month. a. In how many months will both gym memberships cost the same? What will that cost be?
Set up cost equations where m is the number of months enrolled:
[LIST=1]
[*]C(m) = 35m + 150
[*]C(m) = 60m
[/LIST]
Set them equal to each other:
35m + 150 = 60m
[URL='http://www.mathcelebrity.com/1unk.php?num=35m%2B150%3D60m&pl=Solve']Pasting the equation above into our search engine[/URL], we get [B]m = 6[/B].

Chang is serving his child french fries and chicken wings for lunch today. Let f be the number of f

Chang is serving his child french fries and chicken wings for lunch today. Let f be the number of french fries in the lunch, and let c be the number of chicken wings. Each french fry has 25 calories, and each chicken wing has 100 calories. Chang wants the total calorie count from the french fries and chicken wings to be less than 600 calories. Using the values and variables given, write an inequality describing this.
We have [B]25f + 100c < 600[/B] as our inequality.

Charmaine’s fish tank has 16 liters of water in it. she plans to add 6 liters per minute until the t

Charmaine’s fish tank has 16 liters of water in it. she plans to add 6 liters per minute until the tank has at least 58 liters. What are the possible numbers of minutes Charmaine could add water?
This is an algebraic inequality. The phrase [I]at least[/I] means greater than or equal to. So we have:
6m + 16 >= 58 <-- This is our algebraic expression/inequality.
To solve this, [URL='https://www.mathcelebrity.com/1unk.php?num=6m%2B16%3E%3D58&pl=Solve']we type this into our search engine [/URL]and we get:
[B]m >= 7[/B]

Charrie found a piece of 8 meters rope. She cuts it into equal length. She made three cuts. How long

Charrie found a piece of 8 meters rope. She cuts it into equal length. She made three cuts. How long is each piece of the rope?
Equal length means we divide the length of the rope by the number of equal cuts
[B]8/3 or 2 & 2/3 meters[/B]

Choose the best inequality for this scenario. Casey bought a sandwich and a drink for $3.75. If she

Choose the best inequality for this scenario. Casey bought a sandwich and a drink for $3.75. If she has $6.00 to spend, what is the most she can spend on dessert?
Let dessert spend be d. We have:
d <= $6.00 - $3.75
[B]d <= $2.25[/B]

Clark wants to give some baseball cards to his friends. If he gives 6 cards to each of his friends,

Clark wants to give some baseball cards to his friends. If he gives 6 cards to each of his friends, he will have 5 cards left. If he gives 8 cards to each of his friends, he will need 7 more cards. How many friends is the giving the cards to?
Let the number of friends Clark gives his cards to be f. Let the total amount of cards he gives out be n. We're given 2 equations:
[LIST=1]
[*]6f + 5 = n
[*]8f - 7 = n
[/LIST]
Since both equations equal n, we set these equations equal to each other
6f + 5 = 8f - 7
To solve for f, we [URL='https://www.mathcelebrity.com/1unk.php?num=6f%2B5%3D8f-7&pl=Solve']type this equation into our search engine[/URL] and we get:
f = [B]6
[/B]
To check our work, we plug in f = 6 into each equation:
[LIST=1]
[*]6(6) + 5 = 36 + 5 = 41
[*]8(6) - 7 = 48 - 7 = 41
[/LIST]
So this checks out. Clark has 41 total cards which he gives to 6 friends.

Class A has 8 pupils and class B has 10 pupils. Both classes sit the same maths test. The mean sco

Class A has 8 pupils and class B has 10 pupils. Both classes sit the same maths test. The mean score for class A is 55. The mean score for both classes is 76. What is the mean score (rounded to 1 DP) in the maths test for class B
Mean of the sum equals the sum of the means.
U(A + B) = U(A) + U(B)
76 = 55 + U(B)
Subtract 55 from each side, we get:
[B]U(B) = 21[/B]

Colin was thinking of a number. Colin divides by 8, then adds 1 to get an answer of 2. What was the

Colin was thinking of a number. Colin divides by 8, then adds 1 to get an answer of 2. What was the original number?
Let the number be n.
Divide by 8:
n/8
Then add 1:
n/8 + 1
The phrase [I]get an answer[/I] of means an equation, so we set n/8 + 1 equal to 2:
n/8 + 1 = 2
To solve for n, we subtract 1 from each side to isolate the n term:
n/8 + 1 - 1 = 2 - 1
Cancel the 1's on the left side, we get:
n/8 = 1
Cross multiply:
n = 8*1
n = [B]8[/B]

Company A rents copy machines for $300 a month plus $0.05 per copy. Company B charges $600 plus $0.0

Company A rents copy machines for $300 a month plus $0.05 per copy. Company B charges $600 plus $0.01 per copy. For which number of copies do the two companies charge the same amount?
With c as the number of copies, we have:
Company A Cost = 300 + 0.05c
Company B Cost = 600 + 0.01c
Set them equal to each other
300 + 0.05c = 600 + 0.01c
Use our [URL='http://www.mathcelebrity.com/1unk.php?num=300%2B0.05c%3D600%2B0.01c&pl=Solve']equation solver[/URL] to get:
[B]c = 7,500[/B]

Comparison of Numbers

Free Comparison of Numbers Calculator - Compares two numbers and checks to see if they are equal to one another, if the first number is greater than the second number, or the first number is less than the second number. Minimum and maximum.

Compute a 75% Chebyshev interval around the mean for x values and also for y values.

Compute a 75% Chebyshev interval around the mean for [I]x[/I] values and also for [I]y[/I] values.
[B][U]Grid E: [I]x[/I] variable[/U][/B]
11.92 34.86 26.72 24.50 38.93 8.59 29.31
23.39 24.13 30.05 21.54 35.97 7.48 35.97
[B][U]Grid H: [I]y[/I] variable[/U][/B]
27.86 13.29 33.03 44.31 16.58 42.43
39.61 25.51 39.14 16.58 47.13 14.70 57.47 34.44
According to Chebyshev's Theorem,
[1 - (1/k^2)] proportion of values will fall between Mean +/- (k*SD)
k in this case equal to z
z = (X-Mean)/SD
X = Mean + (z*SD)
1 - 1/k^2 = 0.75
- 1/k^2 = 0.75 - 1= - 0.25
1/k^2 = 0.25
k^2 = 1/0.25
k^2 = 4
k = 2
Therefore, z = k = 2
First, [URL='http://www.mathcelebrity.com/statbasic.php?num1=11.92%2C34.86%2C26.72%2C24.50%2C38.93%2C8.59%2C29.31%2C23.39%2C24.13%2C30.05%2C21.54%2C35.97%2C7.48%2C35.97&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']determine the mean and standard deviation of x[/URL]
Mean(x) = 25.24
SD(x) = 9.7873
Required Interval for x is:
Mean - (z * SD) < X < Mean + (z * SD)
25.24 - (2 * 9.7873) < X < 25.24 - (2 * 9.7873)
25.24 - 19.5746 < X < 25.24 + 19.5746
5.6654 < X < 44.8146
Next, [URL='http://www.mathcelebrity.com/statbasic.php?num1=27.86%2C13.29%2C33.03%2C44.31%2C16.58%2C42.43%2C39.61%2C25.51%2C39.14%2C16.58%2C47.13%2C14.70%2C57.47%2C34.44&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']determine the mean and standard deviation of y[/URL]
Mean(y) = 32.29
SD(y) = 9.7873
Required Interval for y is:
Mean - (z * SD) < Y < Mean + (z * SD)
32.29 - (2 * 13.1932) < Y < 32.29 - (2 * 13.1932)
32.29 - 26.3864 < Y < 32.29 + 26.3864
5.9036 < X < 58.6764

Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. How many chi

Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. How many children will she put in each group?
We take 24 children divided by 4 equal groups = 24/4
24/4 = [B]6 children per group[/B]

Consecutive Integer Word Problems

Free Consecutive Integer Word Problems Calculator - Calculates the word problem for what two consecutive integers, if summed up or multiplied together, equal a number entered.

Craig went bowling with $25 to spend. He rented shoes for $5.25 and paid $4.00 for each game. What w

Craig went bowling with $25 to spend. He rented shoes for $5.25 and paid $4.00 for each game. What was the greatest number of games Craig could have played?
Set up the cost function C(g) where g is the number of games Craig plays:
C(g) = Game fee * number of games (g) + shoe rental fee
C(g) = 4g + 5.25
The problem asks for the maximum number of games Craig can play for $25. So we want an inequality of [I]less than or equal to[/I].
4g + 5.25 <= 25
[URL='https://www.mathcelebrity.com/1unk.php?num=4g%2B5.25%3C%3D25&pl=Solve']Type this inequality into our search engine[/URL], and we get:
g <= 4.9375
We want exact games, so we round this down to [B]4 games[/B].

d is h decreased by 301

d is h decreased by 301
h decreased by 301 means we subtract 301 from h
h - 301
The phrase [I]is[/I] means equal to, so we set d equal to this expression:
[B]d = h - 301[/B]

d squared is greater than or equal to 17

d squared is greater than or equal to 17
d squared means we raise the variable d to the power of 2:
d^2
The phrase [I]greater than or equal to[/I] means an inequality. So we set this up using the >= in relation to 17:
[B]d^2 >= 17[/B]

Dan makes 11 an hour working at the local grocery store. Over the past year he has saved 137.50 towa

Dan makes 11 an hour working at the local grocery store. Over the past year he has saved 137.50 toward a new pair of retro sneakers. If sneakers cost 240, how many hours will he need to be able to buy the sneakers?
Figure out his remaining savings target:
240 - 137.50 = 102.50
Let x equal the number of remaining hours Dan needs to work
11x = 102.50
Divide each side by 11
x = 9.318
We round up for a half-hour to 9.5, or a full hour to 10.

Dan makes 9 dollars for each hour of work. Write an equation to represent his total pay p after work

Dan makes 9 dollars for each hour of work. Write an equation to represent his total pay p after working h hours.
We know that pay (p) on an hourly basis (h) equals:
p = Hourly Rate * h
We're given an hourly rate of 9, so we have:
p = [B]9h[/B]

Dan's school is planning a field trip to an art museum. Bus company A charges a $60 rental fee plus

Dan's school is planning a field trip to an art museum. Bus company A charges a $60 rental fee plus $4 per student. Bus company B charges $150 plus $2 per student. How many students would have to go for the cost to be the same?
[U]Set up Company A's cost equation C(s) where s is the number of students[/U]
C(s) = Cost per student * s + Rental Fee
C(s) = 4s + 60
[U]Set up Company B's cost equation C(s) where s is the number of students[/U]
C(s) = Cost per student * s + Rental Fee
C(s) = 2s + 150
The problem asks for s where both C(s) equations would be equal. So we set Company A and Company B's C(s) equal to each other:
4s + 60 = 2s + 150
To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=4s%2B60%3D2s%2B150&pl=Solve']type this equation into our search engine[/URL] and we get:
s = [B]45[/B]

Dawn has less than $60. She wants to buy 3 sweaters. What price of sweaters can she afford if all th

Dawn has less than $60. She wants to buy 3 sweaters. What price of sweaters can she afford if all the sweaters are the same price? Let s be the price of each sweater. Write this as an inequality.
The phrase [I]less than[/I] means an inequality, so we have the following inequality:
3s < 60
To solve this inequality, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=3s%3C60&pl=Show+Interval+Notation']type it in our math engine[/URL] and we get:
s < [B]20[/B]

Debra buys candy that costs 4 per pound. She will spend less than 20 on candy. What are the possible

Debra buys candy that costs 4 per pound. She will spend less than 20 on candy. What are the possible numbers of pounds she will buy?
Set up an inequality using less than < and p for pounds:
4p < 20
Divide each side by 4:
4p/4 < 20/4
[B]p < 5[/B]

Dedra took a total of 6 pages of notes during 2 hours of class. After attending 3 hours of class, ho

Dedra took a total of 6 pages of notes during 2 hours of class. After attending 3 hours of class, how many total pages of notes will Dedra have in her notebook?
Set up a proportion of pages of notes to hours of class where p equals the number of pages of notes Dedra takes for 3 hours of class:
6/2 = p/3
[URL='https://www.mathcelebrity.com/prop.php?num1=6&num2=p&den1=2&den2=3&propsign=%3D&pl=Calculate+missing+proportion+value']Type this proportion into our search engine[/URL], and we get:
p = [B]9[/B]

Del and his 5 friends eat 4 pizzas. If each has the same amount and there is 1/4 of a pizza left ove

Del and his 5 friends eat 4 pizzas. If each has the same amount and there is 1/4 of a pizza left over, how much did each person eat?
This means 4 full pizzas - 1/4 of a pizza = 3 & 3/4 pizzas eaten
Del and his 5 friends means 6 people total. Since they ate equal amounts, we divide pizzas eaten by total people:
3 & 3/4 / 6
Convert 3 & 3/4 to a mixed fraction:
(4*3 + 3)/4 = 15/4
15/4/6
Divide by a fraction is the same as multiply by a reciprocal:
15/4 * 1/6 = [B]15/24 pizzas per person[/B]

Deon opened his account starting with $650 and he is going to take out $40 per month. Mai opened up

Deon opened his account starting with $650 and he is going to take out $40 per month. Mai opened up her account with a starting amount of $850 and is going to take out $65 per month. When would the two accounts have the same amount of money?
We set up a balance equation B(m) where m is the number of months.
[U]Set up Deon's Balance equation:[/U]
Withdrawals mean we subtract from our current balance
B(m) = Starting Balance - Withdrawal Amount * m
B(m) = 650 - 40m
[U]Set up Mai's Balance equation:[/U]
Withdrawals mean we subtract from our current balance
B(m) = Starting Balance - Withdrawal Amount * m
B(m) = 850 - 65m
When the two accounts have the same amount of money, we can set both balance equations equal to each other and solve for m:
650 - 40m = 850 - 65m
Solve for [I]m[/I] in the equation 650 - 40m = 850 - 65m
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables -40m and -65m. To do that, we add 65m to both sides
-40m + 650 + 65m = -65m + 850 + 65m
[SIZE=5][B]Step 2: Cancel -65m on the right side:[/B][/SIZE]
25m + 650 = 850
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 650 and 850. To do that, we subtract 650 from both sides
25m + 650 - 650 = 850 - 650
[SIZE=5][B]Step 4: Cancel 650 on the left side:[/B][/SIZE]
25m = 200
[SIZE=5][B]Step 5: Divide each side of the equation by 25[/B][/SIZE]
25m/25 = 200/25
m = [B]8[/B]

Determine a conversion ratio that could be used to convert miles to inches

Determine a conversion ratio that could be used to convert miles to inches.
We know that 1 mile equals 5,280 feet.
We know that 1 foot equals 12 inches.
So 1 miles = 5,280 feet * 12 inches per foot= [B]63,360 inches[/B]

Diegos savings increased by 9 is 68 . Use the variable to represent Diegos savings.

Diegos savings increased by 9 is 68 . Use the variable to represent Diegos savings.
Let Diego's savings be s.
The phrase [I]increased by[/I] means add, so we add 9 to s
s + 9
The phrase [I]is [/I]means equal to, so we set 2 + 9 = 68
[B]s + 9 = 68[/B]

difference between 2 positive numbers is 3 and the sum of their squares is 117

difference between 2 positive numbers is 3 and the sum of their squares is 117
Declare variables for each of the two numbers:
[LIST]
[*]Let the first variable be x
[*]Let the second variable be y
[/LIST]
We're given 2 equations:
[LIST=1]
[*]x - y = 3
[*]x^2 + y^2 = 117
[/LIST]
Rewrite equation (1) in terms of x by adding y to each side:
[LIST=1]
[*]x = y + 3
[*]x^2 + y^2 = 117
[/LIST]
Substitute equation (1) into equation (2) for x:
(y + 3)^2 + y^2 = 117
Evaluate and simplify:
y^2 + 3y + 3y + 9 + y^2 = 117
Combine like terms:
2y^2 + 6y + 9 = 117
Subtract 117 from each side:
2y^2 + 6y + 9 - 117 = 117 - 117
2y^2 + 6y - 108 = 0
This is a quadratic equation:
Solve the quadratic equation 2y2+6y-108 = 0
With the standard form of ax2 + bx + c, we have our a, b, and c values:
a = 2, b = 6, c = -108
Solve the quadratic equation 2y^2 + 6y - 108 = 0
The quadratic formula is denoted below:
y = -b ± sqrt(b^2 - 4ac)/2a
[U]Step 1 - calculate negative b:[/U]
-b = -(6)
-b = -6
[U]Step 2 - calculate the discriminant ?:[/U]
? = b2 - 4ac:
? = 62 - 4 x 2 x -108
? = 36 - -864
? = 900 <--- Discriminant
Since ? is greater than zero, we can expect two real and unequal roots.
[U]Step 3 - take the square root of the discriminant ?:[/U]
?? = ?(900)
?? = 30
[U]Step 4 - find numerator 1 which is -b + the square root of the Discriminant:[/U]
Numerator 1 = -b + ??
Numerator 1 = -6 + 30
Numerator 1 = 24
[U]Step 5 - find numerator 2 which is -b - the square root of the Discriminant:[/U]
Numerator 2 = -b - ??
Numerator 2 = -6 - 30
Numerator 2 = -36
[U]Step 6 - calculate your denominator which is 2a:[/U]
Denominator = 2 * a
Denominator = 2 * 2
Denominator = 4
[U]Step 7 - you have everything you need to solve. Find solutions:[/U]
Solution 1 = Numerator 1/Denominator
Solution 1 = 24/4
Solution 1 = 6
Solution 2 = Numerator 2/Denominator
Solution 2 = -36/4
Solution 2 = -9
[U]As a solution set, our answers would be:[/U]
(Solution 1, Solution 2) = (6, -9)
Since one of the solutions is not positive and the problem asks for 2 positive number, this problem has no solution

Difference between 23 and y is 12

Difference between 23 and y
23 - y
Is, means equal to, so we set 23 - y equal to 12
[B]23 - y = 12
[/B]
If you need to solve this algebraic expression, use our [URL='http://www.mathcelebrity.com/1unk.php?num=23-y%3D12&pl=Solve']equation calculator[/URL]:
[B]y = 11[/B]

Division Equality Property

Free Division Equality Property Calculator - Demonstrates the Division Equality Property Calculator
Numerical Properties

Division Property Of Inequality

Free Division Property Of Inequality Calculator - Demonstrates the Division Property Of Inequality
Numerical Properties

Dunder Mifflin will print business cards for $0.10 each plus setup charge of $15. Werham Hogg offers

Dunder Mifflin will print business cards for $0.10 each plus setup charge of $15. Werham Hogg offers business cards for $0.15 each with a setup charge of $10. What numbers of business cards cost the same from either company
Declare variables:
[LIST]
[*]Let b be the number of business cards.
[/LIST]
[U]Set up the cost function C(b) for Dunder Mifflin:[/U]
C(b) = Cost to print each business card * b + Setup Charge
C(b) = 0.1b + 15
[U]Set up the cost function C(b) for Werham Hogg:[/U]
C(b) = Cost to print each business card * b + Setup Charge
C(b) = 0.15b + 10
The phrase [I]cost the same[/I] means we set both C(b)'s equal to each other and solve for b:
0.1b + 15 = 0.15b + 10
Solve for [I]b[/I] in the equation 0.1b + 15 = 0.15b + 10
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 0.1b and 0.15b. To do that, we subtract 0.15b from both sides
0.1b + 15 - 0.15b = 0.15b + 10 - 0.15b
[SIZE=5][B]Step 2: Cancel 0.15b on the right side:[/B][/SIZE]
-0.05b + 15 = 10
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 15 and 10. To do that, we subtract 15 from both sides
-0.05b + 15 - 15 = 10 - 15
[SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE]
-0.05b = -5
[SIZE=5][B]Step 5: Divide each side of the equation by -0.05[/B][/SIZE]
-0.05b/-0.05 = -5/-0.05
b = [B]100[/B]

Dylans mother tells Dylan he must spend less time playing electronic games. On the weekends he spend

Dylans mother tells Dylan he must spend less time playing electronic games. On the weekends he spends 9.5 hours playing electronic games. If he plays between 13 and 19 hours each week, how many hours does he play games on weekdays?
Let x equal the number of hours Dylan plays electronic games per week.
[U]Set up our inequality:[/U]
13 <= x <= 19
[U]To see how much he plays during weekdays, subtract off the weekend time[/U]
13 - 9.5 <= x <= 19 - 9.5
[B]3.5 <= x <= 9.5[/B]

eana paid $5 for 20 lbs of bananas. how much were each pound?

$5 for 20 pounds equal = $x for 1 pound
Set up a proportion:
5/20 = x/1
[URL='http://www.mathcelebrity.com/prop.php?num1=5&num2=x&den1=20&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']Use our proportion calculator[/URL]
x = 0.25

Emil bought a camera for $268.26, including tax. He made a down payment of $12.00 and paid the balan

Emil bought a camera for $268.26, including tax. He made a down payment of $12.00 and paid the balance in 6 equal monthly payments. What was Emil’s monthly payment for this camera?
Calculate remaining balance
268.26 - 12 = 256.26
Determine monthly payment:
256.26/6 = [B]21.36[/B]

Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30.

Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30. What is the greatest age Mary could be?
Let e = Emily's age and m = Mary's age.
We have the equation e = 2m + 3 and the inequality e + m < 30
Substitute the equation for e into the inequality:
2m + 3 + m < 30
Add the m terms
3m + 3 < 30
Subtract 3 from each side of the inequality
3m < 27
Divide each side of the inequality by 3 to isolate m
m < 9
Therefore, the [B]greatest age[/B] Mary could be is 8, since less than 9 [U]does not include[/U] 9.

Equation and Inequalities

Free Equation and Inequalities Calculator - Solves an equation or inequality with 1 unknown variable and no exponents as well as certain absolute value equations and inequalities such as |x|=c and |ax| = c where a and c are constants. Solves square root, cube root, and other root equations in the form ax^2=c, ax^2 + b = c. Also solves radical equations in the form asqrt(bx) = c. Also solves open sentences and it will solve one step problems and two step equations. 2 step equations and one step equations and multi step equations

Estimate a 15% tip on a dinner bell of $49.76 by first round the bill amount to the nearest $10

Estimate a 15% tip on a dinner bell of $49.76 by first round the bill amount to the nearest $10.
Round the bill to the nearest $10
[LIST]
[*]$49.76 is in between ($40, $50)
[*]1/2 of that interval is (40 + 50)/2 = 90/2 = 45
[*]Since $49.76 is greater than or equal to 45, we round up to $50
[/LIST]
Add a 15% tip
50(1 + 0.15) = 50 + 7.50 = [B]$57.50[/B]

Ethan has $9079 in his retirement account, and Kurt has $9259 in his. Ethan is adding $19per day, wh

Ethan has $9079 in his retirement account, and Kurt has $9259 in his. Ethan is adding $19per day, whereas Kurt is contributing $1 per day. Eventually, the two accounts will contain the same amount. What balance will each account have? How long will that take?
Set up account equations A(d) where d is the number of days since time 0 for each account.
Ethan A(d): 9079 + 19d
Kurt A(d): 9259 + d
The problems asks for when they are equal, and how much money they have in them. So set each account equation equal to each other:
9079 + 19d = 9259 + d
[URL='https://www.mathcelebrity.com/1unk.php?num=9079%2B19d%3D9259%2Bd&pl=Solve']Typing this equation into our search engine[/URL], we get [B]d = 10[/B].
So in 10 days, both accounts will have equal amounts in them.
Now, pick one of the account equations, either Ethan or Kurt, and plug in d = 10. Let's choose Kurt's since we have a simpler equation:
A(10) = 9259 + 10
A(10) = $[B]9,269
[/B]
After 10 days, both accounts have $9,269 in them.

evaluate 16 raised to 1/4

evaluate 16 raised to 1/4
What number raised to the 4th power equals 16?
[B]2[/B], since 2 * 2 * 2 * 2 = 16

Every 6 customers receive a soda, every 8 a hot dog there are 329 customers . how many received both

This is a least common multiple problem.
[URL='http://www.mathcelebrity.com/gcflcm.php?num1=6&num2=8&num3=&pl=LCM']The least common multiple of 6 and 8 is 24[/URL]
So every 24th person, less than or equal to 329 receives both a soda [U]and[/U] a hot dog.
Using our multiples calculator, we find there are [URL='http://www.mathcelebrity.com/multiple.php?num=24&pl=Multiplication+Multiples']13 multiples of 24 less than or equal to 329[/URL].
24,48,72,96,120,144,168,192,216,240,264,288,312

Express the fact that x differs from 3 by less than 2/7 as an inequality involving absolute value. S

Express the fact that x differs from 3 by less than 2/7 as an inequality involving absolute value. Solve for x.
Let's build this algebraic expression in pieces:
The phrase [I]differs from[/I] means a difference.
x - 3
By less than 2/7 means we use the < sign compared to 2/7
x - 3 < 2/7
Finally, the problem says we involve absolute value. So we write this as:
[B]|x - 3| < 2/7[/B]

f of X equals three to the x

f of X equals three to the x
[B]f(x) = 3^x[/B]

f(x)=a(b)^x and we know that f(3)=17 and f(7)=3156. what is the value of b

f(x)=a(b)^x and we know that f(3)=17 and f(7)=3156. what is the value of b
Set up both equations with values
When x = 3, f(3) = 17, so we have a(b)^3 = 17
When x = 7, f(7) = 3156, so we have a(b)^7 = 3156
Isolate a in each equation
a = 17/(b)^3
a = 3156/(b)^7
Now set them equal to each other
17/(b)^3 = 3156/(b)^7
Cross Multiply
17b^7 = 3156b^3
Divide each side by b^3
17b^4 = 3156
Divide each side by 17
b^4 = 185.6471
[B]b = 3.6912[/B]

Find Mean 106 and standard deviation 10 of the sample mean which is 25

Do you mean x bar?
mean of 106 inches and a standard deviation of 10 inches and for sample of size is 25. Determine the mean and the standard deviation of /x
If so, x bar equals the population mean. So it's [B]106[/B].
Sample standard deviation = Population standard deviation / square root of n
10/Sqrt(25)
10/5
[B]2[/B]

Find the elements on the principal diagonal of matrix B

Find the elements on the principal diagonal of matrix B
Matrix B:
|0 0 8|
|-1 3 0|
|2 -5 -7|
The main diagonal is any entry where row equals column
|[B]0[/B] 0 8|
|-1 [B]3 [/B] 0|
|2 -5 [B]-7[/B]|
In this case, it is [B]0, 3, -7[/B]

Find the largest of three consecutive even integers when six times the first integers is equal to fi

Find the largest of three consecutive even integers when six times the first integers is equal to five times the middle integer.
Let the first of the 3 consecutive even integers be n.
The second consecutive even integer is n + 2.
The third (largest) consecutive even integer is n + 4.
We are given 6n = 5(n + 2).
Multiply through on the right side, and we get:
6n = 5n + 10
[URL='https://www.mathcelebrity.com/1unk.php?num=6n%3D5n%2B10&pl=Solve']Typing 6n = 5n + 10 into the search engine[/URL], we get n = 10.
Remember, n was our smallest of 3 consecutive even integers. So the largest is:
n + 4
10 + 4
[B]14[/B]

Find two consecutive odd integers such that the sum of their squares is 290

Find two consecutive odd integers such that the sum of their squares is 290.
Let the first odd integer be n.
The next odd integer is n + 2
Square them both:
n^2
(n + 2)^2 = n^2 + 4n + 4 from our [URL='https://www.mathcelebrity.com/expand.php?term1=%28n%2B2%29%5E2&pl=Expand']expansion calculator[/URL]
The sum of the squares equals 290
n^2 + n^2 + 4n + 4 = 290
Group like terms:
2n^2 + 4n + 4 = 290
[URL='https://www.mathcelebrity.com/quadratic.php?num=2n%5E2%2B4n%2B4%3D290&pl=Solve+Quadratic+Equation&hintnum=+0']Enter this quadratic into our search engine[/URL], and we get:
n = 11, n = -13
Which means the two consecutive odd integer are:
11 and 11 + 2 = 13. [B](11, 13)[/B]
-13 and -13 + 2 = -11 [B](-13, -11)[/B]

Find two consecutive positive integers such that the difference of their square is 25

Find two consecutive positive integers such that the difference of their square is 25.
Let the first integer be n. This means the next integer is (n + 1).
Square n: n^2
Square the next consecutive integer: (n + 1)^2 = n^2 + 2n + 1
Now, we take the difference of their squares and set it equal to 25:
(n^2 + 2n + 1) - n^2 = 25
Cancelling the n^2, we get:
2n + 1 = 25
[URL='https://www.mathcelebrity.com/1unk.php?num=2n%2B1%3D25&pl=Solve']Typing this equation into our search engine[/URL], we get:
n = [B]12[/B]

Find two consecutive positive integers such that the sum of their squares is 25

Find two consecutive positive integers such that the sum of their squares is 25.
Let the first integer be x. The next consecutive positive integer is x + 1.
The sum of their squares equals 25. We write this as::
x^2 + (x + 1)^2
Expanding, we get:
x^2 + x^2 + 2x + 1 = 25
Group like terms:
2x^2 + 2x + 1 = 25
Subtract 25 from each side:
2x^2 + 2x - 24 = 0
Simplify by dividing each side by 2:
x^2 + x - 12 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-12%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get x = 3 or x = -4. The problem asks for positive integers, so we discard -4, and use 3.
This means, our next positive integer is 3 + 1 = 4. So we have [B](3, 4) [/B]as our answers.
Let's check our work:
3^2 + 4^2 = 9 + 16 = 25

Find x

Find x
[IMG]https://mathcelebrity.com/community/data/attachments/0/cong-angles.jpg[/IMG]
Since both angles are congruent, we set them equal to each other:
6x - 20 = 4x
To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=6x-20%3D4x&pl=Solve']type this equation into our math engine[/URL] and we get:
x = [B]10[/B]

Five less than a number is at least -7 and at most 7.

Five less than a number is at least -7 and at most 7.
A number signifies an arbitrary variable, let's call it x.
Five less than a number: x - 5
Is at least -7 means greater than or equal to and at most 7 means less than or equal to, so we have a joint inequality:
[B]-7 <= x - 5 <= 7[/B]

Five times Kim's age plus 13 equals 58. How old is Kim?

Five times Kim's age plus 13 equals 58. How old is Kim?
Let Kim's age be a. We have:
Five times Kim's age:
5a
Plus 13 means we add 13
5a + 13
Equals 58 means we set the expression 5a + 13 equal to 58
5a + 13 = 58 <-- This is our algebraic expression
To solve this equation for a, [URL='https://www.mathcelebrity.com/1unk.php?num=5a%2B13%3D58&pl=Solve']we type it into our search engine[/URL] and get:
a = [B]9[/B]

Fixed cost 500 marginal cost 8 item sells for 30

fixed cost 500 marginal cost 8 item sells for 30.
Set up Cost Function C(x) where x is the number of items sold:
C(x) = Marginal Cost * x + Fixed Cost
C(x) = 8x + 500
Set up Revenue Function R(x) where x is the number of items sold:
R(x) = Revenue per item * items sold
R(x) = 30x
Set up break even function (Cost Equals Revenue)
C(x) = R(x)
8x + 500 = 30x
Subtract 8x from each side:
22x = 500
Divide each side by 22:
x = 22.727272 ~ 23 units for breakeven

Flight is $295 and car rental is $39 a day, if a competition charges $320 and $33 a day car rental,

Flight is $295 and car rental is $39 a day, if a competition charges $320 and $33 a day car rental, which is cheaper?
Set up cost function where d is the number of days:
[LIST]
[*]Control business: C(d) = 39d + 295
[*]Competitor business: C(d) = 33d + 320
[/LIST]
Set the [URL='http://www.mathcelebrity.com/1unk.php?num=39d%2B295%3D33d%2B320&pl=Solve']cost functions equal to each other[/URL]:
We get d = 4.1667.
The next integer day up is 5. Now plug in d = 1, 2, 3, 4. For the first 4 days, the control business is cheaper. However, starting at day 5, the competitor business is now cheaper forever.

For her phone service, Maya pays a monthly fee of $27 , and she pays an additional $0.04 per minu

For her phone service, Maya pays a monthly fee of $27 , and she pays an additional $0.04 per minute of use. The least she has been charged in a month is $86.04 . What are the possible numbers of minutes she has used her phone in a month? Use m for the number of minutes, and solve your inequality for m .
Maya's cost function is C(m), where m is the number of minutes used.
C(m) = 0.04m + 27
We are given C(m) = $86.04. We want her cost function [I]less than or equal[/I] to this.
0.04m + 27 <= 86.04
[URL='https://www.mathcelebrity.com/1unk.php?num=0.04m%2B27%3C%3D86.04&pl=Solve']Type this inequality into our search engine[/URL], and we get [B]m <= 1476[/B].

Four cousins were born at two-year intervals. The sum of their ages is 36. What are their ages?

Four cousins were born at two-year intervals. The sum of their ages is 36. What are their ages?
So the last cousin is n years old. this means consecutive cousins are n + 2 years older than the next.
whether their ages are even or odd, we have the sum of 4 consecutive (odd|even) integers equal to 36. We [URL='https://www.mathcelebrity.com/sum-of-consecutive-numbers.php?num=sumof4consecutiveevenintegersis36&pl=Calculate']type this into our search engine[/URL] and we get the ages of:
[B]6, 8, 10, 12[/B]

Four more then double a number is greater than 2

Four more then double a number is greater than 2
Double a number:
A number implies an arbitrary variable, let's call it "x". Double means multiply this by 2
2x
Four more than this:
2x + 4
Now, we set this expression as an inequality greater than 2
[B]2x + 4 > 2[/B]

g equals 232 subtracted from the quantity 377 times g

g equals 232 subtracted from the quantity 377 times g
377 times g:
377g
232 subtracted from 377 times g:
377g - 232
We set the variable g equal to this expression:
[B]g = 377g - 232[/B]

g less than 143 is equal to 39 reduced by w

g less than 143 is equal to 39 reduced by w
g less than 143 means we subtract g from 143
143 - g
39 reduced by w means we subtract w from 39
39 - w
We set these 2 expressions equal to each other:
[B]143 - g = 39 - w[/B]

Gary is buying chips. Each bag costs $3.50. He has $40 to spend. Write an inequality to represent th

Gary is buying chips. Each bag costs $3.50. He has $40 to spend. Write an inequality to represent the number of chip bags, c, he can afford.
Gary's spend is found by this inequality:
[B]3.50c <= 40
[/B]
To solve this inequality, [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=3.50c%3C%3D40&pl=Show+Interval+Notation']we type it in our search engine[/URL] and we get:
[B]c <= 11.43[/B]

Geocache puzzle help

Let x equal the number of sticks he started with. We have:
The equation is 4/5 * (3/4 * (2/3 * (0.5x - 0.5) -1/3) - 0.75) - 0.2 = 19
Add 0.2 to each side:
4/5 * (3/4 * (2/3 * (0.5x - 0.5) -1/3) - 0.75) = 19.2
Multiply each side by 5/4
(3/4 * (2/3 * (0.5x - 0.5) - 1/3) - 0.75) = 24
Multiply the inside piece first:
2/6x - 2/6 - 1/3
2/6x - 4/6
Now subtract 0.75 which is 3/4
2/6x - 4/6 - 3/4
4/6 is 8/12 and 3/4 is 9/12, so we have:
2/6x - 17/12
Now multiply by 3/4
6/24x - 51/48 = 24
Simplify:
1/4x - 17/16 = 24
Multiply through by 4
x - 17/4 = 96
Since 17/4 = 4.25, add 4.25 to each side
x = 100.25
Since he did not cut up any sticks, he has a full stick to start with:
So x = [B]101[/B]

Geocache puzzle help

Let x equal the number of sticks he started with. We have:
The equation is 4/5 * (3/4 * (2/3 * (0.5x - 0.5) -1/3) - 0.75) - 0.2 = 19
Add 0.2 to each side:
4/5 * (3/4 * (2/3 * (0.5x - 0.5) -1/3) - 0.75) = 19.2
Multiply each side by 5/4
(3/4 * (2/3 * (0.5x - 0.5) - 1/3) - 0.75) = 24
Multiply the inside piece first:
2/6x - 2/6 - 1/3
2/6x - 4/6
Now subtract 0.75 which is 3/4
2/6x - 4/6 - 3/4
4/6 is 8/12 and 3/4 is 9/12, so we have:
2/6x - 17/12
Now multiply by 3/4
6/24x - 51/48 = 24
Simplify:
1/4x - 17/16 = 24
Multiply through by 4
x - 17/4 = 96
Since 17/4 = 4.25, add 4.25 to each side
x = 100.25
Since he did not cut up any sticks, he has a full stick to start with:
So x = [B]101[/B]

Gino has 7 whole pineapples. He cuts each whole into 4 equal parts. Write an improper fraction for t

Gino has 7 whole pineapples. He cuts each whole into 4 equal parts. Write an improper fraction for the cut parts of pineapples.
Take our whole pineapples divided by the number of equal parts:
[B]7/4[/B]

Giovanni is thinking of a number. If he adds 2 to it, then divides that sum by 3, he gets 7. What is

Giovanni is thinking of a number. If he adds 2 to it, then divides that sum by 3, he gets 7. What is the number?
Let the number be n:
[LIST]
[*]n
[*]Add 2: n + 2
[*]Divide the sum by 3: (n + 2)/3
[*]The word "gets" means an equation, so we set (n + 2)/3 equal to 7
[/LIST]
(n + 2)/3 = 7
Cross multiply:
n + 2 = 21
To solve for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=n%2B2%3D21&pl=Solve']type this equation into our search engine[/URL] and we get:
n = [B]19[/B]

Given: BC = EF AC = EG AB = 10 BC = 3 Prove FG = 10

Given: BC = EF
AC = EG
AB = 10
BC = 3
Prove FG = 10
[LIST]
[*]AC = AB + BC (Segment Addition Postulate)
[*]AB = 10, BC = 3 (Given)
[*]AC = 10 + 3 (Substitution Property of Equality)
[*]AC = 13 (Simplify)
[*]AC = EG, BC = EF (Given)
[*]EG = 13, EF = 3 (Segment Equivalence)
[*]EG = EF + FG (Segment Addition Postulate)
[*]13 = 3 + FG (Substitution Property of Equality)
[*]FG = 10 (Subtraction Property)
[/LIST]

Goal is to take at least 10,000 steps per day. According to your pedometer you have walked 5,274 ste

Goal is to take at least 10,000 steps per day. According to your pedometer you have walked 5,274 steps. Write and solve an inequality to find the possible numbers of steps you can take to reach your goal.
[U]
Subtract off the existing steps (s) from your goal of 10,000[/U]
g >= 10000 - 5274
[B]g >= 4726[/B]
[I]Note: we use >= since 10,000 steps meets the goal as well as anytihng above it[/I]

Graham is hiking at an altitude of 14,040 feet and is descending 50 feet each minute.Max is hiking a

Graham is hiking at an altitude of 14,040 feet and is descending 50 feet each minute.Max is hiking at an altitude of 12,500 feet and is ascending 20 feet each minute. How many minutes will it take until they're at the same altitude?
Set up the Altitude function A(m) where m is the number of minutes that went by since now.
Set up Graham's altitude function A(m):
A(m) = 14040 - 50m <-- we subtract for descending
Set up Max's altitude function A(m):
A(m) = 12500 + 20m <-- we add for ascending
Set the altitudes equal to each other to solve for m:
14040 - 50m = 12500 + 20m
[URL='https://www.mathcelebrity.com/1unk.php?num=14040-50m%3D12500%2B20m&pl=Solve']We type this equation into our search engine to solve for m[/URL] and we get:
m = [B]22[/B]

Gym A: $75 joining fee and $35 monthly charge. Gym B: No joining fee and $60 monthly charge. (Think

Gym A: $75 joining fee and $35 monthly charge. Gym B: No joining fee and $60 monthly charge. (Think of the monthly charges paid at the end of the month.) Enter the number of months it will take for the total cost for both gyms to be equal.
Gym A cost function C(m) where m is the number of months:
C(m) = Monthly charge * months + Joining Fee
C(m) = 35m + 75
Gym B cost function C(m) where m is the number of months:
C(m) = Monthly charge * months + Joining Fee
C(m) = 60m
Set them equal to each other:
35m + 75 = 60m
To solve for m, [URL='https://www.mathcelebrity.com/1unk.php?num=35m%2B75%3D60m&pl=Solve']we type this equation into our search engine[/URL] and get:
m = [B]3[/B]

Half of g multiplied by t squared is equal to d.

Half of g multiplied by t squared is equal to d.
Half of g:
g/2
t squared:
t^2
Half of g multiplied by t squared:
gt^2/2
The phrase [I]is equal to[/I] mean we set gt^2/2 equal to d:
[B]gt^2/2 = d[/B]

Happy Paws charges $16.00 plus $1.50 per hour to keep a dog during the day. Woof Watchers charges $1

Happy Paws charges $16.00 plus $1.50 per hour to keep a dog during the day. Woof Watchers charges $11.00 plus $2.75 per hour. Complete the equation and solve it to find for how many hours the total cost of the services is equal. Use the variable h to represent the number of hours.
Happy Paws Cost: C = 16 + 1.5h
Woof Watchers: C = 11 + 2.75h
Setup the equation where there costs are equal
16 + 1.5h = 11 + 2.75h
Subtract 11 from each side:
5 + 1.5h = 2.75h
Subtract 1.5h from each side
1.25h = 5
Divide each side by 1.25
[B]h = 4[/B]

Happy Paws charges $19.00 plus $5.50 per hour to keep a dog during the day. Woof Watchers charges $1

Happy Paws charges $19.00 plus $5.50 per hour to keep a dog during the day. Woof Watchers charges $11.00 plus $6.75 per hour. Complete the equation and solve it to find for how many hours the total cost of the services is equal. Use the variable h to represent the number of hours.
[B]Happy Paws cost equation:[/B]
5.50h + 19
[B]Woof Watchers cost equation:[/B]
6.75h + 11
[B]Set them equal to each other:[/B]
5.50h + 19 = 6.75h + 11
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=5.50h%2B19%3D6.75h%2B11&pl=Solve']equation solver[/URL], we get [B]h = 6.4[/B].

Hari planted 324 plants in such a way that there were as many rows of plants as there were number of

Hari planted 324 plants in such a way that there were as many rows of plants as there were number of columns. Find the number of rows and columns.
Let r be the number of rows and c be the number of columns. We have the area:
rc = 324
Since rows equal columns, we have a square, and we can set r = c.
c^2 = 324
Take the square root of each side:
[B]c = 18[/B]
Which means [B]r = 18[/B] as well.
What we have is a garden of 18 x 18.

harley had $500 in his bank account at the beginning of the year. he spends $20 each week on food, c

harley had $500 in his bank account at the beginning of the year. he spends $20 each week on food, clothing, and movie tickets. he wants to have more than $100 at the end of summer to make sure he has enough to purchase some new shoes before school starts. how many weeks, w, can harley withdraw money from his savings account and still have more than $100 to buy new shoes?
Let the number of weeks be w. Harley needs $100 (or more) for shoes. We have the balance in Harley's account as:
500 - 20w >= 100
To solve this inequality for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=500-20w%3E%3D100&pl=Solve']type it in our search engine[/URL] and we get:
[B]w <= 20[/B]

HomeWork Help Please Respond ASAP!!!

The phrase a number means an arbitrary variable, let's call it x.
Three times a number:
3x
And 18 means we add 18
3x + 18
The word is means equal to, so we set 3x + 18 equal to -39
3x + 18 = -39
This is your algebraic expression. If you want to solve for x, plug it into the [URL='http://www.mathcelebrity.com/1unk.php?num=3x%2B18%3D-39&pl=Solve']search engine[/URL] and you get x = -19

Hope it's okay to ask this here?

a) 1800 is the cost to run the business for a day. To clarify, when you plug in x = 0 for 0 candy bars sold, you are left with -1,800, which is the cost of doing business for one day.
b) Maximum profit is found by taking the derivative of the profit equation and setting it equal to 0.
P'(x) = -0.002x + 3
With P'(x) = 0, we get:
-0.002x + 3 = 0
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=-0.002x%2B3%3D0&pl=Solve']equation solver[/URL], we get:
x = 1,500
To get maximum profit, we plug in x = 1,500 to our [I]original profit equation[/I]
P(1,500) = ? 0.001(1,500)^2 + 3(1,500) ? 1800
P(1,500) = -2,250 + 4,500 - 1,800
P(1,500) = $[B]450[/B]

How many hours are in d days

How many hours are in d days
Since 1 day equals 24 hours, we have:
[B]24d[/B]

how many sixths equal one-third

how many sixths equal one-third
We have a variable x where we want to solve for in the following equation:
x/6 = 1/3
[URL='https://www.mathcelebrity.com/prop.php?num1=x&num2=1&den1=6&den2=3&propsign=%3D&pl=Calculate+missing+proportion+value']Typing this proportion into our math engine[/URL], we get:
x = [B]2[/B]

How many twelfths equal three-sixths?

How many twelfths equal three-sixths?
We set up the equation below where x is the number of twelfths in three-sixths:
1/12x = 3/6
Cross multiply, and we get:
12x * 3 = 6 * 1
36x = 6
To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=36x%3D6&pl=Solve']type this in our math engine[/URL] and we get:
x = [B]1/6 or 0.16667[/B]

How old am I if 400 reduced by 3 times my age is 124?

How old am I if 400 reduced by 3 times my age is 124?
Let my age be a. We're given an algebraic expression:
[LIST]
[*]3 times my age means we multiply a by 3: 3a
[*]400 reduced by 3 times my age means we subtract 3a from 400:
[*]400 - 3a
[*]The word [I]is[/I] mean an equation, so we set 400 - 3a equal to 124
[/LIST]
400 - 3a = 124
To solve for a, [URL='https://www.mathcelebrity.com/1unk.php?num=400-3a%3D124&pl=Solve']we type this equation into our search engine[/URL] and we get:
a = [B]92[/B]

How old am I if: 210 reduced by 3 times my current age is 4 times my current age?

How old am I if: 210 reduced by 3 times my current age is 4 times my current age?
Let your current age be a. We're given:
[LIST]
[*]210 reduced by 3 times current age = 210 - 3a
[*]4 times current age = 4a
[*]The word [I]is[/I] means equal to. So we set 210 - 3a equal to 4a
[/LIST]
210 - 3a = 4a
To solve for a, we [URL='https://www.mathcelebrity.com/1unk.php?num=210-3a%3D4a&pl=Solve']type this equation into our search engine [/URL]and we get:
a = [B]30[/B]

How old am I of 400 reduced by 2 times my age is 224

How old am I of 400 reduced by 2 times my age is 224
[LIST=1]
[*]Let my age be a.
[*]2 times my age: 2a
[*]400 reduced by 2 times my age: 400 - 2a
[*]The phrase [I]is [/I]means an equation. So we set 400 - 2a equal to 224 for our algebraic expression
[/LIST]
[B]400 - 2a = 224
[/B]
If the problem asks you to solve for a, we [URL='https://www.mathcelebrity.com/1unk.php?num=400-2a%3D224&pl=Solve']type this equation into our search engine[/URL] and we get:
a = [B]88[/B]

I am thinking of a number. I multiply it by 14 and add 13. I get the same answer if I multiply by 5

I am thinking of a number. I multiply it by 14 and add 13. I get the same answer if I multiply by 5 and add 283. What is my number?
Let the number be n. We're given two expressions:
[LIST=1]
[*]Multiply it by 14 and add 13: 14n + 13
[*]Multiply by 5 and add 283: 5n + 283
[/LIST]
The phrase [I]I get the same answer[/I] means an equation. So we set expression 1 equal to expression 2:
14n + 13 = 5n + 283
To solve for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=14n%2B13%3D5n%2B283&pl=Solve']type this equation into our search engine[/URL] and we get:
n = [B]30[/B]

I am thinking of a number. I multiply it by 14 and add 21. I get the same answer if I multiply by 4

I am thinking of a number. I multiply it by 14 and add 21. I get the same answer if I multiply by 4 and add 141.
Let the number be n.
We have two expressions:
[LIST=1]
[*]Multiply by 14 and add 21 is written as: 14n + 21
[*]Multiply by 4 and add 141 is written as: 4n + 141
[/LIST]
The phrase [I]get the same expression[/I] means they are equal. So we set (1) and (2) equal to each other and solve for n:
14n + 21 = 4n + 141
[URL='https://www.mathcelebrity.com/1unk.php?num=14n%2B21%3D4n%2B141&pl=Solve']Type this equation into our search engine [/URL]to solve for n and we get:
n = [B]12[/B]

I am Thinking of a number. I multiply it by 3 and add 67. I get the same answer If i multiply by 6 s

I am Thinking of a number. I multiply it by 3 and add 67. I get the same answer If i multiply by 6 subtract 8.
Let the number be n. We're given two equal expressions:
[LIST=1]
[*]3n + 67
[*]6n - 8
[/LIST]
Set the expressions equal to each other since they give the [B]same answer[/B]:
3n + 67 = 6n - 8
We have an equation. [URL='https://www.mathcelebrity.com/1unk.php?num=3n%2B67%3D6n-8&pl=Solve']Type this equation into our search engine and we get[/URL]:
n = [B]25[/B]

I am thinking of a number. I multiply it by 7 and add 25. I get the same answer if I multiply by 3 a

I am thinking of a number. I multiply it by 7 and add 25. I get the same answer if I multiply by 3 and add 93. What is my number?
Let the number be n. We're given two expressions:
[LIST]
[*]Multiply the number by 7: 7n
[*]add 25: 7n + 25. <-- Expression 1
[*]Multiply by 3: 3n
[*]Add 93: 3n + 93 <-- Expression 2
[*]The phrase [I]get the same answer[/I] means both expression 1 and expression 2 are equal. So we set them equal to each other:
[/LIST]
7n + 25 = 3n + 93
To solve for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=7n%2B25%3D3n%2B93&pl=Solve']type this equation into our search engine[/URL] and we get:
n = [B]17[/B]

I am thinking of a number.i multiply it by 5 and add 139. I get the same number if I multiply by 13

I am thinking of a number.i multiply it by 5 and add 139. I get the same number if I multiply by 13 and subtract 13.What is my number?
Take a number (n):
The first operation is multiply 5 times n, and then add 39:
5n + 139
The second operation is multiply 13 times n and subtract 13:
13n - 13
Set both operations equal to each other since they result in [I]the same number[/I]
5n + 139 = 13n - 13
[URL='https://www.mathcelebrity.com/1unk.php?num=5n%2B139%3D13n-13&pl=Solve']Type this equation into our search engine[/URL] and we get:
[B]n = 19[/B]

I only own blue blankets and red blankets. 8 out of every 15 blankets I have are red.

I only own blue blankets and red blankets. 8 out of every 15 blankets I have are red. If have i 45 blankets, how many are blue?
If 8 out of 15 blankets are red, then 15 - 8 = 7 are blue
So 7 out of every 15 blankets are blue.
Set up a proportion of blue blankets to total blankets where b is the number of blue blankets in 45 blankets
7/15 = b/45
Cross multiply:
If 2 proportions are equal, then we can do the following:
Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2
15b = 45 * 7
15b = 315
To solve for b, divide each side of the equation by 15:
15b/15 = 315/15
Cancel the 15's on the left side and we get:
b = [B]21[/B]

I think of a number. I multiply it by 6 and add 3. If my answer is 75, calculate the number I starte

I think of a number. I multiply it by 6 and add 3. If my answer is 75, calculate the number I started with.
Let the number be n.
Multiply it by 6:
6n
Add 3:
6n + 3
If the answer is 75, we set 6n + 3 equal to 75:
6n + 3 = 75
We have an equation. To solve for n, [URL='https://www.mathcelebrity.com/1unk.php?num=6n%2B3%3D75&pl=Solve']we type this equation into our search engine[/URL] and get:
[B]n = 12[/B]

If .75 inches on a map are equal to 6 miles, how many miles is one inch equal to?

If .75 inches on a map are equal to 6 miles, how many miles is one inch equal to?
Using the unit measurement, we have:
6 miles / 0.75 inches = [B]8 miles per inch[/B]

If 11 times a number is added to twice the number, the result is 104

If 11 times a number is added to twice the number, the result is 104
Let [I]the number[/I] be an arbitrary variable we call x.
11 times a number:
11x
Twice the number (means we multiply x by 2):
2x
The phrase [I]is added to[/I] means we add 2x to 11x:
11x + 2x
Simplify by grouping like terms:
(11 + 2)x = 13x
The phrase [I]the result is[/I] means an equation, so we set 13x equal to 104:
13x = 104 <-- This is our algebraic expression
To solve this equation for x, [URL='https://www.mathcelebrity.com/1unk.php?num=13x%3D104&pl=Solve']we type it in our search engine[/URL] and we get:
x = [B]8[/B]

If 3.75 inches on a map are equal to 18.75 miles, how many miles are 5 inches equal to?

If 3.75 inches on a map are equal to 18.75 miles, how many miles are 5 inches equal to?
Set up a proportion of inches to miles where m is the number of miles for 5 inches:
3.75/18.75 = 5/m
Using our [URL='https://www.mathcelebrity.com/proportion-calculator.php?num1=3.75&num2=5&den1=18.75&den2=m&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator,[/URL] we get:
m = [B]25 miles[/B]

If 3/5 of a bun is shared equally between 4 people, what fraction of the bun would each receive?

If 3/5 of a bun is shared equally between 4 people, what fraction of the bun would each receive?
[URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F5&frac2=1%2F4&pl=Multiply']We divide 3/5 by 4[/URL] to get [B]3/20[/B]

If 4 times a number is added to 9, the result is 49

If 4 times a number is added to 9, the result is 49.
[I]A number[/I] means an arbitrary variable, let's call it x.
4 [I]times a number[/I] means we multiply x by 4
4x
[I]Added to[/I] 9 means we add 9 to 4x
4x + 9
[I]The result is[/I] means we have an equation, so we set 4x + 9 equal to 49
[B]4x + 9 = 49[/B] <-- This is our algebraic expression
To solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=4x%2B9%3D49&pl=Solve']we type it in the search engine[/URL] and get x = 10

If 7 times the square of an integer is added to 5 times the integer, the result is 2. Find the integ

If 7 times the square of an integer is added to 5 times the integer, the result is 2. Find the integer.
[LIST]
[*]Let the integer be "x".
[*]Square the integer: x^2
[*]7 times the square: 7x^2
[*]5 times the integer: 5x
[*]Add them together: 7x^2 + 5x
[*][I]The result is[/I] means an equation, so we set 7x^2 + 5x equal to 2
[/LIST]
7x^2 + 5x = 2
[U]This is a quadratic equation. To get it into standard form, we subtract 2 from each side:[/U]
7x^2 + 5x - 2 = 2 - 2
7x^2 + 5x - 2 = 0
[URL='https://www.mathcelebrity.com/quadratic.php?num=7x%5E2%2B5x-2%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Type this problem into our search engine[/URL], and we get two solutions:
[LIST=1]
[*]x = 2/7
[*]x= -1
[/LIST]
The problem asks for an integer, so our answer is x[B] = -1[/B].
[U]Let's check our work by plugging x = -1 into the quadratic:[/U]
7x^2 + 5x - 2 = 0
7(-1)^2 + 5(-1) - 2 ? 0
7(1) - 5 - 2 ? 0
0 = 0
So we verified our answer, [B]x = -1[/B].

If 72 is added to a number it will be 4 times as large as it was originally

If 72 is added to a number it will be 4 times as large as it was originally
The phrase [I]a number[/I] means an arbitrary variable. Let's call it x.
x
72 added to a number:
x + 72
4 times as large as it was originally means we take the original number x and multiply it by 4:
4x
Now, the phrase [I]it will be[/I] means an equation, so we set x + 72 equal to 4x to get our final algebraic expression:
[B]x + 72 = 4x[/B]
[B][/B]
If the problem asks you to solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=x%2B72%3D4x&pl=Solve']type this equation into our search engine[/URL] and we get:
x = [B]24[/B]

If 800 feet of fencing is available, find the maximum area that can be enclosed.

If 800 feet of fencing is available, find the maximum area that can be enclosed.
Perimeter of a rectangle is:
2l + 2w = P
However, we're given one side (length) is bordered by the river and the fence length is 800, so we have:
So we have l + 2w = 800
Rearranging in terms of l, we have:
l = 800 - 2w
The Area of a rectangle is:
A = lw
Plug in the value for l in the perimeter into this:
A = (800 - 2w)w
A = 800w - 2w^2
Take the [URL='https://www.mathcelebrity.com/dfii.php?term1=800w+-+2w%5E2&fpt=0&ptarget1=0&ptarget2=0&itarget=0%2C1&starget=0%2C1&nsimp=8&pl=1st+Derivative']first derivative[/URL]:
A' = 800 - 4w
Now set this equal to 0 for maximum points:
4w = 800
[URL='https://www.mathcelebrity.com/1unk.php?num=4w%3D800&pl=Solve']Typing this equation into the search engine[/URL], we get:
w = 200
Now plug this into our perimeter equation:
l = 800 - 2(200)
l = 800 - 400
l = 400
The maximum area to be enclosed is;
A = lw
A = 400(200)
A = [B]80,000 square feet[/B]

If 9 is added to 1/3 of a number, the result is 15. What is the number?

If 9 is added to 1/3 of a number, the result is 15. What is the number?
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
1/3 of a number means we multiply x by 1/3:
x/3
9 is added to 1/3 of a number:
x/3 + 9
The phrase [I]the result is[/I] means an equation. so we set x/3 + 9 equal to 15
x/3 + 9 = 15
To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=x%2F3%2B9%3D15&pl=Solve']type it in our search engine[/URL] and we get:
x = [B]18[/B]

if 9 times a number is decreased by 6, the result is 111

if 9 times a number is decreased by 6, the result is 111
A number means an arbitrary variable, let's call it x.
9 times a number:
9x
Decreased by 6
9x - 6
The result is 11, this means we set 9x - 6 equal to 11
[B]9x - 6 = 11
[/B]
To solve this equation for x, use our [URL='http://www.mathcelebrity.com/1unk.php?num=9x-6%3D11&pl=Solve']equation calculator[/URL]

if a number is added to its square, it equals 20

if a number is added to its square, it equals 20.
Let the number be an arbitrary variable, let's call it n.
The square of the number means we raise n to the power of 2:
n^2
We add n^2 to n:
n^2 + n
It equals 20 so we set n^2 + n equal to 20
n^2 + n = 20
This is a quadratic equation. So [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn%3D20&pl=Solve+Quadratic+Equation&hintnum=+0']we type this equation into our search engine[/URL] to solve for n and we get two solutions:
[B]n = (-5, 4)[/B]

if a number is decreased by 5, and then the result is multiplied by 2, the result is 26

If a number is decreased by 5, and then the result is multiplied by 2, the result is 26
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
[I]Decreased by[/I] means we subtract 5 from x:
x - 5
Multiply the result by 2:
2(x - 5)
The result is 26 means we set 2(x - 5) equal to 26:
[B]2(x - 5) = 26[/B]

if a number is tripled the result is 60

if a number is tripled the result is 60
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
Triple the number means we multiply by 3:
3x
The phrase [I]the result is[/I] means an equation, so we set 3x equal to 60:
[B]3x = 60 <-- This is our algebraic expression
[/B]
If you want to solve this equation, then [URL='https://www.mathcelebrity.com/1unk.php?num=3x%3D60&pl=Solve']you type in 3x = 60 into the search engine[/URL] and get:
x = 20

if a train travels at 80 mph for 15 mins, what is the distance traveled?

if a train travels at 80 mph for 15 mins, what is the distance traveled?
Let d = distance, r = rate, and t = time, we have the distance equation:
D = rt
Plugging in our values for r and t, we have:
D = 80mph * 15 min
Remember our speed is in miles per hour, so 15 min equal 1/4 of an hour
D = 80mph * 1/4
D = [B]20 miles[/B]

If a=-9 and b=-6, show that (a-b) unequal (b-a)

If a=-9 and b=-6, show that (a-b) unequal (b-a)
[U]a - b:[/U]
a - b = -9 - -6
a - b = -9 + 6
a - b = -3
[U]b - a:[/U]
b - a = -6 - -9
b - a = -6 + 9
b - a = 3
[B]Since -3 <> 3, then a - b <> b - a[/B]

If Distance equals Speed times Time (D = S x T), then what does time equal in terms of speed and dis

If Distance equals Speed times Time (D = S x T), then what does time equal in terms of speed and distance?
Divide each side by S to isolate T:
D/S = S x T/S
Cancel the S's on the right side:
[B]T = D/S[/B]

if f(x)=-5x+11 and f(n)=21 what does n equal

if f(x)=-5x+11 and f(n)=21 what does n equal
f(n) = -5(n) + 11
Since f(n) = 21, we have:
-5(n) + 11 = 21
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=-5n%2B11%3D21&pl=Solve']equation solver[/URL], we get [B]n = -2[/B].

If from twice a number you subtract four, the difference is twenty

The phrase [I]a number[/I] means an arbitrary variable. We can pick any letter a-z except for i and e.
Let's choose x.
Twice a number means we multiply x by 2:
2x
Subtract four:
2x - 4
The word [I]is [/I]means equal to. We set 2x - 4 equal to 20 for our algebraic expression:
[B]2x - 4 = 20
[/B]
If the problem asks you to solve for x:
we [URL='https://www.mathcelebrity.com/1unk.php?num=2x-4%3D20&pl=Solve']plug this equation into our calculator [/URL]and get x = [B]12[/B]

If n is odd, then 3n + 2 is odd

Look at the Contrapositive: If n is even, then 3n + 2 is even...
Suppose that the conclusion is false, i.e., that n is even.
Then n = 2k for some integer k.
Then we have:
3n + 2 = 3(2k) + 2
3n + 2 = 6k + 2
3n + 2 = 2(3k + 1).
Thus 3n + 2 is even, because it equals 2j for an integer j = 3k + 1.
So 3n + 2 is not odd.
We have shown that ¬(n is odd) ? ¬(3n + 2 is odd),
therefore, the contrapositive (3n + 2 is odd) ? (n is odd) is also true.

If p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equ

If p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equal to 2.
We set up the variation equation with a constant k such that:
p = k/q^2 [I](inversely proportional means we divide)
[/I]
When q is 4 and p is 2, we have:
2 = k/4^2
2 = k/16
Cross multiply:
k = 2 * 16
k = 32
Now, the problem asks for p when q = 2:
p = 32/2^2
p = 32/4
p = [B]8
[MEDIA=youtube]Mro0j-LxUGE[/MEDIA][/B]

If sin(26)=x what does cos(64) equal?

If sin(26)=x what does cos(64) equal?
Using our cofunction calculator, we see the cofunction of sin(26) = cos(64).
Therefore, sin(26) = cos(64), so cos(64) = [B]x[/B]

If the difference of a number and 4 is multiplied by 3 the result is 19

If the difference of a number and 4 is multiplied by 3 the result is 19
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
The difference of a number and 4:
x - 4
The phrase [I]is multiplied by[/I] means we multiply x - 4 by 3:
3(x - 4)
The phrase [I]the result is[/I] means equals, so we set 3(x - 4) equal to 19
[B]3(x - 4) = 19
[MEDIA=youtube]Q8bnVJuWeVk[/MEDIA][/B]

If there are 36 candies and they are distributed equally among 6 children, how many candies will eac

If there are 36 candies and they are distributed equally among 6 children, how many candies will each child get?
36 candies / 6 children = [B]6 candies per child[/B]

If thrice a number is increased by 11,the result is 35. What is the number

If thrice a number is increased by 11,the result is 35. What is the number?
[LIST]
[*]The phrase [I]a number [/I]means an arbitrary variable. Let's call it x.
[*]Thrice means multiply by 3, so we have 3x
[*]Increased by 11 means we add 11, so we have 3x + 11
[*]The [I]result is[/I] means an equation, so we set 3x + 11 equal to 35
[/LIST]
3x + 11 = 35 <-- This is our algebraic expression
The problem ask us to solve the algebraic expression.
[URL='https://www.mathcelebrity.com/1unk.php?num=3x%2B11%3D35&pl=Solve']Typing this problem into our search engine[/URL], we get [B]x = 8[/B].

If twice a number is divided by 7, the result is -28

If twice a number is divided by 7, the result is -28.
The phrase [I]a number[/I] means an arbitrary variable, let's call it "x".
Twice x means we multiply x by 2: 2x
Divide this by 7: 2x/7
We set this equal to -28, and we have our algebraic expression:
[B]2x/7 = -28 [/B]

if two angles are supplementary and congruent then they are right angles

if two angles are supplementary and congruent then they are right angles
Let the first angle be x. Let the second angle be y.
Supplementary angles means their sum is 180:
x + y = 180
We're given both angles are congruent, meaning equal. So we set x = y:
y + y = 180
To solve for y, we [URL='https://www.mathcelebrity.com/1unk.php?num=y%2By%3D180&pl=Solve']type this equation into our search engine[/URL] and we get:
y = [B]90. <-- 90 degrees is a right angle, so this is TRUE[/B]

If two consecutive even numbers are added, the sum is equal to 226. What is the smaller of the two n

If two consecutive even numbers are added, the sum is equal to 226. What is the smaller of the two numbers?
Let the smaller number be n.
The next consecutive even number is n + 2.
Add them together to equal 226:
n + n + 2 = 226
Solve for [I]n[/I] in the equation n + n + 2 = 226
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 1)n = 2n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2n + 2 = + 226
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 2 and 226. To do that, we subtract 2 from both sides
2n + 2 - 2 = 226 - 2
[SIZE=5][B]Step 4: Cancel 2 on the left side:[/B][/SIZE]
2n = 224
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2n/2 = 224/2
n = [B]112
[URL='https://www.mathcelebrity.com/1unk.php?num=n%2Bn%2B2%3D226&pl=Solve']Source[/URL][/B]

If x=5 and y=6-x, what does y equal?

If x=5 and y=6-x, what does y equal?
For x = 5, we have:
y = 6 - 5
y = [B]1[/B]

If y varies inversely as X and Y equals 5 when x equals 2 find X when Y is 4

If y varies inversely as X and Y equals 5 when x equals 2 find X when Y is 4.
Using our [URL='http://www.mathcelebrity.com/variation.php?var1=y&cmeth=varies+inversely+as&var2=x&init1=y%3D5&init2=x%3D2&g1=y%3D4&pl=Calculate+Variation']inverse variation calculator[/URL], we get x = 2.5

If y=2x and y=18, what is the value of x

If y=2x and y=18, what is the value of x
Since y = 2x [B]and[/B] y = 18, we set 2x equals to 18 since they both equal y
2x = 18
[URL='https://www.mathcelebrity.com/1unk.php?num=2x%3D18&pl=Solve']Type this equation into our search engine[/URL] and we get:
x = [B]9[/B]

if you add 35 to twice a number, the result is 17. What is the number?

if you add 35 to twice a number, the result is 17. What is the number?
A number is represented by a variable, let's call it "x".
Twice a number means we multiply by 2 --> 2x
Add 35
2x + 35
Now set that entire expression equal to 17
2x + 35 = 17
[URL='http://www.mathcelebrity.com/1unk.php?num=2x%2B35%3D17&pl=Solve']Plug that into the search engine to solve for x[/URL]
[B]x = -9[/B]

If you are running 6 miles per hour, then it takes you 10 minutes to run 1 mile. If you are running

If you are running 6 miles per hour, then it takes you 10 minutes to run 1 mile. If you are running 8 miles per hour, it takes you 7.5 minutes to run a mile. What does your speed have to be for your speed in miles per hour to be equal to your mile time in minutes?
From above, we have:
[LIST]
[*]6mph x 10 minutes = 1 mile
[*]8mph x 7.5 minutes = 1 mile
[/LIST]
Notice that mph x minutes = 60 since there are 60 minutes in 1 hour?
So we have x mph x y minutes = 60.
Since we want mph and y (minutes) = x (mph), we have
x^2 = 60
x = sqrt(60)
[B]x = 7.746 mph[/B]

If you buy a computer directly from the manufacturer for $3,509 and agree to repay it in 36 equal in

If you buy a computer directly from the manufacturer for $3,509 and agree to repay it in 36 equal installments at 1.73% interest per month on the unpaid balance, how much are your monthly payments? How much total interest will be paid?
[U]Determine the monthly payment[/U]
The monthly payment is [B]$114.87[/B] using our [URL='http://www.mathcelebrity.com/annimmpv.php?pv=3059&av=&pmt=&n=36&i=1.73&check1=1&pl=Calculate']annuity calculator[/URL]
[U]Determine the total payments made[/U]
Total payment is 36 times $114.87 = $4,135.37
[U]Now determine the total interest paid[/U]
Take the total payments of $4,135.37 and subtract the original loan of $3,059 to get interest paid of [B]$1,076.37[/B]

If you have $272, and you spend $17 each day, how long would it be until you had no money left?

If you have $272, and you spend $17 each day, how long would it be until you had no money left?
Let d be the number of days. We have a balance expression of:
272 - 17d
We want to know when the balance is 0, so we set 272 - 17d equal to 0.
272 - 17d = 0
To solve for d, we [URL='http://272 - 17d = 0']type this equation into our search engine[/URL] and we get:
d = [B]16[/B]

If you multiply me by 33 and subtract 20, the result is 46. Who am I?

If you multiply me by 33 and subtract 20, the result is 46. Who am I?
[LIST]
[*]Start with the variable x
[*]Multiply me by 33 = 33x
[*]Subtract 20: 33x - 20
[*]The result is 46, means we set this expression equal to 46: 33x - 20 = 46
[/LIST]
Run this through our [URL='http://www.mathcelebrity.com/1unk.php?num=33x-20%3D46&pl=Solve']equation calculator[/URL], and we get:
[B]x = 2[/B]

In 8 years kelly's age will be twice what it is now. How old is kelly?

In 8 years kelly's age will be twice what it is now. How old is kelly?
Let Kelly's age be a.
In 8 years means we add 8 to a:
a + 8
Twice means we multiply a by 2:
2a
The phrase [I]will be[/I] means equal to, so we set a + 8 equal to 2a
a + 8 = 2a
To solve this equation, we [URL='https://www.mathcelebrity.com/1unk.php?num=a%2B8%3D2a&pl=Solve']type it in our math engine[/URL] and we get:
a = [B]8
[/B]
[U]Evaluate a = 8 and check our work[/U]
8 + 8 ? 2(8)
16 = 16
[MEDIA=youtube]y4jaQpkaJEw[/MEDIA]

In rolling a die, the event E is getting a number greater than or equal to 3. What is the complement

In rolling a die, the event E is getting a number greater than or equal to 3. What is the complement of the event?
The complement E' is everything but the event. So we have:
E = P(n >= 3)
E' = [B]P(n < 3)[/B]

Ina has $40 in her bank account and saves $8 a week. Ree has $200 in her bank account and spends $12

Ina has $40 in her bank account and saves $8 a week. Ree has $200 in her bank account and spends $12 a week. Write an equation to represent each girl.
Let w equal the number of weeks, and f(w) be the amount of money in the account after w weeks:
[LIST]
[*]Ina: [B]f(w) = 40 + 8w[/B]
[LIST]
[*]We add because Ina saves money, so her account grows
[/LIST]
[*]Ree: [B]f(w) = 200 - 12w[/B]
[LIST]
[*]We subtract because Ree saves
[/LIST]
[/LIST]

Interval Notation and Set Builder Notation

Free Interval Notation and Set Builder Notation Calculator - This calculator translates the following inequality statements to interval notation and set builder notation:

* x < 5

* y <= 5

* z > 5

* a >= 5

* b < 5 or b > 20

* Compound Inequality such as 0 <= c < 4

* |x|<3

* Reverse Interval Notation to Inequality Statement such as (-7,5]

* {x|x<1}

* Word representations of interval notations such as 2 is less than or equal to x is less than or equal to 8

* x < 5

* y <= 5

* z > 5

* a >= 5

* b < 5 or b > 20

* Compound Inequality such as 0 <= c < 4

* |x|<3

* Reverse Interval Notation to Inequality Statement such as (-7,5]

* {x|x<1}

* Word representations of interval notations such as 2 is less than or equal to x is less than or equal to 8

Is 20% equivalent to 2/5?

Is 20% equivalent to 2/5?
Let's compare fractions to fractions:
20% equals 1/5 from our [URL='https://www.mathcelebrity.com/perc.php?num=+5&den=+8&num1=+16&pct1=+80&pct2=+90&den1=+80&pct=20&pcheck=4&decimal=+65.236&astart=+12&aend=+20&wp1=20&wp2=30&pl=Calculate']percentage-decimal-fraction calculator[/URL].
1/5 < 2/5 so these fractions are [B][I]not equivalent[/I][/B].

Is it correct to word "10% * 50 + 50" as "10% upper 50"?

Yes, it's close to the upper bound. I just wonder what we interpret the below inequality is?
y > 10% x + x
My two cents: y is greater than 10% upper x
What do you say?

Is the point (4,7) a solution of the equation yequals15xminus8?

Is the point (4,7) a solution of the equation y equals 15x minus 8?
Plug in x = 4:
15(4) - 8
60 - 8
52
Since 52 <> 4, (4,7) is [U][B]not[/B][/U] a solution of the equation y equals 15x minus 8

Isabel earns $7.50 per hour on the weekends. Write and solve an inequality to find how many hours sh

Isabel earns $7.50 per hour on the weekends. Write and solve an inequality to find how many hours she needs to work to earn at least $120.
A few things to note:
[LIST]
[*]Earnings = Rate * time
[*]Let h be the number of hours worked
[*]The phrase [I]at least[/I] means greater than or equal to, so we have the following inequality.
[/LIST]
We represent this with the following inequality:
7.5h < 120
To solve this inequality for h, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=7.5h%3C120&pl=Show+Interval+Notation']type it into our math engine[/URL] and we get:
[B]h < 16[/B]

Isabel is making face mask. She spends $50 on supplies and plans on selling them for $4 per mask. Ho

Isabel is making face mask. She spends $50 on supplies and plans on selling them for $4 per mask. How many mask does have to make in order to make a profit equal to $90?
[U]Set up the cost function C(m) where m is the number of masks:[/U]
C(m) = supply cost
C(m) = 50
[U]Set up the cost function R(m) where m is the number of masks:[/U]
R(m) = Sale Price * m
R(m) = 4m
[U]Set up the profit function P(m) where m is the number of masks:[/U]
P(m) = R(m) - C(m)
P(m) = 4m - 50
The problems asks for profit of 90, so we set P(m) = 90:
4m - 50 = 90
To solve this equation for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=4m-50%3D90&pl=Solve']type it in our search engine[/URL] and we get:
m = [B]35[/B]

Isabel will run less than 36 minutes today. So far, she has run 22 minutes. What are the possible nu

Isabel will run less than 36 minutes today. So far, she has run 22 minutes. What are the possible numbers of additional minutes she will run?
Set up our inequality. If she ran 22 minutes, we need to find an expression to find out the remaining minutes
x + 22 < 36
Subtract 22 from each side:
x < 14
Remember, she cannot run negative minutes, so our lower bound is 0, so we have:
[B]0 < x < 14
[/B]

Jack is making snack bags. He has 18 baby carrots and 42 pretzels. He wants to divide them equally a

Jack is making snack bags. He has 18 baby carrots and 42 pretzels. He wants to divide them equally among the bags. What is the greatest number of snack bags he can make?
Find the [URL='http://www.mathcelebrity.com/gcflcm.php?num1=18&num2=42&num3=&pl=GCF']Greatest Common Factor[/URL] of (18, 42) = 6
6 bags for 18 carrots = 3 carrots per bag
6 bags for 42 pretzels = 7 pretzels per bag
[B]6 bags is the answer[/B]

Jake Callahan stars in the new TV series Stone Simons: Kid Astronaut. The day after the first episod

Jake Callahan stars in the new TV series Stone Simons: Kid Astronaut. The day after the first episode airs, Jake receives a bunch of fan mail. He splits all the letters into 3 equal stacks to open with his mom and his sister. Each stack contains 21 letters. Which equation can you use to find the number of letters n Jake receives?
The number of letters n is represented by number of stacks (s) times letter per stack (l). We're given s = 3 and l = 21, so we have:
n = 21(3)
n = [B]63[/B]

Jamie spent $15.36 on several items at the store. he spent an equal amount on each item. if jamie sp

Jamie spent $15.36 on several items at the store. he spent an equal amount on each item. if jamie spent $1.92 on each item, how many items did he buy?
Let x equal the number of items Jamie bought. We have:
1.92x = 15.36
Divide each side by 1.92
[B]x = 8[/B]

jane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an ine

jane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an inequality to show how many sodas he can buy.
Let s be the number of sodas.
Cost for the day is:
Price per soda * s + Admission Price
4.25s + 42
We're told that Jane has 55, which means Jane cannot spend more than 55. Jane can spend up to or less than 55. We write this as an inequality using <= 55
[B]4.25s + 42 <= 55[/B]
[B][/B]
If the problems asks you to solve for s, we type it in our math engine and we get:
Solve for [I]s[/I] in the inequality 4.25s + 42 ? 55
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 42 and 55. To do that, we subtract 42 from both sides
4.25s + 42 - 42 ? 55 - 42
[SIZE=5][B]Step 2: Cancel 42 on the left side:[/B][/SIZE]
4.25s ? 13
[SIZE=5][B]Step 3: Divide each side of the inequality by 4.25[/B][/SIZE]
4.25s/4.25 ? 13/4.25
[B]s ? 3.06[/B]

Jason has an equal number of nickels and dimes. The total value of his nickels and dimes is $2.25. H

Jason has an equal number of nickels and dimes. The total value of his nickels and dimes is $2.25. How many nickels does Jason have?
Let the number of nickels be n
Let the number of dimes be d
We're given two equations:
[LIST=1]
[*]d = n
[*]0.05n + 0.1d = 2.25
[/LIST]
Substitute equation (1) for d into equation (2):
0.05n + 0.1n = 2.25
Solve for [I]n[/I] in the equation 0.05n + 0.1n = 2.25
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(0.05 + 0.1)n = 0.15n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
0.15n = + 2.25
[SIZE=5][B]Step 3: Divide each side of the equation by 0.15[/B][/SIZE]
0.15n/0.15 = 2.25/0.15
n = [B]15[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=0.05n%2B0.1n%3D2.25&pl=Solve']Source[/URL]

Jason is 9 miles ahead of Joe running at 5.5 miles per hour and Joe is running at the speed of 7 mil

[SIZE=6]Jason is 9 miles ahead of Joe running at 5.5 miles per hour and Joe is running at the speed of 7 miles per hour. How long does it take Joe to catch Jason?
A. 3 hours
B. 4 hours
C. 6 hours
D. 8 hours
Distance formula is d = rt
Jason's formula (Add 9 since he's ahead 9 miles):
d = 5.5t + 9
Joe's formula:
d = 7t
Set both distance formulas equal to each other:
5.5t + 9 = 7t
Subtract 5.5t from each side:
5.5t - 5.5t + 9 = 7t - 5.5t
1.5t = 9
Divide each side by 1.5:
1.5t/1.5 = 9/1.5
t = [B]6 hours[/B]
[U]Check our work with t = 6[/U]
Joe = 7(6) = 42
Jason = 5.5(6) + 9= 33 + 9 = 42
[MEDIA=youtube]qae3WCq9wzM[/MEDIA]
[/SIZE]

Jay earns S amount per day for working in a company. His total expenses per day is equal to the amou

Jay earns S amount per day for working in a company. His total expenses per day is equal to the amount E. Write an expression to show how much he earned per day in a month. Suppose he is working for 20 days per month.
[LIST=1]
[*]Each day, Jay earns a profit of S - E.
[*]For one month (30 days), he earns 30(S - E)
[*]For 20 working days in a month, he earns 20(S - E)
[/LIST]

Jennifer is playing cards with her bestie when she draws a card from a pack of 25 cards numbered fro

Jennifer is playing cards with her bestie when she draws a card from a pack of 25 cards numbered from 1 to 25. What is the probability of drawing a number that is square?
The squares from 1 - 25 less than or equal to 25 are as follows:
[LIST=1]
[*]1^2 = 1
[*]2^2 = 4
[*]3^2 = 9
[*]4^2 = 16
[*]5^2 = 25
[/LIST]
So the following 5 cards are squares:
{1, 4, 9, 16, 25}
Therefore, our probability of drawing a square is:
P(square) = Number of Squares / Number of Cards
P(square) = 5/25
This fraction can be simplified. So [URL='https://www.mathcelebrity.com/fraction.php?frac1=5%2F25&frac2=3%2F8&pl=Simplify']we type in 5/25 into our search engine, choose simplify[/URL], and we get:
P(square) = [B]1/5[/B]

Jenny has $1200 and is spending $40 per week. Kelsey has $120 and is saving $50 a week. In how many

Jenny has $1200 and is spending $40 per week. Kelsey has $120 and is saving $50 a week. In how many weeks will Jenny and Kelsey have the same amount of money?
Jenny: Let w be the number of weeks. Spending means we subtract, so we set up a balance equation B(w):
B(w) = 1200 - 40w
Kelsey: Let w be the number of weeks. Saving means we add, so we set up a balance equation B(w):
B(w) = 120 + 50w
When they have the same amount of money, we set the balance equations equal to each other:
1200 - 40w = 120 + 50w
To solve for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=1200-40w%3D120%2B50w&pl=Solve']type this equation into our search engine[/URL] and we get:
w = [B]12[/B]

Jill made 122 muffins. She put them into 3 boxes and has two muffins left. How many are in each box

Jill made 122 muffins. She put them into 3 boxes and has two muffins left. How many are in each box if they all contain the same amount of muffins?
Let m equal the number of muffins per box.
We're told that we have 3 boxes and 2 muffins left after filling up all 3 boxes.
3m + 2 = 122
To solve for m, we subtract 2 from each side:
3m + 2 - 2 = 122 - 2
Cancel the 2's on the left side and we get:
3m = 120
Divide each side by 3 to isolate m:
3m/3 = 120/3
Cancel the 3's on the left side and we get:
m = [B]40[/B]

Jim has $440 in his savings account and adds $12 per week to the account. At the same time, Rhonda h

Jim has $440 in his savings account and adds $12 per week to the account. At the same time, Rhonda has $260 in her savings account and adds $18 per week to the account. How long will it take Rhonda to have the same amount in her account as Jim?
[U]Set up Jim's savings function S(w) where w is the number of weeks of savings:[/U]
S(w) = Savings per week * w + Initial Savings
S(w) = 12w + 440
[U]Set up Rhonda's savings function S(w) where w is the number of weeks of savings:[/U]
S(w) = Savings per week * w + Initial Savings
S(w) = 18w + 260
The problems asks for w where both savings functions equal each other:
12w + 440 = 18w + 260
To solve for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=12w%2B440%3D18w%2B260&pl=Solve']type this equation into our math engine[/URL] and we get:
w = [B]30[/B]

Jim was thinking of a number. Jim adds 20 to it, then doubles it and gets an answer of 99.2. What wa

Jim was thinking of a number. Jim adds 20 to it, then doubles it and gets an answer of 99.2. What was the original number?
Start with x.
Add 20 to it
x + 20
Double it
2(x + 20)
Set this equal to 99.2
2(x + 20) = 99.2
Divide each side by 2:
x + 20 = 49.6
Subtract 20 from each side:
x = [B]29.6[/B]

Jinas final exam has true/false questions, worth 3 points each, and multiple choice questions, worth

Jinas final exam has true/false questions, worth 3 points each, and multiple choice questions, worth 4 points each. Let x be the number of true/false questions she gets correct, and let y be the number of multiple choice questions she gets correct. She needs at least 76 points on the exam to get an A in the class. Using the values and variables given, write an inequality describing this.
At least means greater than or equal to, so we have:
[B]3x + 4y >= 76[/B]

Jody is buying a scrapbook and sheets of designer paper. She has $40 and needs at least $18.25 to bu

Jody is buying a scrapbook and sheets of designer paper. She has $40 and needs at least $18.25 to buy the scrapbook. Each sheet of paper costs $0.34. How many sheets of paper can she buy?
Set up a cost equation for the number of pieces of paper (p):
0.34p + 18.25 <= 40 <-- we have an inequality since we can't go over 40
[URL='https://www.mathcelebrity.com/1unk.php?num=0.34p%2B18.25%3C%3D40&pl=Solve']Type this inequality into our search engine[/URL] and we get:
p <= 63.97
We round down, so we get p = [B]63[/B].

Joe opens a bank account that starts with $20 and deposits $10 each week. Bria has a different accou

Joe opens a bank account that starts with $20 and deposits $10 each week. Bria has a different account that starts with $1000 but withdraws $15 each week. When will Joe and Bria have the same amount of money?
Let w be the number of weeks. Deposits mean we add money and withdrawals mean we subtract money.
[U]Joe's Balance function B(w) where w is the number of weeks:[/U]
20 + 10w
[U]Bria's Balance function B(w) where w is the number of weeks:[/U]
1000 - 15w
[U]The problem asks for when both balances will be the same. So we set them equal to each other and solve for w:[/U]
20 + 10w = 1000 - 15w
To solve for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=20%2B10w%3D1000-15w&pl=Solve']type this equation into our search engine[/URL] and we get:
w = 39.2
We round up to full week and get:
w = [B]40[/B]

Joel bought 88 books. Some books cost $13 each and some cost $17 each. In all, he spent $128. Which

Joel bought 88 books. Some books cost $13 each and some cost $17 each. In all, he spent $128. Which system of linear equations represents the given situation?
Let a be the number of the $13 book, and b equal the number of $17 books. We have the following system of linear equations:
[LIST=1]
[*][B]a + b = 88[/B]
[*][B]13a + 17b = 128[/B]
[/LIST]
To solve this system, use our calculator for the following methods:
[LIST]
[*][URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+b+%3D+88&term2=13a+%2B+17b+%3D+128&pl=Substitution']Substitution[/URL]
[*][URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+b+%3D+88&term2=13a+%2B+17b+%3D+128&pl=Elimination']Elimination[/URL]
[*][URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=a+%2B+b+%3D+88&term2=13a+%2B+17b+%3D+128&pl=Cramers+Method']Cramers Method[/URL]
[/LIST]

Joey and Romnick play in the same soccer team. Last Saturday, Romnick scored 3 more goals than Joey,

Joey and Romnick play in the same soccer team. Last Saturday, Romnick scored 3 more goals than Joey,but together they scored less than 9 goals. What are the possible number of goal Romnick scored?
Let j be Joey's goals
Let r by Romnick's goals
We're given 1 equation and 1 inequality:
[LIST=1]
[*]r = j + 3
[*]r + j < 9
[/LIST]
Rearranging equation 1 for j, we have:
[LIST=1]
[*]j = r - 3
[*]r + j < 9
[/LIST]
Substitute equation (1) into inequality (2) for j:
r + r - 3 < 9
2r - 3 < 9
[URL='https://www.mathcelebrity.com/1unk.php?num=2r-3%3C9&pl=Solve']Typing this inequality into our math engine[/URL], we get:
[B]r < 6[/B]

Jonathan earns a base salary of $1500 plus 10% of his sales each month. Raymond earns $1200 plus 15%

Jonathan earns a base salary of $1500 plus 10% of his sales each month. Raymond earns $1200 plus 15% of his sales each month. How much will Jonathan and Raymond have to sell in order to earn the same amount each month?
[U]Step 1: Set up Jonathan's sales equation S(m) where m is the amount of sales made each month:[/U]
S(m) = Commission percentage * m + Base Salary
10% written as a decimal is 0.1. We want decimals to solve equations easier.
S(m) = 0.1m + 1500
[U]Step 2: Set up Raymond's sales equation S(m) where m is the amount of sales made each month:[/U]
S(m) = Commission percentage * m + Base Salary
15% written as a decimal is 0.15. We want decimals to solve equations easier.
S(m) = 0.15m + 1200
[U]The question asks what is m when both S(m)'s equal each other[/U]:
The phrase [I]earn the same amount [/I]means we set Jonathan's and Raymond's sales equations equal to each other
0.1m + 1500 = 0.15m + 1200
We want to isolate m terms on one side of the equation.
Subtract 1200 from each side:
0.1m + 1500 - 1200 = 0.15m + 1200 - 1200
Cancel the 1200's on the right side and we get:
0.1m - 300 = 0.15m
Next, we subtract 0.1m from each side of the equation to isolate m
0.1m - 0.1m + 300 = 0.15m - 0.1m
Cancel the 0.1m terms on the left side and we get:
300 = 0.05m
Flip the statement since it's an equal sign to get the variable on the left side:
0.05m = 300
To solve for m, we divide each side of the equation by 0.05:
0.05m/0.05 = 300/0.05
Cancelling the 0.05 on the left side, we get:
m = [B]6000[/B]

Jordan has already scored 153 points this basketball season. If he scores 17 points per game, which

Jordan has already scored 153 points this basketball season. If he scores 17 points per game, which inequality represents the number of addional games he needs to play in order to score at least 255 points for the season?
Let g be the number of games Jordan plays. Total points per game is 17g. And he’s already scored 153. So we need 17g + 153 to be [I]at least [/I]255. The phrase at least means greater than or equal to, so we use the >= operator for our inequality:
[B]17g + 153 >= 255[/B]

Jose has scored 556 points on his math tests so far this semester. To get an A for the semester, he

Jose has scored 556 points on his math tests so far this semester. To get an A for the semester, he must score at least 660 points. Write and solve an inequality to find the minimum number of points he must score on the remaining tests, n, in order to get an A.
We want to know n, such that 556 + n >= 660. <-- We use >= symbol since at least means greater than or equal to.
556 + n >= 660
Use our [URL='http://www.mathcelebrity.com/1unk.php?num=556%2Bn%3E%3D660&pl=Solve']equation/inequality calculator[/URL], we get [B]n >= 104[/B]

JP's age is twice the age of Reyna. The sum of their ages does not exceed 51

JP's age is twice the age of Reyna. The sum of their ages does not exceed 51
Let JP's age be j. Let Reyna's age be r. We're given two expressions:
[LIST=1]
[*]w = 2r
[*]r + w <= 51. ([I]Does not exceed means less than or equal to)[/I]
[/LIST]
We substitute (1) into (2) for w to get the inequality:
r + 2r <= 51
To solve this inequality, we type it in our search engine and we get:
[B]r <= 17[/B]

Juan has d dimes and q quarters in his pocket. The total value of the coins is less than $14.75. Whi

Juan has d dimes and q quarters in his pocket. The total value of the coins is less than $14.75. Which inequality models this situation?
[U]Let d be the number of dimes and q be the number of quarters[/U]
[B]0.1d + 0.25q < 14.75[/B]

Juan has d dimes and q quarters in his pocket. The total value of the coins is less than $14.75. Whi

Juan has d dimes and q quarters in his pocket. The total value of the coins is less than $14.75. Which inequality models this situation?
Since dimes are worth $0.10 and quarters are worth $0.25, we have:
[B]0.10d + 0.25q < 14.75[/B]

Juan spent at most $2.50 on apples and oranges. He bought 5 apples at $0.36 each. What is the most h

Juan spent at most $2.50 on apples and oranges. He bought 5 apples at $0.36 each. What is the most he spent on oranges?
Let a be spending apples and o be spending on oranges, we have:
[LIST=1]
[*]a + o <= 2.36 <-- At most means less than or equal to
[*]a = 5 * 0.36 = 1.8
[/LIST]
Substitute (2) into (1)
1.8 + o <= 2.36
Subtract 1.8 from each side
[B]o <= 0.56[/B]

Juliana answered 42 times correctly on an 80-item quiz, while her sister Angela answered 21 items co

Juliana answered 42 times correctly on an 80-item quiz, while her sister Angela answered 21 items correctly on a 40-item quiz. Do they have the same portion of correct answers?
Let's compare based on correct answers to questions:
Juliana = 42/80 = 0.525
Angela = 21/40 = 0.525
So yes, they do have the same portion of correct answers.
But there's another way to solve this:
[LIST=1]
[*]Divide Juliana's the top and bottom of Juliana's fraction by 2.
[*]We picked 2 as a GCF shown in our calculator.
[*]Type [URL='https://www.mathcelebrity.com/gcflcm.php?num1=42&num2=80&num3=&pl=GCF']GCF of 42 and 80[/URL].
[/LIST]
Divide top and bottom of Juliana's fraction by the GCF of 2
42/2 = 80/2 = 21/40
This ratio equals Angela's.

Julie has $300 to plan a dance. There is a one-time fee of $75 to reserve a room. It also costs $1.5

Julie has $300 to plan a dance. There is a one-time fee of $75 to reserve a room. It also costs $1.50 per person for food and drinks. What is the maximum number of people that can come to the dance?
Let each person be p. We have the following relationship for cost:
1.50p + 75 <=300
We use the <= sign since we cannot go over the $300 budget.
[URL='https://www.mathcelebrity.com/1unk.php?num=1.50p%2B75%3C%3D300&pl=Solve']We type this inequality into our search engine[/URL], and we get:
p <= 150
Since we have the equal sign within the inequality, the maximum number of people that can come to the dance is [B]150.[/B]

k equals the sum of h and 23

The sum of h and 23 means we add:
h + 23
k equals means we set our expression above equal to k
h + 23 = k

Karen bought a bucket of popcorn at the movies for $5. She also bought some candy for $2 each. Karen

Karen bought a bucket of popcorn at the movies for $5. She also bought some candy for $2 each. Karen has to spend less than $15 on the popcorn and candy. Which inequality can be used to find c, the number of candies that Karen could have bought?
Since the candy cost is the product of price and quantity, we have:
2c + 5 < 15
To solve this inequality for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=2c%2B5%3C15&pl=Solve']type it in our math engine [/URL]and we get:
[B]c < 5[/B]

Karmen just got hired to work at Walmart. She spent $15 on her new uniform and she gets paid $8 per

Karmen just got hired to work at Walmart. She spent $15 on her new uniform and she gets paid $8 per hour. Write an equation that represents how much money she profits after working for a certain number of hours. How many hours will she have to work for in order to buy a new snowboard ( which costs $450)
Her profit equation P(h) where h is the number of hours worked is:
[B]P(h) = 8h - 15[/B]
Note: [I]We subtract 15 as the cost of Karmen's uniform.
[/I]
Next, we want to see how many hours Karmen must work to buy a new snowboard which costs $450.
We set the profit equation equal to $450
8h - 15 = 450
[URL='https://www.mathcelebrity.com/1unk.php?num=8h-15%3D450&pl=Solve']Typing 8h - 15 = 450 into the search engine[/URL], we get h = 58.13. We round this up to 59 hours.

Kate spent at most $2.50 on apples and oranges. She bought 5 apples at $0.36 each. What is the most

Kate spent at most $2.50 on apples and oranges. She bought 5 apples at $0.36 each. What is the most She spent on oranges
[U]Assumptions and givens:[/U]
[LIST]
[*]Let a be the total cost of apples
[*]Let o be the total cost of oranges
[/LIST]
The phrase [I]at most[/I] means less than or equal to, so we have:
a + o <= 2.50
[U]Find the cost of apples (a)[/U]
a = price per apple * quantity of apples
a = 0.36 * 5
a = 1.8
Our new inequality with a = 1.8 is:
1.8 + o <= 2.50
[URL='https://www.mathcelebrity.com/1unk.php?num=1.8%2Bo%3C%3D2.50&pl=Solve']Typing this inequality into our search engine[/URL], we get:
[B]o <= 0.7[/B]

Kaylee had $197 in her savings now her savings is $429 . How much was her paycheck

Kaylee had $197 in her savings now her savings is $429 . How much was her paycheck
Her paycheck equals the increase in savings from $197 to $429. We want the difference:
Paycheck = Savings Now - Savings Before Paycheck
Paycheck = $429 - $197
Paycheck = [B]$232[/B]

keisha is babysitting at 8$ per hour to earn money for a car. So far she has saved $1300. The car th

keisha is babysitting at 8$ per hour to earn money for a car. So far she has saved $1300. The car that keisha wants to buy costs at least $5440. How many hours does Keisha need to babysit to earn enough to buy the car
Set up the Earning function E(h) where h is the number of hours Keisha needs to babysit:
E(h) = 8h + 1300
The question asks for h when E(h) is at least 5440. The phrase [I]at least[/I] means an inequality, which is greater than or equal to. So we have:
8h + 1300 >= 5440
To solve this inequality, we [URL='https://www.mathcelebrity.com/1unk.php?num=8h%2B1300%3E%3D5440&pl=Solve']type it in our search engine[/URL] and we get:
h >= [B]517.5[/B]

Keith is going to Renaissance Festival with $120 to pay for his admission, food and the cost of game

Keith is going to Renaissance Festival with $120 to pay for his admission, food and the cost of games. He spends a total of $85 on admission and food. Games cost $5 each. Which inequality models the maximum number of games Keith can play.
Let the number of games be g. Keith can spend less than or equal to 120. So we have
[B]5g + 85 <= 120
[/B]
If we want to solve the inequality for g, we [URL='https://www.mathcelebrity.com/1unk.php?num=5g%2B85%3C%3D120&pl=Solve']type it in our search engine[/URL] and we have:
g <= 7

Kellie has only $5.25 to buy breakfast. She wants to buy as many carrot muffins as she can. Each muf

Kellie has only $5.25 to buy breakfast. She wants to buy as many carrot muffins as she can. Each muffin costs $0.75. What’s an equation?
Let m be the number of muffins. Cost equals price * quantity, so we have:
[B]0.75m = 5.25
[/B]
To solve the equation for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.75m%3D5.25&pl=Solve']type the equation into our search engine[/URL] and we get:
m = [B]7[/B]

Kierra had $35 to spend at the movies. If it was $11 to get in and snacks were 2$ each, how many sna

Kierra had $35 to spend at the movies. If it was $11 to get in and snacks were 2$ each, how many snacks could she buy?
Subtract off cover charge:
35 - 11 = 24
Let s equal the number of snacks Kierra can buy. With each snack costing $2, we have the following equation:
2s = 24
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2s%3D24&pl=Solve']equation calculator[/URL], we have:
[B]s = 12[/B]

kim and jason just had business cards made. kim’s printing company charged a one time setup fee of $

kim and jason just had business cards made. kim’s printing company charged a one time setup fee of $8 and then $20 per box of cards. jason,meanwhile ordered his online. they cost $8 per box. there was no setup fee, but he had to pay $20 to have his order shipped to his house. by coincidence, kim and jason ended up spending the same amount on their business cards. how many boxes did each buy? how much did each spend?
Set up Kim's cost function C(b) where b is the number of boxes:
C(b) = Cost per box * number of cards + Setup Fee + Shipping Fee
C(b) = 20c + 8 + 0
Set up Jason's cost function C(b) where b is the number of boxes:
C(b) = Cost per box * number of cards + Setup Fee + Shipping Fee
C(b) = 8c + 0 + 20
Since Kim and Jason spent the same amount, set both cost equations equal to each other:
20c + 8 = 8c + 20
[URL='https://www.mathcelebrity.com/1unk.php?num=20c%2B8%3D8c%2B20&pl=Solve']Type this equation into our search engine[/URL] to solve for c, and we get:
c = 1
How much did they spend? We pick either Kim's or Jason's cost equation since they spent the same, and plug in c = 1:
Kim:
C(1) = 20(1) + 8
C(1) = 20 + 8
C(1) = [B]28
[/B]
Jason:
C(1) = 8(1) + 20
C(1) = 8 + 20
C(1) = [B]28[/B]

kim wants to buy candy for 4 dollars a pound. if she wants to spend less than 20 dollars, how many b

kim wants to buy candy for 4 dollars a pound. if she wants to spend less than 20 dollars, how many bags can she buy
Since cost = price * quantity, we have the following inequality with b as the number of bags:
4b < 20
To solve this inequality for b, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=4b%3C20&pl=Show+Interval+Notation']type it in our search engine[/URL] and we get:
[B]b < 5[/B]

Kimberly is taking three online classes during the summer. She spends 10 hours each week studying fo

Kimberly is taking three online classes during the summer. She spends 10 hours each week studying for her marketing class, 12 hours studying for her statistics class, and 8 hours studying for her business law class. What percent of her study time does she spend for her statistics class?
The percentage equals hours spent on statistics divided by total hours spent studying for everything.
[U]Calculate total study hours:[/U]
Total Study Hours = Marketing Class Study Hours + Statistics Class Study Hours + Business Law Study Hours
Total Study Hours = 10 + 8 + 12
Total Study Hours = [B]30[/B]
[U]Calculate Statistics Study Hours Percentage:[/U]
Statistics Study Hours Percentage = Statistics Class Study Hours / Total Study Hours
Statistics Class Study Hours = 8/30
Using our [URL='https://www.mathcelebrity.com/perc.php?num=8&den=30&pcheck=1&num1=16&pct1=80&pct2=70&den1=80&idpct1=10&hltype=1&idpct2=90&pct=82&decimal=+65.236&astart=12&aend=20&wp1=20&wp2=30&pl=Calculate']fraction to decimal calculator[/URL], we get
Statistics Class Study Hours = [B]26.67%[/B]

kira will spend less than 27 on gifts. so far, she has spent 12$. what are the possible additional a

kira will spend less than 27 on gifts. so far, she has spent 12$. what are the possible additional amounts she will spend?
The key word in this problem is [I]less than[/I]. So we know this is an inequality.
Let s be Kira's possible spend. We have:
s + 12 < 27
To solve for s in this inequality, we subtract 12 from each side:
s + 12 - 12 < 27 - 12
Cancel the 12's on the left side, and we get:
[B]s < 15
[/B]
[I]The summary here is Kira can spend anything up to [U]but not including[/U] 15[/I]

Krutika was thinking of a number. Krutika doubles it and adds 8.7 to get an answer of 64.9. Form an

Krutika was thinking of a number. Krutika doubles it and adds 8.7 to get an answer of 64.9. Form an equation with x from the information.
[LIST=1]
[*]The number we start with is x.
[*]Double it means we multiply by 2: 2x
[*]Add 8.7: 2x + 8.7
[*][I]Get an answer[/I] means we have an equation, so we set (3) above equal to 64.9
[*][B]2x + 8.7 = 64.9[/B]
[/LIST]
If you want to solve for x, use our [URL='http://www.mathcelebrity.com/1unk.php?num=2x%2B8.7%3D64.9&pl=Solve']equation calculator[/URL].

Laura found a roll of fencing in her garage. She couldn't decide whether to fence in a square garden

Laura found a roll of fencing in her garage. She couldn't decide whether to fence in a square garden or a round garden with the fencing.
Laura did some calculations and found that a circular garden would give her 1380 more square feet than a square garden. How many feet of fencing were in the roll that Laura found? (Round to the nearest foot.)
Feet of fencing = n
Perimeter of square garden = n
Each side of square = n/4
Square garden's area = (n/4)^2 = n^2/16
Area of circle garden with circumference = n is:
Circumference = pi * d
n = pi * d
Divide body tissues by pi:
d = n/pi
Radius = n/2pi
Area = pi * n/2pi * n/2pi
Area = pin^2/4pi^2
Reduce by canceling pi:
n^2/4pi
n^2/4 * 3.14
n^2/12.56
The problem says that the difference between the square's area and the circle's area is equal to 1380 square feet.
Area of Circle - Area of Square = 1380
n^2/12.56 - n^2/16 = 1380
Common denominator = 200.96
(16n^2 - 12.56n^2)/200.96 = 1380
3.44n^2/200.96 = 1380
Cross multiply:
3.44n^2 = 277,324.8
n^2 = 80,617.7
n = 283.9
Nearest foot = [B]284[/B]

Laura spent half of her money on a necklace. She spent 14.60 of what was left having dinner with car

Laura spent half of her money on a necklace. She spent 14.60 of what was left having dinner with carolyn. if she had 3.90 left, how much money did she start out with?
Let x equal Laura's starting money
1/2x = 14.60 + 3.90
1/2x = 18.5
Divide each side by 1/2
[B]x = $37[/B]

Lauren wrote a total of 6 pages over 2 hours. How many hours will Lauren have to spend writing this

Lauren wrote a total of 6 pages over 2 hours. How many hours will Lauren have to spend writing this week in order to have written a total of 9 pages? Solve using unit rates.
6 pages per 2 hours equals 6/2 = 3 pages per hour as a unit rate
Set up equation using h hours:
3h = 9
Divide each side by 3
[B]h = 3[/B]

Let f(x) = 3x - 6 and g(x)= -2x + 5. Which of the following is f(x) - g(x)?

Let f(x) = 3x - 6 and g(x)= -2x + 5. Which of the following is f(x) - g(x)?
f(x) - g(x) = 3x - 6 - (-2x + 5)
Distribute the negative sign where double negative equals a plus:
f(x) - g(x) = 3x - 6 + 2x - 5
Combine like terms:
f(x) - g(x) = (3 + 2)x - 6 - 5
f(x) - g(x) = [B]5x - 11[/B]

Let p be what Peter earns hourly. Peter earns less than 9 dollars an hour.

p < 9 is our inequality.

Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For

Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23) = 6 and S(23) = 5. Suppose N is a two-digit number such that N = P(N) + S(N). What could N be? Is there more than one answer?
For example, for 23 P(23) = 6 and S(23) = 5, but 23 could not be the N that we want since 23 <> 5 + 6
Let t = tens digit and o = ones digit
P(n) = to
S(n) = t + o
P(n) + S(n) = to + t + o
N = 10t + o
Set them equal to each other N = P(N) + S(N)
10t + o = to + t + o
o's cancel, so we have
10t = to + t
Subtract t from each side, we have
9t = to
Divide each side by t
o = 9
So any two-digit number with 9 as the ones digit will work:
[B]{19,29,39,49,59,69,79,89,99}[/B]

let x be the variable, an age that is at least 57 years old

let x be the variable, an age that is at least 57 years old
At least means greater than or equal to
x >= 57

Lily needs an internet connectivity package for her firm. She has a choice between CIVISIN and GOMI

Lily needs an internet connectivity package for her firm. She has a choice between CIVISIN and GOMI with the following monthly billing policies. Each company's monthly billing policy has an initial operating fee and charge per megabyte.
Operating Fee charge per Mb
CIVSIN 29.95 0.14
GOMI 4.95 0.39
(i) Write down a system of equations to model the above situation
(ii) At how many Mb is the monthly cost the same? What is the equal monthly cost of the two plans?
(i) Set up a cost function C(m) for CIVSIN where m is the number of megabytes used:
C(m) = charge per Mb * m + Operating Fee
[B]C(m) = 0.14m + 29.95[/B]
Set up a cost function C(m) for GOMI where m is the number of megabytes used:
C(m) = charge per Mb * m + Operating Fee
[B]C(m) = 0.39m + 4.95
[/B]
(ii) At how many Mb is the monthly cost the same?
Set both cost functions equal to each other:
0.14m + 29.95 = 0.39m + 4.95
We [URL='https://www.mathcelebrity.com/1unk.php?num=0.14m%2B29.95%3D0.39m%2B4.95&pl=Solve']type this equation into our search engine[/URL] and we get:
m = [B]100[/B]
(ii) What is the equal monthly cost of the two plans?
CIVSIN - We want C(100) from above where m = 100
C(100) = 0.14(100) + 29.95
C(100) = 14 + 29.95
C(100) = [B]43.95[/B]
GOMI - We want C(100) from above where m = 100
C(100) = 0.39(100) + 4.95
C(100) = 39 + 4.95
C(100) = [B]43.95[/B]

Line m passes through points (7, 5) and (9, 10). Line n passes through points (3, 1) and (7, 10). Ar

Line m passes through points (7, 5) and (9, 10). Line n passes through points (3, 1) and (7, 10). Are line m and line n parallel or perpendicular
[U]Slope of line m is:[/U]
(y2 - y1)/(x2 - x1)
(10 - 5)/(9 - 7)
5/2
[U]Slope of line n is:[/U]
(y2 - y1)/(x2 - x1)
(10 - 1)/(7 - 3)
9/4
Run 3 checks on the slopes:
[LIST=1]
[*]Lines that are parallel have equal slopes. Since 5/2 does not equal 9/4, these lines [B]are not parallel[/B]
[*]Lines that are perpendicular have negative reciprocal slopes. Since 9/4 is not equal to -2/5 (the reciprocal of the slope of m), these lines [B]are not perpendicular[/B]
[*][B]Therefore, since the lines are not parallel and not perpendicular[/B]
[/LIST]

Lisa has $150 at most to spend on clothes. She wants to buy a pair of jeans for $58 and will spend t

Lisa has $150 at most to spend on clothes. She wants to buy a pair of jeans for $58 and will spend the rest on t-shirts that cost $14 each.
Let the number of t-shirts be t. Lisa can spend up to, but not more than 150. We have the following inequality:
14t + 58 <= 150
To solve this inequality, we [URL='https://www.mathcelebrity.com/1unk.php?num=14j%2B58%3C%3D150&pl=Solve']type it in our search engine[/URL] and we get:
t <= 6.57
To round to a whole number, we round down to [B]t = 6 [/B]

Lisa wants to rent a boat and spend less than $52. The boat costs $7 per hour, and Lisa has a discou

Lisa wants to rent a boat and spend less than $52. The boat costs $7 per hour, and Lisa has a discount coupon for $4 off. What are the possible numbers of hours Lisa could rent the boat?
Calculate discounted cost:
Discounted cost = Full Cost - Coupon
Discounted cost = 52 - 7
Discounted cost = 45
Since price equals rate * hours (h), and we want the inequality (less than) we have:
7h < 52
Using our [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=7h%3C52&pl=Show+Interval+Notation']inequality calculator,[/URL] we see that:
[B]h < 7.42[/B]

Lucas is offered either 15% or $21 off his total shopping bill. How much would have to be spent to m

Lucas is offered either 15% or $21 off his total shopping bill. How much would have to be spent to make the 15% option the best one?
Let the total bill be b. We have:
0.15b > 21 <-- Since 15% is 0.15
Using our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=0.15b%3E21&pl=Show+Interval+Notation']inequality calculator[/URL], we get [B]b>140[/B].
So any bill greater than $140 will make the 15% off option the best one, since the discount will be higher than $21.

M decreased by the sum of 13 and the number P is less than 12

M decreased by the sum of 13 and the number P is less than 12
The sum of 13 and the number P
13 + P
M decreased by the sum of 13 and the number P
M - (13 + P)
Less than 12 means we set this entire expression less than 12 as an inequality
[B]M - (13 + P) < 12[/B]

M is the midpoint of AB. Prove AB = 2AM

M is the midpoint of AB. Prove AB = 2AM
M is the midpoint of AB (Given)
AM = MB (Definition of Congruent Segments)
AM + MB = AB (Segment Addition Postulate)
AM + AM = AB (Substitution Property of Equality)
2AM = AB (Distributive property)
[MEDIA=youtube]8BNo_4kvBzw[/MEDIA]

M is the set of integers that are greater than or equal to -1 and less than or equal to 2

M is the set of integers that are greater than or equal to -1 and less than or equal to 2
We include -1 on the left, and include 2 on the right
[B]M = {-1, 0, 1, 1, 2)[/B]

M is the sum of a and its reciprocal

M is the sum of a and its reciprocal
The reciprocal of a variable is 1 divided by the variable
1/a
The sum of a and its reciprocal means we add:
a + 1/a
The phrase [I]is[/I] means an equation, so we set M equal to the sum of a + 1/a:
[B]M = 1 + 1/a[/B]

maggie has two job offers. The first job offers to pay her $50 per week and 10 1/2 cents per flier.

maggie has two job offers. The first job offers to pay her $50 per week and 10 1/2 cents per flier. The second job offer will pay only $30 per week but gives 20 cents per flier. Write and solve an equation to find how many fliers must she deliver so that the two offers pay the same per week?
Let the number of fliers be f.
First job:
0.105f + 50
Second job:
20f + 30
Set them equal to each other:
0.105f + 50 = 20f + 30
[URL='https://www.mathcelebrity.com/1unk.php?num=0.105f%2B50%3D20f%2B30&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]f = 1[/B]

Maggie is shopping for her friend's party. She has a budget of $40 to spend. She needs to get a bann

Maggie is shopping for her friend's party. She has a budget of $40 to spend. She needs to get a banner for $25 and candy necklaces that cost $1.25 each. Write an inequality for the budget.
Let n be the necklaces. Since Maggie can spend [I]up to[/I] $40, we have the following inequality:
[B]1.25n + 25 <=40
[/B]
If you have to solve for n in the inequality, we [URL='https://www.mathcelebrity.com/1unk.php?num=1.25n%2B25%3C%3D40&pl=Solve']type it in our math engine[/URL] and we get:
[B]n < = 12[/B]

Marcela is having a presidential debate watching party with all of her friends, She will be making c

Marcela is having a presidential debate watching party with all of her friends, She will be making chicken wings and hot dogs. Each chicken wing costs $2 to make and each hot dog costs $3. She needs to spend at least $500. Marcela knows that she will make more than 50 chicken wings and hot dogs combined. She also knows that she will make less than 120 chicken wings and less that 100 hot dogs. What are her inequalities?
Let c be the number of chicken wings and h be the number of hot dogs. Set up the given inequalities:
[LIST=1]
[*]c + h > 50 [I]Marcela knows that she will make more than 50 chicken wings and hot dogs combined.[/I]
[*]2c + 3h >= 500 [I]She needs to spend at least $500[/I]
[*]c < 120 [I]She also knows that she will make less than 120 chicken wings[/I]
[*]h < 100 [I]and less that 100 hot dogs[/I]
[/LIST]

Marco takes 2 quizzes each week. Write an equation that shows the relationship between the number of

Marco takes 2 quizzes each week. Write an equation that shows the relationship between the number of weeks x and the total number of quizzes y. Write your answer as an equation with y first, followed by an equals sign.
Our total quizzes equal 2 times the number of weeks (x):
[B]y = 2x[/B]

Margaret earns 15 an hour at her job. Each week she earns at least 450 and no more than 600. How man

Margaret earns 15 an hour at her job. Each week she earns at least 450 and no more than 600. How many hours does Margaret work each week?
Let h be the hours worked
We know that hourly rate * h equals total earnings.
The phrases at least and no more than signify inequalities, so we have:
450 <= 15h <= 600
Divide each entry by 15:
[B]30 <= h <= 40[/B]
This means Margaret works at least 30 hours a week and no more than 40

Maria bought seven boxes. A week later half of all her boxes were destroyed in a fire. There are now

Maria bought seven boxes. A week later half of all her boxes were destroyed in a fire. There are now only 22 boxes left. With how many did she start?
Let the number of boxes Maria started with be b. We're given the following pieces:
[LIST]
[*]She starts with b
[*]She bought 7 boxes. So we add 7 to b: b + 7
[*]If half the boxes were destroyed, she's left with 1/2. So we divide (b + 7)/2
[*]Only 22 boxes left means we set (b + 7)/2 equal to 22
[/LIST]
(b + 7)/2 = 22
Cross multiply:
b + 7 = 22 * 2
b + 7 = 44
[URL='https://www.mathcelebrity.com/1unk.php?num=b%2B7%3D44&pl=Solve']Type this equation into our search engine[/URL] to solve for b and we get:
b = [B]37[/B]

Marla wants to rent a bike Green Lake Park has an entrance fee of $8 and charges $2 per hour for bik

Marla wants to rent a bike Green Lake Park has an entrance fee of $8 and charges $2 per hour for bike Oak Park has an entrance fee of $2 and charges $5 per hour for bike rentals she wants to know how many hours are friend will make the costs equal
[U]Green Lake Park: Set up the cost function C(h) where h is the number of hours[/U]
C(h) = Hourly Rental Rate * h + Entrance Fee
C(h) = 2h + 8
[U]Oak Park: Set up the cost function C(h) where h is the number of hours[/U]
C(h) = Hourly Rental Rate * h + Entrance Fee
C(h) = 5h + 2
[U]Marla wants to know how many hours make the cost equal, so we set Green Lake Park's cost function equal to Oak Parks's cost function:[/U]
2h + 8 = 5h + 2
To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=2h%2B8%3D5h%2B2&pl=Solve']type this equation into our search engine[/URL] and we get:
h = [B]2[/B]

Matilda needs at least $112 to buy an new dress. She has already saved $40. She earns $9 an hour bab

Matilda needs at least $112 to buy an new dress. She has already saved $40. She earns $9 an hour babysitting. Write and solve and inequality to find how many hours she will need to babysit to buy the dress.
Subtract remaining amount needed after savings:
112 - 40 = 72
Let h be her hourly wages for babysitting. We have the equation:
[B]9h = 72[/B]
Divide each side by 9
[B]h = 8[/B]

Matt has $100 dollars in a checking account and deposits $20 per month. Ben has $80 in a checking ac

Matt has $100 dollars in a checking account and deposits $20 per month. Ben has $80 in a checking account and deposits $30 per month. Will the accounts ever be the same balance? explain
Set up the Balance account B(m), where m is the number of months since the deposit.
Matt:
B(m) = 20m + 100
Ben:
B(m) = 80 + 30m
Set both balance equations equal to each other to see if they ever have the same balance:
20m + 100 = 80 + 30m
To solve for m, [URL='https://www.mathcelebrity.com/1unk.php?num=20m%2B100%3D80%2B30m&pl=Solve']we type this equation into our search engine[/URL] and we get:
m = [B]2
So yes, they will have the same balance at m = 2[/B]

Megan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week. After how many

Megan has $50 and saves $5.50 each week. Connor has $18.50 and saves $7.75 each week. After how many weeks will megan and connor have saved the same amount
[U]Set up the Balance function B(w) where w is the number of weeks for Megan:[/U]
B(w) = savings per week * w + Current Balance
B(w) = 5.50w + 50
[U]Set up the Balance function B(w) where w is the number of weeks for Connor:[/U]
B(w) = savings per week * w + Current Balance
B(w) = 7.75w + 18.50
The problem asks for w when both B(w) are equal. So we set both B(w) equations equal to each other:
5.50w + 50 = 7.75w + 18.50
To solve this equation for w, we[URL='https://www.mathcelebrity.com/1unk.php?num=5.50w%2B50%3D7.75w%2B18.50&pl=Solve'] type it in our search engine[/URL] and we get:
w = [B]14[/B]

Miguel has $80 in his bank and saves $2 a week. Jesse has $30 in his bank but saves $7 a week. In ho

Miguel has $80 in his bank and saves $2 a week. Jesse has $30 in his bank but saves $7 a week. In how many weeks will Jesse have more in his bank than Miguel?
[U]Set up the Bank value B(w) for Miguel where w is the number of weeks[/U]
B(w) = Savings Per week * w + Current Bank Balance
B(w) = 2w + 80
[U]Set up the Bank value B(w) for Jesse where w is the number of weeks[/U]
B(w) = Savings Per week * w + Current Bank Balance
B(w) = 7w + 30
The problem asks when Jesse's account will be more than Miguel's. So we set up an inequality where:
7w + 30 > 2w + 80
To solve this inequality, we [URL='https://www.mathcelebrity.com/1unk.php?num=7w%2B30%3E2w%2B80&pl=Solve']type it in our search engine[/URL] and we get:
[B]w > 10[/B]

Milan plans to watch 2 movies each month. Write an equation to represent the total number of movies

Milan plans to watch 2 movies each month. Write an equation to represent the total number of movies n that he will watch in m months.
Number of movies equals movies per month times the number of months. So we have:
[B]n = 2m[/B]

Mr. Smith wants to spend less than $125 at a zoo. A ticket cost $7 he is taking 2 kids with him. Use

Mr. Smith wants to spend less than $125 at a zoo. A ticket cost $7 he is taking 2 kids with him. Use p to represent the other money he can spend there.
2 kids and Mr. Smith = 3 people.
Total Ticket Cost is 3 people * 7 per ticket = 21
If he has 125 to spend, we have the following inequality using less than or equal to (<=) since he can spend up to or less than 125:
p + 21 <= 125
Subtract 21 from each side:
[B]p <= 104[/B]

Mrs. Evans has 120 crayons and 30 pieces of paper to give her students. What is the largest number o

Mrs. Evans has 120 crayons and 30 pieces of paper to give her students. What is the largest number of students she can have her class so that each student gets an equal number of crayons and equal number of paper?
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=30&num2=120&num3=&pl=GCF+and+LCM']Using our GCF calculator for the GCF(30, 120)[/URL], we get 30.
So 30 people get the following:
[B]30/30 = 1 piece of paper
120/30 = 4 crayons[/B]

Mrs. Lowe charges $45 an hour with a $10 flat fee for tutoring. Mrs. Smith charges $40 an hour wit

Mrs. Lowe charges $45 an hour with a $10 flat fee for tutoring. Mrs. Smith charges $40 an hour with a $15 flat fee to tutor. Write an equation that represents the situation when the cost is the same to be tutored by Mrs. Lowe and Mrs. Smith.
[U]Set up cost equation for Mrs. Lowe where h is the number of hours tutored:[/U]
Cost = Hourly Rate * number of hours + flat fee
Cost = 45h + 10
[U]Set up cost equation for Mrs. Smith where h is the number of hours tutored:[/U]
Cost = Hourly Rate * number of hours + flat fee
Cost = 40h + 15
[U]Set both cost equations equal to each other:[/U]
45h + 10 = 40h + 15 <-- This is our equation
To solve for h if the problem asks, we [URL='https://www.mathcelebrity.com/1unk.php?num=45h%2B10%3D40h%2B15&pl=Solve']type this equation into our search engine[/URL] and we get:
h = 1

Multiplication Equality Property

Free Multiplication Equality Property Calculator - Demonstrates the Multiplication Equality Property
Numerical Properties

Multiplication Property Of Inequality

Free Multiplication Property Of Inequality Calculator - Demonstrates the Multiplication Property Of Inequality
Numerical Properties

Multiply a number by 6 and subtracting 6 gives the same result as multiplying the number by 3 and su

Multiply a number by 6 and subtracting 6 gives the same result as multiplying the number by 3 and subtracting 4. Find the number
The phrase [I]a number [/I]means an arbitrary variable, let's call it x.
multiply a number by 6 and subtract 6:
6x - 6
Multiply a number by 3 and subtract 4:
3x - 4
The phrase [I]gives the same result[/I] means an equation. So we set 6x - 6 equal to 3x - 4
6x - 6 = 3x - 4
To solve this equation for x, we type it in our search engine and we get:
x = [B]2/3[/B]

Multiplying a number by 6 is equal to the number increased by 9

Multiplying a number by 6 is equal to the number increased by 9.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
Multiply it by 6 --> 6x
We set this equal to the same number increased by 9. Increased by means we add:
[B]6x = x + 9 <-- This is our algebraic expression
[/B]
To solve this equation, we [URL='https://www.mathcelebrity.com/1unk.php?num=6x%3Dx%2B9&pl=Solve']type it into the search engine [/URL]and get x = 1.8.

n and m are congruent and supplementary. prove n and m are right angles

n and m are congruent and supplementary. prove n and m are right angles
Given:
[LIST]
[*]n and m are congruent
[*]n and m are supplementary
[/LIST]
If n and m are supplementary, that means we have the equation:
m + n = 180
We're also given n and m are congruent, meaning they are equal. So we can substitute n = m into the supplementary equation:
m + m = 180
To solve this equation for m, [URL='https://www.mathcelebrity.com/1unk.php?num=m%2Bm%3D180&pl=Solve']we type it in our search engine[/URL] and we get:
m = 90
This means m = 90, n = 90, which means they are both right angles since by definition, a right angle is 90 degrees.

n is equal to 135 less than the quantity 61 times n

n is equal to 135 less than the quantity 61 times n
61 times n:
61n
135 less than the quantity 61 times n
61n - 135
We set n equal to this expression:
[B]n = 61n - 135[/B]

n is equal to the product of 7 and the sum of m and 6

n is equal to the product of 7 and the sum of m and 6
The sum of m and 6:
m + 6
The product of 7 and this sum:
7(m + 6)
We set this expression equal to n:
[B]7(m + 6) = n[/B]

n is the sum of twenty-five and fifteen

n is the sum of twenty-five and fifteen
The sum of twenty-five and fifteen:
25 + 15
The word [I]is[/I] means an equal to, so we set 25 + 15 equal to n:
[B]n = 25 + 15
n = 40[/B]

N reduced by 2 is the same as Z increased by 7

N reduced by 2 is the same as Z increased by 7
[LIST]
[*]N reduced by 2 means subtract --> n - 2
[*]z increased by 7 means add --> z + 7
[*][I]Is the same as[/I] means equal to, so we set these expressions equal to each other
[*][B]n - 2 = z + 7[/B]
[/LIST]

name five coins that equal 18 cents

name five coins that equal 18 cents
Here are the five coins:
[LIST]
[*][B]1 dime = [/B]10 cents
[*][B]1 nickel = [/B]5 cents
[*][B]3 pennies = [/B]3 cents
[*]5 coins = 10 + 5 + 3 = 18 cents
[/LIST]

Nancy started the year with $435 in the bank and is saving $25 a week. Shane started with $875 and i

Nancy started the year with $435 in the bank and is saving $25 a week. Shane started with $875 and is spending $15 a week. [I]When will they both have the same amount of money in the bank?[/I]
[I][/I]
Set up the Account equation A(w) where w is the number of weeks that pass.
Nancy (we add since savings means she accumulates [B]more[/B]):
A(w) = 25w + 435
Shane (we subtract since spending means he loses [B]more[/B]):
A(w) = 875 - 15w
Set both A(w) equations equal to each other to since we want to see what w is when the account are equal:
25w + 435 = 875 - 15w
[URL='https://www.mathcelebrity.com/1unk.php?num=25w%2B435%3D875-15w&pl=Solve']Type this equation into our search engine to solve for w[/URL] and we get:
w =[B] 11[/B]

Narda has $250 and Ding has $170. How much money must Narda give to Ding so that each of them will h

Narda has $250 and Ding has $170. How much money must Narda give to Ding so that each of them will have an equal amount of money?
Find the difference of Narda and Ding's money:
Difference = Narda - Ding
Difference = 250 - 170
Difference = 80
Find half the difference:
Half the difference = 80/2
Half the difference = 40
So Narda must give Ding [B]$40[/B] to have equal amounts:
Narda's new total = 250 - 40 = 210
Ding's new total = 1760 + 40 = 210

Natalie made a deal with a farmer. She agreed to work for an entire year and in return, the farmer w

Natalie made a deal with a farmer. She agreed to work for an entire year and in return, the farmer would give her $10,200 plus a prize pig.
After working for 5 months, Natalie decided to quit. The farmer determined that 5 months of work was equal to $3375 plus the pig. How much money was the pig worth?
The value of a year's work is $10,200 plus a pig of unknown value. The farmer took away $6825 because Natalie worked 5 months. If Natalie worked 7 more months, she would have been paid the additional $6825.
6825/7 months work = $975 per month
A full year's work is $975 * 12 = $11,700
Pig value = Full years work - payout
Pig value = 11,700 - 10,200
Pig value = [B]1,500[/B]

Nia is trying to decide between two possible jobs. Job A pays $2000 a month with a 2% annual raise.

Nia is trying to decide between two possible jobs. Job A pays $2000 a month with a 2% annual raise. Job B pays 24,000 a year with a $500 annual raise. Write a function to represent the annual salary for Job A after x years. Write a function to represent the annual salary for Job B after x years. After how many years would Nia have a greater salary at Job A?
Nia Job A salary at time t: S(t)
$2,000 per month equals $24,000 per year.
So we have S(t) = 24,000(1.o2)^t
Nia Job B salary at time t: S(t)
$24,000 per year.
So we have S(t) = 24,000 + 500t
We want to know t when Job A salary is greater than Job B Salary:
24,000(1.o2)^t > 24,000 + 500t
Time | A | B
0 | 24000 | 24000
1 | 24480 | 24500
2 | 24969.6 | 25000
3 | 25468.99 | 25500
4 | 25978.37 | 26000
5 | 26497.94 | 26500
6 | 27027.9 | 27000
7 | 27568.46 | 27500
8 | 28119.83 | 28000
9 | 28682.22 | 28500
10 | 29255.87 | 29000
11 | 29840.98 | 29500
12 | 30437.8 | 30000
13 | 31046.56 | 30500

Nine less than a number is no more than 8 and no less than 3

Nine less than a number is no more than 8 and no less than 3
A number is denoted as an arbitrary variable, let's call it x.
We have a double inequality:
[LIST=1]
[*]No more than 8 means less than or equal to 8
[*]No less than 3 means greater than or equal to 3
[/LIST]
[B]3 <= x <= 8[/B]

Nine less than the product of 2 and y is not less than 15

The product of 2 and y means we multiply
2y
Nine less than that product means we subtract 9
2y - 9
Finally, the entire expression is not less than 15, which means 15 or more. We use greater than or equal to
[B]2y - 9 >= 15
[/B]
If you want to solve this inequality and write the interval notation, use [URL='http://www.mathcelebrity.com/1unk.php?num=2y-9%3E%3D15&pl=Solve']our calculator[/URL].

nine times x is twice the sum of x and five

nine times x is twice the sum of x and five
Take this algebraic expression in 4 pieces:
[U]Step 1: nine time x:[/U]
9x
[U]Step 2: The sum of x and five means we add 5 to x:[/U]
x + 5
[U]Step 3: The word [I]twice[/I] means we multiply the sum x + 5 by 2:[/U]
2(x + 5)
[U]Step 4: The word [I]is[/I] means equal to, so we set 9x equal to 2(x + 5) to get our final algebraic expression of:[/U]
[B]9x = 2(x + 5)[/B]

Normal body temperature is 98.6 ? F. Write an inequality that describes the temperature

Normal body temperature is 98.6 ? F. Write an inequality that describes the temperature, T, of people with above normal temperatures.
Above means greater than, so we set up the inequality:
[B]T > 98.6 ?[/B]

Notebooks cost $1.39 each. What are the possible numbers of notebooks that can be purchased with $10

Notebooks cost $1.39 each. What are the possible numbers of notebooks that can be purchased with $10?
Let n be the number of notebooks you can purchase. We have the following inequality:
1.39n <= 10
Divide each side by 1.39
n <= 7.194
We want whole notebooks, we cannot buy fractions of notebooks, so we have:
n <= 7
The question asks for the possible numbers of notebooks we can buy. This implies we buy at least 1, but our inequality says not more than 7. So our number set is:
[B]N = {1, 2, 3, 4, 5, 6, 7}[/B]

Number of cents in q quarters is 275

Number of cents in q quarters is 275
Each quarter makes 25 cents. We write this as 0.25q.
Now set this equal to 275
0.25q = 275
Typing this [URL='http://www.mathcelebrity.com/1unk.php?num=0.25q%3D275&pl=Solve']equation in the search engine[/URL], we get [B]q = 1,100[/B].

On a Friday evening a pizza shop had orders for 4 pepperoni, 97 vegetable, and 335 cheese pizzas. If

On a Friday evening a pizza shop had orders for 4 pepperoni, 97 vegetable, and 335 cheese pizzas. If the 4 cooks each made an equal number of pizzas, how many pizzas did each cook make?
Total Pizzas Made = 4 pepperoni + 97 vegetable + 335 cheese
Total Pizzas Made = 436
Equal number of pizzas per cook = 436 pizzas / 4 cooks
Equal number of pizzas per cook = [B]109[/B]

One number is equal to the square of another. Find the numbers if both are positive and their sum is

One number is equal to the square of another. Find the numbers if both are positive and their sum is 650
Let the number be n. Then the square is n^2. We're given:
n^2 + n = 650
Subtract 650 from each side:
n^2 + n - 650 = 0
We have a quadratic equation. [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2%2Bn-650%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']We type this into our search engine[/URL] and we get:
n = 25 and n = -26
Since the equation asks for a positive solution, we use [B]n = 25[/B] as our first solution.
the second solution is 25^2 = [B]625[/B]

One positive number is one-fifth of another number. The difference between the two numbers is 192, f

One positive number is one-fifth of another number. The difference between the two numbers is 192, find the numbers.
Let the first number be x and the second number be y. We're given two equations:
[LIST=1]
[*]x = y/5
[*]x + y = 192
[/LIST]
Substitute equation 1 into equation 2:
y/5 + y = 192
Since 1 equals 5/5, we rewrite our equation like this:
y/5 = 5y/5 = 192
We have fractions with like denominators, so we add the numerators:
(1 + 5)y/5 = 192
6y/5 = 192
[URL='https://www.mathcelebrity.com/prop.php?num1=6y&num2=192&den1=5&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']Type this equation into our search engine[/URL], and we get:
[B]y = 160[/B]
Substitute this value into equation 1:
x = 160/5
x = [B]32[/B]

One-half a number is fifty

The phrase [I]a number[/I] means an arbitrary variable. We can pick any letter a-z except for i and e.
Let's choose x.
One-half a number means we divide x by2:
x/2
The word [I]is[/I] means equal to. We set x/2 equal to 50 for our algebraic expression
[B]x/2 = 50
[/B]
If the problem asks us to solve for x, we cross multiply:
x = 2 * 50
x = [B]100[/B]

Orange Theory is currently offering a deal where you can buy a fitness pass for $100 and then each c

Orange Theory is currently offering a deal where you can buy a fitness pass for $100 and then each class is $13, otherwise it is $18 for each class. After how many classes is the total cost with the fitness pass the same as the total cost without the fitness pass?
Let the number of classes be c.
For the fitness pass plan, we have the total cost of:
13c + 100
For the flat rate plan, we have the total cost of:
18c
The question asks for c when both plans are equal. So we set both costs equal and solve for c:
13c + 100 = 18c
We [URL='https://www.mathcelebrity.com/1unk.php?num=13c%2B100%3D18c&pl=Solve']type this equation into our math engine[/URL] and we get:
c = [B]20[/B]

p decreased by 65 is the same as the total of f and 194

p decreased by 65 is the same as the total of f and 194
p decreased by 65
p - 65
The total of f and 194
f + 194
The phrase [I]is the same as[/I] means equal to, so we set the expressions above equal to each other
[B]p - 65 = f + 194[/B]

p is equal to r plus 2 times q

p is equal to r plus 2 times q
2 times q:
2q
r plus 2 times q:
r + 2q
is equal to means we set p equal to r + 2q
[B]p = r + 2q[/B]

Penelope and Owen work at a furniture store. Penelope is paid $215 per week plus 3.5% of her total s

Penelope and Owen work at a furniture store. Penelope is paid $215 per week plus 3.5% of her total sales in dollars, xx, which can be represented by g(x)=215+0.035x. Owen is paid $242 per week plus 2.5% of his total sales in dollars, xx, which can be represented by f(x)=242+0.025x. Determine the value of xx, in dollars, that will make their weekly pay the same.
Set the pay functions of Owen and Penelope equal to each other:
215+0.035x = 242+0.025x
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=215%2B0.035x%3D242%2B0.025x&pl=Solve']equation calculator[/URL], we get:
[B]x = 2700[/B]

People with a drivers license are at least 16 years old and no older than 85 years old

People with a drivers license are at least 16 years old and no older than 85 years old.
Set up the inequality, where p represents the people:
[LIST=1]
[*]The phrase [I]at least[/I] means greater than or equal to. So we use the >= sign. 16 <= p
[*]The phrase [I]no older than[/I] means less than or equal to. So we use the <= sign. p <= 85
[/LIST]
Combine these inequalities, and we get:
[B]16 <= p <= 85[/B]
To see the interval notation for this inequality and all possible values, visit the [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=16%3C%3Dp%3C%3D85&pl=Show+Interval+Notation']interval notation calculator[/URL].

Pet supplies makes a profit of $5.50 per bag, if the store wants to make a profit of no less than $5

Pet supplies makes a profit of $5.50 per bag, if the store wants to make a profit of no less than $5225, how many bags does it need to sell?
5.5ob >= $5,225
Divide each side of the inequality by $5.50
b >=9.5 bags, so round up to a whole number of 10 bags.

Peter was thinking of a number. Peter doubles it and adds 0.8 to get an answer of 31. Form an equati

Peter was thinking of a number. Peter doubles it and adds 0.8 to get an answer of 31. Form an equation with x from the information.
Take this algebraic expression in parts, starting with the unknown number x:
[LIST]
[*]x
[*][I]Double it [/I]means we multiply x by 2: 2x
[*]Add 0.8: 2x + 0.8
[*]The phrase [I]to get an answer of[/I] means an equation. So we set 2x + 0.8 equal to 31
[/LIST]
Build our final algebraic expression:
[B]2x + 0.8 = 31[/B]
[B][/B]
If you have to solve for x, then we [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B0.8%3D31&pl=Solve']type this equation into our search engine[/URL] and we get:
x = 15.1

Positive numbers less than 4

Update, this has been added to our shortcuts.
You can type any expression in the form, positive numbers less than x where x is any integer.
You can also type positive numbers greater than x where x is any integer.
Same with less than or equal to and greater than or equal to.

Pound of strawberries for $4.00. What is the price, in dollars, per ounce of strawberries?

Pound of strawberries for $4.00. What is the price, in dollars, per ounce of strawberries?
1 pound equals 16 ounces. So the pounds per ounce equals:
$4.00/16 ounces
Divide top and bottom by 16, we get:
[B]$0.25 per ounce[/B]

Power is equal to:

Power is equal to:
a. ?
b. ?
c. 1 - ?
d. 1 - ?
[B]d. 1 - ?[/B]
[B][/B]
[I]Correct Decision 1 - ? = Power of a Test[/I]

Probability of getting either a sum of 8 or at least one 4 in the roll of a pair dice

Sum of 8 equal to 5/36 shown [URL='http://www.mathcelebrity.com/2dice.php?gl=1&pl=8&opdice=1&rolist=+2%2C3%2C9%2C10&dby=2%2C3%2C5&ndby=4%2C5&montect=+100']here[/URL].
At least one 4 means one of three scenarios:
[LIST=1]
[*](4, not 4) = 1/6 * 5/6 = 5/36
[*](not 4, 4) = 5/6 * 1/6 = 5/36
[*](4, 4) = 1/6 * 1/6 = 1/36
[/LIST]
The phrase "or", means we add both probabilities (sum of 8) and (at least one 4):
5/36 + (5/36 + 5/36 + 1/36)
16/36
Simplify by dividing each part of the fraction by 4
[B]4/9[/B]

Prove 0! = 1

Prove 0! = 1
Let n be a whole number, where n! represents the product of n and all integers below it through 1.
The factorial formula for n is:
n! = n · (n - 1) * (n - 2) * ... * 3 * 2 * 1
Written in partially expanded form, n! is:
n! = n * (n - 1)!
[U]Substitute n = 1 into this expression:[/U]
n! = n * (n - 1)!
1! = 1 * (1 - 1)!
1! = 1 * (0)!
For the expression to be true, 0! [U]must[/U] equal 1. Otherwise, 1! <> 1 which contradicts the equation above

Prove 0! = 1

[URL='https://www.mathcelebrity.com/proofs.php?num=prove0%21%3D1&pl=Prove']Prove 0! = 1[/URL]
Let n be a whole number, where n! represents:
The product of n and all integers below it through 1.
The factorial formula for n is
n! = n · (n - 1) · (n - 2) · ... · 3 · 2 · 1
Written in partially expanded form, n! is:
n! = n · (n - 1)!
[SIZE=5][B]Substitute n = 1 into this expression:[/B][/SIZE]
n! = n · (n - 1)!
1! = 1 · (1 - 1)!
1! = 1 · (0)!
For the expression to be true, 0! [U]must[/U] equal 1.
Otherwise, 1! ? 1 which contradicts the equation above
[MEDIA=youtube]wDgRgfj1cIs[/MEDIA]

Prove there is no integer that is both even and odd

Let us take an integer x which is both even [I]and[/I] odd.
[LIST]
[*]As an even integer, we write x in the form 2m for some integer m
[*]As an odd integer, we write x in the form 2n + 1 for some integer n
[/LIST]
Since both the even and odd integers are the same number, we set them equal to each other
2m = 2n + 1
Subtract 2n from each side:
2m - 2n = 1
Factor out a 2 on the left side:
2(m - n) = 1
By definition of divisibility, this means that 2 divides 1.
But we know that the only two numbers which divide 1 are 1 and -1.
Therefore, our original assumption that x was both even and odd must be false.
[MEDIA=youtube]SMM9ubEVcLE[/MEDIA]

Q is 5% less than P

The phrase is means equal to, so we have:
Q = P - 5%
5% is written as 0.05, so we have:
Q = P - 0.05

q is equal to 207 subtracted from the quantity 4 times q

q is equal to 207 subtracted from the quantity 4 times q
4 time q
4q
207 subtracted from 4 times q:
4q - 207
Set this equal to q:
[B]4q - 207 = q [/B]<-- This is our algebraic expression
To solve for q, [URL='https://www.mathcelebrity.com/1unk.php?num=4q-207%3Dq&pl=Solve']type this equation into the search engine[/URL]. We get:
[B]q = 69[/B]

Quadratic Equations and Inequalities

Free Quadratic Equations and Inequalities Calculator - Solves for quadratic equations in the form ax^{2} + bx + c = 0. Also generates practice problems as well as hints for each problem.

* Solve using the quadratic formula and the discriminant Δ

* Complete the Square for the Quadratic

* Factor the Quadratic

* Y-Intercept

* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)^{2} + k

* Concavity of the parabola formed by the quadratic

* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.

* Solve using the quadratic formula and the discriminant Δ

* Complete the Square for the Quadratic

* Factor the Quadratic

* Y-Intercept

* Vertex (h,k) of the parabola formed by the quadratic where h is the Axis of Symmetry as well as the vertex form of the equation a(h - h)

* Concavity of the parabola formed by the quadratic

* Using the Rational Root Theorem (Rational Zero Theorem), the calculator will determine potential roots which can then be tested against the synthetic calculator.

r less than 164 is 248 more than the product of 216 and r

r less than 164 is 248 more than the product of 216 and r
[U]r less than 164:[/U]
164 - r
[U]The product of 216 and r:[/U]
216r
[U]248 more than the product of 216 and r[/U]
216r + 248
[I]The word is means an equation, so we set 164 - r equal to 216r + 248[/I]
[B]164 - r = 216r + 248[/B]

Rachel runs 2 miles during each track practice. Write an equation that shows the relationship betwe

Rachel runs 2 miles during each track practice. Write an equation that shows the relationship between the practices p and the distance d.
Distance equals rate * practicdes, so we have:
[B]d = 2p[/B]

Rachel runs each lap in 6 minutes. She will run less than 8 laps today. What are the possible number

Rachel runs each lap in 6 minutes. She will run less than 8 laps today. What are the possible numbers of minutes she will run today
Less than means an inequality.
6 minutes per lap * 8 laps = 48 minutes.
If m is the number of minutes Rachel runs, then we have:
[B]m < 48[/B]

Rachel saved $200 and spends $25 each week. Roy just started saving $15 per week. At what week will

Rachel saved $200 and spends $25 each week. Roy just started saving $15 per week. At what week will they have the same amount?
Let Rachel's account value R(w) where w is the number of weeks be:
R(w) = 200 - 25w <-- We subtract -25w because she spends it every week, decreasing her balance.
Let Roy's account value R(w) where w is the number of weeks be:
R(w) = 15w
Set them equal to each other:
200 - 25w = 15w
To solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=200-25w%3D15w&pl=Solve']we type it into our search engine[/URL] and get:
[B]w = 5[/B]

rectangle abcd prove: triangle adc is congruent to triangle bcd

rectangle abcd prove: triangle adc is congruent to triangle bcd
1. Given: ABCD is a rectangle
2. AB = CD since opposite sides of rectangle are congruent
3. BC = AD since opposite sides of rectangle are congruent
4. AC = AC by the Reflexive Property of Equality
5. triangle ADC = triangle CBA by the Side-Side-Side (SSS) Property

Rhonda raised $245 for her softball team's fundraiser.She wants to raise no less than $455.Write and

Rhonda raised $245 for her softball team's fundraiser.She wants to raise no less than $455.Write and solve an inequality to determine how much more money Rhonda must raise to reach her goal. Let d represent the amount of money in dollars Rhonda must raise to reach her goal.
The phrase [I]no less than[/I] is an inequality using the greater than or equal sign:
d >= 455 - 245
d >= [B]210[/B]

Rita has spent $16 so far on gifts. The additional amount she will spend will be less than $14. What

Rita has spent $16 so far on gifts. The additional amount she will spend will be less than $14. What are the possible total amounts she will spend?
Rita will spend at least another cent on other gifts above the $16 she spent so far, but no more than $14. Also, the problem says less than 14. 16 + 14 is 30, so that is the top end of her spending.
Let's say her remaining spending is s. Set up the inequality for possible spending values.
[B]16 < s < 30[/B]

Ryan buys candy that costs $4 per pound. He will buy at least 12 pounds of candy. What are the possi

Ryan buys candy that costs $4 per pound. He will buy at least 12 pounds of candy. What are the possible amounts he will spend on candy?
Clue for you: the phrase [I]at least[/I] means an inequality.
Let s be the spend on candy.
Cost = Price * quantity
Cost = 4 * 12
Cost = 48
The phrase [I]at least[/I] means greater than or equal to:
[B]s >= 48[/B]

S equals the quotient of r and the sum of r and 8.

S equals the quotient of r and the sum of r and 8.
A quotient means a fraction, so we have:
[B]S = r/(r + 8)[/B]

Sally and Adam works a different job. Sally makes $5 per hour and Adam makes $4 per hour. They each

Sally and Adam works a different job. Sally makes $5 per hour and Adam makes $4 per hour. They each earn the same amount per week but Adam works 2 more hours. How many hours a week does Adam work?
[LIST]
[*]Let [I]s[/I] be the number of hours Sally works every week.
[*]Let [I]a[/I] be the number of hours Adam works every week.
[*]We are given: a = s + 2
[/LIST]
Sally's weekly earnings: 5s
Adam's weekly earnings: 4a
Since they both earn the same amount each week, we set Sally's earnings equal to Adam's earnings:
5s = 4a
But remember, we're given a = s + 2, so we substitute this into Adam's earnings:
5s = 4(s + 2)
Multiply through on the right side:
5s = 4s + 8 <-- [URL='https://www.mathcelebrity.com/expand.php?term1=4%28s%2B2%29&pl=Expand']multiplying 4(s + 2)[/URL]
[URL='https://www.mathcelebrity.com/1unk.php?num=5s%3D4s%2B8&pl=Solve']Typing this equation into the search engine[/URL], we get s = 8.
The problem asks for Adam's earnings (a). We plug s = 8 into Adam's weekly hours:
a = s + 2
a = 8 + 2
[B]a = 10[/B]

Sally is 4 years older than Mark. Twice Sally's age plus 5 times Mark's age is equal to 64.

Sally is 4 years older than Mark. Twice Sally's age plus 5 times Mark's age is equal to 64.
Let Sally's age be s. Let Mark's age be m. We're given two equations:
[LIST=1]
[*]s = m + 4
[*]2s + 5m = 64 <-- [I]Since Twice means we multiply by 2[/I]
[/LIST]
Substitute equation (1) into equation (2):
2(m + 4) + 5m = 64
Multiply through:
2m + 8 + 5m = 64
Group like terms:
(2 + 5)m + 8 = 64
7m + 8 = 64
[URL='https://www.mathcelebrity.com/1unk.php?num=7m%2B8%3D64&pl=Solve']Type this equation into the search engine[/URL] and we get:
m = [B]8[/B]

Sara opened an account with $800 and withdrew $20 per week. Jordan opened an account with $500 and d

Sara opened an account with $800 and withdrew $20 per week. Jordan opened an account with $500 and deposited $30 per week. In how many weeks will their account be equal?
Each week, Sara's account value is:
800 - 20w <-- Subtract because Sara withdraws money each week
Each week, Jordan's account value is:
500 + 30w <-- Add because Jordan deposits money each week
Set them equal to each other:
800 - 20w = 500 + 30w
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=800-20w%3D500%2B30w&pl=Solve']equation solver[/URL], we get w = 6.
Check our work:
800 - 20(6)
800 - 120
680
500 + 30(6)
500 + 180
680

Sarah has $250 in her account. She withdraws $25 per week. How many weeks can she withdraw money fro

Sarah has $250 in her account. She withdraws $25 per week. How many weeks can she withdraw money from her account and still have money left?
Let w be the number of weeks. We have the following equation for the Balance after w weeks:
B(w) = 250 - 25w [I]we subtract for withdrawals[/I]
The ability to withdrawal money means have a positive or zero balance after withdrawal. So we set up the inequality below:
250 - 25w >= 0
To solve this inequality for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=250-25w%3E%3D0&pl=Solve']type it in our search engine[/URL] and we get:
w <= [B]10
So Sarah can withdrawal for up to 10 weeks[/B]

Sarah makes $9 per hour working at a daycare center and $12 per hour working at a restaurant. Next

Sarah makes $9 per hour working at a daycare center and $12 per hour working at a restaurant. Next week, Sarah is scheduled to work 8 hours at the daycare center. Which of the following inequalities represents the number of hours (h) that Sandra needs to work at the restaurant next week to earn at least $156 from these two jobs?
Set up Sarah's earnings function E(h) where h is the hours Sarah must work at the restaurant:
12h + 9(8) >= 156 <-- The phrase [I]at least[/I] means greater than or equal to, so we set this up as an inequality. Also, the daycare earnings are $9 per hour * 8 hours
Multiplying through and simplifying, we get:
12h + 72 >= 156
We [URL='https://www.mathcelebrity.com/1unk.php?num=12h%2B72%3E%3D156&pl=Solve']type this inequality into the search engine[/URL], and we get:
[B]h>=7[/B]

Savannah is a salesperson who sells computers at an electronics store. She makes a base pay of $90 e

Savannah is a salesperson who sells computers at an electronics store. She makes a base pay of $90 each day and is also paid a commission for each sale she makes. One day, Savannah sold 4 computers and was paid a total of $100. Write an equation for the function P(x), representing Savannah's total pay on a day on which she sells x computers.
If base pay is $90 per day, then the total commission Savannah made for selling 4 computers is:
Commission = Total Pay - Base Pay
Commission = 100 - 90
Commission = $10
Assuming the commission for each computer is equal, we need to find the commission per computer:
Commission per computer = Total Commission / Number of Computers Sold
Commission per computer = 10/4
Commission per computer = $2.50
Now, we build the Total pay function P(x):
Total Pay = Base Pay + Commission * Number of Computers sold
[B]P(x) = 90 + 2.5x[/B]

Set C contains all counting numbers less than 20

C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}
Notice, we do not include 20 since it said less than, and not less than or equal to.

Seven less than 1/4 of a number is 9.

Seven less than 1/4 of a number is 9.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
1/4 of a number means we multiply x by 1/4:
x/4
Seven less than this means we subtract 7 from x/4:
x/4 - 7
The word [I]is[/I] means an equation, so we set x/4 - 7 equal to 9:
[B]x/4 - 7 = 9[/B]

Seven subtracted from the product of 3 and a number is greater than or equal to -26

Seven subtracted from the product of 3 and a number is greater than or equal to -26
[LIST=1]
[*]A number means an arbitrary variable, let's call it x.
[*]The product of 3 and a number is written as 3x
[*]Seven subtracted from 3x is written as 3x - 7
[*]Finally, that entire expression is greater than or equal to -26: [B]3x - 7 >= - 26[/B]
[/LIST]

Simplify sin^2(x)/(1 - sin^2(x))

We know from the pythagorean theorem:
[SIZE=5][B]sin^2(x) + cos^2(x) = 1[/B]
[B]Subtract sin^2(x) from each side and we get:[/B]
[B][B]cos^2(x) = 1 - [B]sin^2(x)[/B][/B][/B]
[B][B][B]We can rewrite our original expression as:[/B][/B][/B]
[B][B][B]sin^2(x)/cos^2(x)[/B][/B][/B]
[/SIZE]
[B][B][B][SIZE=5]But this expression [/SIZE][SIZE=4]equals[/SIZE][SIZE=5] tan^2(x)[/SIZE][/B][/B][/B]
[MEDIA=youtube]zqYg0VRq5Ak[/MEDIA]

Six less than twice a number is at least -1 and at most 1

First, the phrase [I]a number[/I] means we choose an arbitrary variable, let's call it x.
Twice a number means we multiply it by 2.
2x
Six less than that means we subtract 6
2x - 6
Now, the last piece, we set up an inequality. At least -1 means greater than or equal to 1. At most 1 means less than or equal to 1. Notice, for both points, we include the number.
-1 <= 2x - 6 <= 1

Sixteen subtracted from five times a number equals the number plus four

The phrase [I]a number[/I] means an arbitrary variable. We can pick any letter a-z except for i and e.
Let's choose x.
5 times a number
5x
Sixteen subtracted from five times a number
5x - 16
the number plus 4:
x + 4
Equals means we set 5x - 16 equals to x + 4 for our algebraic expression:
[B]5x - 16 = x + 4[/B]
[B][/B]
If you have to solve for x, [URL='https://www.mathcelebrity.com/1unk.php?num=5x-16%3Dx%2B4&pl=Solve']type this expression into our math solver[/URL] and we get:
x = [B]5[/B]

Solve for x

[IMG]https://mathcelebrity.com/community/data/attachments/0/supp-angles.jpg[/IMG]
The angle with measurements of 148 degrees lies on a straight line, which means it's supplementanry angle is:
180 - 148 = 32
Since the angle of 2x - 16 and 32 lie on a straight line, their angle sum equals 180:
2x + 16 + 32 = 180
To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B16%2B32%3D180&pl=Solve']type it in our math engine [/URL]and we get:
x = [B]66[/B]

Sophie and Claire are having a foot race. Claire is given a 100-foot head-start. If Sophie is runn

Sophie and Claire are having a foot race. Claire is given a 100-foot head-start. If Sophie is running at 5 feet per second and Claire is running at 3 feet per second.
i. After how many seconds will Sophie catch Claire?
ii. If the race is 500 feet, who wins?
i.
Sophie's distance formula is given as D = 5s
Claire's distance formula is given as D = 3s + 100
Set them equal to each other
5s = 3s + 100
Subtract 3s from both sides:
2s = 100
Divide each side by 2
[B]s = 50[/B]
ii. [B]Sophie since after 50 seconds, she takes the lead and never gives it back.[/B]

Squaring a number equals 5 times that number

Squaring a number equals 5 times that number.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
Squaring this number:
x^2
5 times this number means we multiply by 5:
5x
The phrase [I]equals[/I] means we set both expressions equal to each other:
[B]x^2 = 5x [/B] <-- This is our algebraic expression
If you want to solve for x, then we subtract 5x from each side:
x^2 - 5x = 5x - 5x
Cancel the 5x on the right side, leaving us with 0:
x^2 - 5x = 0
Factor out x:
x(x - 5)
So we get x = 0 or [B]x = 5[/B]

Stacy sells art prints for $12 each. Her expenses are $2.50 per print, plus $38 for equipment. How m

Stacy sells art prints for $12 each. Her expenses are $2.50 per print, plus $38 for equipment. How many prints must she sell for her revenue to equal her expenses?
Let the art prints be p
Cost function is 38 + 2p
Revenue function is 12p
Set cost equal to revenue
12p = 38 + 2p
Subtract 2p from each side
10p = 38
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=10p%3D38&pl=Solve']equation calculator[/URL] gives us [B]p = 3.8[/B]

Stock A is worth 4.5. Stock B is worth 8.0. Stock C is worth 10.0. She purchased half as many shares

Stock A is worth 4.5. Stock B is worth 8.0. Stock C is worth 10.0. She purchased half as many shares of B as A and half as many shares of C as B. If her investments are worth 660, how many shares of each stock does she own?
Let s be the number of shares in Stock A. We have:
[LIST=1]
[*]A: 4.5s
[*]B: 8s/2 = 4s
[*]C: 10s/4 = 2.5s
[/LIST]
Value equation: 4.5s + 4s + 2.5s = 660
Combining like terms:
11s = 660
Using the [URL='http://www.mathcelebrity.com/1unk.php?num=11s%3D660&pl=Solve']equation calculator[/URL], we get [B]s = 60[/B] for Stock A
Stock B shares is equal to 1/2A = [B]30[/B]
Stock C shares is equal to 1/2B = [B]15[/B]

Students stuff envelopes for extra money. Their initial cost to obtain the information for the job w

Students stuff envelopes for extra money. Their initial cost to obtain the information for the job was $140. Each envelope costs $0.02 and they get paid $0.03per envelope stuffed. Let x represent the number of envelopes stuffed. (a) Express the cost C as a function of x. (b) Express the revenue R as a function of x. (c) Determine analytically the value of x for which revenue equals cost.
a) Cost Function
[B]C(x) = 140 + 0.02x[/B]
b) Revenue Function
[B]R(x) = 0.03x[/B]
c) Set R(x) = C(x)
140 + 0.02x = 0.03x
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=140%2B0.02x%3D0.03x&pl=Solve']equation solver[/URL], we get x = [B]14,000[/B]

Subtraction Equality Property

Free Subtraction Equality Property Calculator - Demonstrates the Subtraction Equality Property
Numerical Properties

Subtraction Property Of Inequality

Free Subtraction Property Of Inequality Calculator - Demonstrates the Subtraction Property Of Inequality
Numerical Properties

sum of 3 consecutive odd integers equals 1 hundred 17

sum of 3 consecutive odd integers equals 1 hundred 17
The sum of 3 consecutive odd numbers equals 117. What are the 3 odd numbers?
1) Set up an equation where our [I]odd numbers[/I] are n, n + 2, n + 4
2) We increment by 2 for each number since we have [I]odd numbers[/I].
3) We set this sum of consecutive [I]odd numbers[/I] equal to 117
n + (n + 2) + (n + 4) = 117
[SIZE=5][B]Simplify this equation by grouping variables and constants together:[/B][/SIZE]
(n + n + n) + 2 + 4 = 117
3n + 6 = 117
[SIZE=5][B]Subtract 6 from each side to isolate 3n:[/B][/SIZE]
3n + 6 - 6 = 117 - 6
[SIZE=5][B]Cancel the 6 on the left side and we get:[/B][/SIZE]
3n + [S]6[/S] - [S]6[/S] = 117 - 6
3n = 111
[SIZE=5][B]Divide each side of the equation by 3 to isolate n:[/B][/SIZE]
3n/3 = 111/3
[SIZE=5][B]Cancel the 3 on the left side:[/B][/SIZE]
[S]3[/S]n/[S]3 [/S]= 111/3
n = 37
Call this n1, so we find our other 2 numbers
n2 = n1 + 2
n2 = 37 + 2
n2 = 39
n3 = n2 + 2
n3 = 39 + 2
n3 = 41
[SIZE=5][B]List out the 3 consecutive odd numbers[/B][/SIZE]
([B]37, 39, 41[/B])
37 ? 1st number, or the Smallest, Minimum, Least Value
39 ? 2nd number
41 ? 3rd or the Largest, Maximum, Highest Value

sum of 5 times h and twice g is equal to 23

sum of 5 times h and twice g is equal to 23
Take this [U]algebraic expressions[/U] problem in pieces.
Step 1: 5 times h:
5h
Step 2: Twice g means we multiply g by 2:
2g
Step 3: sum of 5 times h and twice g means we add 2g to 5h
5h + 2g
Step 4: The phrase [I]is equal to[/I] means an equation, so we set 5h + 2g equal to 23:
[B]5h + 2g = 23[/B]

sum of a number and 7 is subtracted from 15 the result is 6.

Sum of a number and 7 is subtracted from 15 the result is 6.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
We take this expression in pieces. Sum of a number and 7
x + 7
Subtracted from 15
15 - (x + 7)
The result is means an equation, so we set this expression above equal to 6
[B]15 - (x + 7) = 6 <-- This is our algebraic expression[/B]
If the problem asks you to solve for x, we Group like terms
15 - x - 7 = 6
8 - x = 6
[URL='https://www.mathcelebrity.com/1unk.php?num=8-x%3D6&pl=Solve']Type 8 - x = 6 into the search engine[/URL], and we get [B]x = 2[/B]

Sum of a number and it's reciprocal is 6. What is the number?

Sum of a number and it's reciprocal is 6. What is the number?
Let the number be n.
The reciprocal is 1/n.
The word [I]is[/I] means an equation, so we set n + 1/n equal to 6
n + 1/n = 6
Multiply each side by n to remove the fraction:
n^2 + 1 = 6n
Subtract 6n from each side:
[B]n^2 - 6n + 1 = 0 [/B]<-- This is our algebraic expression
If the problem asks you to solve for n, then you [URL='https://www.mathcelebrity.com/quadratic.php?num=n%5E2-6n%2B1%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']type this quadratic equation into our search engine[/URL].

Sum of Five Consecutive Integers

Free Sum of Five Consecutive Integers Calculator - Finds five consecutive integers, if applicable, who have a sum equal to a number.
Sum of 5 consecutive integers

Sum of Four Consecutive Integers

Free Sum of Four Consecutive Integers Calculator - Finds four consecutive integers, if applicable, who have a sum equal to a number.
Sum of 4 consecutive integers

Sum of Three Consecutive Integers

Free Sum of Three Consecutive Integers Calculator - Finds three consecutive integers, if applicable, who have a sum equal to a number.
Sum of 3 consecutive integers

Suppose that 17 inches of wire costs 51 cents at the same rate, how many inches of wire can be bough

Suppose that 17 inches of wire costs 51 cents at the same rate, how many inches of wire can be bought for 42 cents?
Set up a proportion of inches of wire to cost, were w equals the inches of wire at 42 cents. We have:
17/51 = w/42
[URL='https://www.mathcelebrity.com/prop.php?num1=17&num2=w&den1=51&den2=42&propsign=%3D&pl=Calculate+missing+proportion+value']Type this proportion into our search engine[/URL], we get:
[B]w = 14[/B]

Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in ga

Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank. When carrying 20 gallons of fuel, the airplane weighs 2012 pounds. When carrying 55 gallons of fuel, it weighs 2208 pounds. How much does the airplane weigh if it is carrying 65 gallons of fuel?
Linear functions are written in the form of one dependent variable and one independent variable. Using g as the number of gallons and W(g) as the weight, we have:
W(g) = gx + c where c is a constant
We are given:
[LIST]
[*]W(20) = 2012
[*]W(55) = 2208
[/LIST]
We want to know W(65)
Using our givens, we have:
W(20) = 20x + c = 2012
W(55) = 55x + c = 2208
Rearranging both equations, we have:
c = 2012 - 20x
c = 2208 - 55x
Set them both equal to each other:
2012 - 20x = 2208 - 55x
Add 55x to each side:
35x + 2012 = 2208
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=35x%2B2012%3D2208&pl=Solve']equation solver[/URL], we see that x is 5.6
Plugging x = 5.6 back into the first equation, we get:
c = 2012 - 20(5.6)
c = 2012 - 112
c = 2900
Now that we have all our pieces, find W(65)
W(65) = 65(5.6) + 2900
W(65) = 264 + 2900
W(65) = [B]3264[/B]

Suppose we need 4 eggs to make a cake. If there are 24 eggs, write an inequality representing the po

Suppose we need 4 eggs to make a cake. If there are 24 eggs, write an inequality representing the possible number of cakes we can make.
Set up a proportion of eggs to cakes where c is the number of cakes per 24 eggs:
4/1 <= 24/c
[URL='https://www.mathcelebrity.com/prop.php?num1=4&num2=24&den1=1&den2=c&propsign=%3C&pl=Calculate+missing+proportion+value']Typing this proportion inequality into our search engine[/URL], we get:
[B]c <= 6[/B]

Suppose x is a natural number. When you divide x by 7 you get a quotient of q and a remainder of 6.

Suppose x is a natural number. When you divide x by 7 you get a quotient of q and a remainder of 6. When you divide x by 11 you get the same quotient but a remainder of 2. Find x.
[U]Use the quotient remainder theorem[/U]
A = B * Q + R where 0 ? R < B where R is the remainder when you divide A by B
Plugging in our numbers for Equation 1 we have:
[LIST]
[*]A = x
[*]B = 7
[*]Q = q
[*]R = 6
[*]x = 7 * q + 6
[/LIST]
Plugging in our numbers for Equation 2 we have:
[LIST]
[*]A = x
[*]B = 11
[*]Q = q
[*]R = 2
[*]x = 11 * q + 2
[/LIST]
Set both x values equal to each other:
7q + 6 = 11q + 2
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=7q%2B6%3D11q%2B2&pl=Solve']equation calculator[/URL], we get:
q = 1
Plug q = 1 into the first quotient remainder theorem equation, and we get:
x = 7(1) + 6
x = 7 + 6
[B]x = 13[/B]
Plug q = 1 into the second quotient remainder theorem equation, and we get:
x = 11(1) + 2
x = 11 + 2
[B]x = 13[/B]

Suppose you have $28.00 in your bank account and start saving $18.25 every week. Your friend has $16

Suppose you have $28.00 in your bank account and start saving $18.25 every week. Your friend has $161.00 in his account and is withdrawing $15 every week. When will your account balances be the same?
Set up savings and withdrawal equations where w is the number of weeks. B(w) is the current balance
[LIST]
[*]You --> B(w) = 18.25w + 28
[*]Your friend --> B(w) = 161 - 15w
[/LIST]
Set them equal to each other
18.25w + 28 = 161 - 15w
[URL='http://www.mathcelebrity.com/1unk.php?num=18.25w%2B28%3D161-15w&pl=Solve']Type that problem into the search engine[/URL], and you get [B]w = 4[/B].

Suppose you write a book. The printer charges $4 per book to print it, and you spend 5500 on adverti

Suppose you write a book. The printer charges $4 per book to print it, and you spend 5500 on advertising. You sell the book for $15 a copy. How many copies must you sell to break even.
Profit per book is:
P = 15 - 4
P = 11
We want to know the number of books (b) such that:
11b = 5500 <-- Breakeven means cost equals revenue
[URL='https://www.mathcelebrity.com/1unk.php?num=11b%3D5500&pl=Solve']Typing this equation into the search engine[/URL], we get:
b = [B]500[/B]

Susan makes and sells purses. The purses cost her $15 each to make, and she sells them for $30 each.

Susan makes and sells purses. The purses cost her $15 each to make, and she sells them for $30 each. This Saturday, she is renting a booth at a craft fair for $50. Write an equation that can be used to find the number of purses Susan must sell to make a profit of $295
Set up the cost function C(p) where p is the number of purses:
C(p) = Cost per purse * p + Booth Rental
C(p) = 15p + 50
Set up the revenue function R(p) where p is the number of purses:
R(p) = Sale price * p
R(p) = 30p
Set up the profit function which is R(p) - C(p) equal to 295
30p - (15p + 50) = 295
To solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=30p-%2815p%2B50%29%3D295&pl=Solve']we type it into our search engine[/URL] and we get:
p = [B]23[/B]

Taylor is playing a game using a die and a spinner. The spinner is divided into 4 equal parts with c

Taylor is playing a game using a die and a spinner. The spinner is divided into 4 equal parts with colors green, red, yellow, and purple. Taylor rolls the die and spins the spinner. What is the probability the die shows a 2 and the spinner lands on purple?
Probability of rolling a 2 on the die is 1/6
Probability of getting a purple on the spinner is 1/4
Since each event is independent, our joint probability is:
P(2 on the die and Purple on the spinner) = P(2 on the die) x P(Purple on the Spinner)
P(2 on the die and Purple on the spinner) = 1/6 x 1/4
P(2 on the die and Purple on the spinner) = [B]1/24[/B]

The ages of three siblings are all consecutive integers. The sum of of their ages is 39.

The ages of three siblings are all consecutive integers. The sum of of their ages is 39.
Let the age of the youngest sibling be n. This means the second sibling is n + 1. This means the oldest/third sibling is n + 2.
So what we want is the[URL='https://www.mathcelebrity.com/sum-of-consecutive-numbers.php?num=sumof3consecutiveintegersequalto39&pl=Calculate'] sum of 3 consecutive integers equal to 39[/URL]. We type this command into our search engine. We get:
n = 12. So the youngest sibling is [B]12[/B].
The next sibling is 12 + 1 = [B]13[/B]
The oldest/third sibling is 12 + 2 = [B]14[/B]

The arithmetic mean (average) of 17, 26, 42, and 59 is equal to the arithmetic mean of 19 and N. Wha

The arithmetic mean (average) of 17, 26, 42, and 59 is equal to the arithmetic mean of 19 and N. What is the value of N ?
Average of the first number set is [URL='https://www.mathcelebrity.com/statbasic.php?num1=17%2C26%2C42%2C59&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']using our average calculator[/URL] is:
36
Now, the mean (average) or 19 and N is found by adding them together an dividing by 2:
(19 + N)/2
Since both number sets have equal means, we set (19 + N)/2 equal to 36:
(19 + N)/2 = 36
Cross multiply:
19 + N = 36 * 2
19 + n = 72
To solve for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=19%2Bn%3D72&pl=Solve']type this equation into our search engine[/URL] and we get:
n = [B]53[/B]

The auditorium can hold a maximum of 150 people

The auditorium can hold a maximum of 150 people
We want an inequality for the number of people (p) in the auditorium.
The word [I]maximum[/I] means [I]no more than[/I] or [I]less than or equal to[/I]. So we have:
[B]p <= 150[/B]

The average of a number and double the number is 25.5

Let x equal "a number".
Double the number is 2x.
The average is (x + 2x)/2
Combine the terms in the numerator:
3x/2
The word [I]is[/I] means equal to, so we set 3x/2 equal to 25.5
3x/2 = 25.5
Cross multiply the 2:
3x = 51
Divide each side by 3
[B]x = 17[/B]

the average, a, is at least 85

the average, a, is at least 85
At least is an inequality. It also means greater than or equal to, so we have:
[B]a >= 85[/B]

The base of a triangle with a height of 7 units is represented by the formula b=2/7A. The base of th

The base of a triangle with a height of 7 units is represented by the formula b=2/7A. The base of the triangle is less than 10 units. Write and solve an inequality that represents the possible area A of the triangle
We're given:
b=2/7A
We're also told that b is less than 10. So we have:
2/7A < 10
2A/7 < 10
Cross multiply:
2A < 7 * 10
2A < 70
Divide each side of the inequality by 2 to isolate A
2A/2 < 70/2
Cancel the 2's on the left side and we get:
A < [B]35[/B]

The bigger of 2 numbers in 5 larger than the smaller. Twice the smaller, increased by, twice the lar

The bigger of 2 numbers in 5 larger than the smaller. Twice the smaller, increased by, twice the larger, is equal to 50. Find each number.
Let the big number be b. Let the small number be s. We're given two equations:
[LIST=1]
[*]b = s + 5
[*]2s + 2b = 50
[/LIST]
Substitute equation (1) into equation (2)
2s + 2(s + 5) = 50
[URL='https://www.mathcelebrity.com/1unk.php?num=2s%2B2%28s%2B5%29%3D50&pl=Solve']Type this equation into our search engine[/URL], and we get:
[B]s = 10[/B]
Now substitute s = 10 into equation (1) to solve for b:
b = 10 + 5
[B]b = 15[/B]

The blue star publishing company produces daily "Star news". It costs $1200 per day to operate regar

The blue star publishing company produces daily "Star news". It costs $1200 per day to operate regardless of whether any newspaper are published. It costs 0.20 to publish each newspaper. Each daily newspaper has $850 worth of advertising and each newspaper is sold for $.30. Find the number of newspaper required to be sold each day for the Blue Star company to 'break even'. I.e all costs are covered.
Build our cost function where n is the number of newspapers sold:
C(n) = 1200+ 0.2n
Now build the revenue function:
R(n) = 850 + 0.3n
Break even is where cost and revenue are equal, so set C(n) = R(n)
1200+ 0.2n = 850 + 0.3n
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=1200%2B0.2n%3D850%2B0.3n&pl=Solve']equation solver[/URL], we get:
[B]n = 3,500[/B]

The cost of 25 apples is less than $9.50. The cost of 12 apples is more than 3.60. What are the poss

The cost of 25 apples is less than $9.50. The cost of 12 apples is more than 3.60. What are the possible prices of one apple?
Let a be the price of each apple. We're given 2 inequalities:
[LIST=1]
[*]25a < 9.50
[*]12a > 3.60
[/LIST]
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=25a%3C9.50&pl=Show+Interval+Notation']Typing 25a < 9.50 into our search engine[/URL], we get a < 0.38
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=12a%3E3.60&pl=Show+Interval+Notation']Typing 12a > 360 into our search engine[/URL], we get a > 0.3
Therefore, the possible prices a of one apple are expressed as the inequality:
[B]0.3 < a < 0.38[/B]

The cost of a field trip is $220 plus $7 per student. If the school can spend at most $500, how many

The cost of a field trip is $220 plus $7 per student. If the school can spend at most $500, how many students can go on the field trip?
Set up the inequality where s is the number of students:
C(s) = 220 + 7s
We want C(s) <= 500, since at most means no more than
220 + 7s <= 500
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=220%2B7s%3C%3D500&pl=Solve']inequality calculator[/URL], we get:
[B]s <= 40[/B]

The cost of a gallon of milk (m) is .50 more than 5 times the cost of a gallon of water (w). If a ga

The cost of a gallon of milk (m) is .50 more than 5 times the cost of a gallon of water (w). If a gallon of milk cost 3.75, what is the cost of a gallon of water?
We're given:
m = 5w + 0.50
m = $3.75
Set them equal to each other:
5w + 0.50 = 3.75
[URL='https://www.mathcelebrity.com/1unk.php?num=5w%2B0.50%3D3.75&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]w = 0.65[/B]

The cost of renting a rototiller is $19.50 for the first hour and $7.95 for each additional hour. Ho

The cost of renting a rototiller is $19.50 for the first hour and $7.95 for each additional hour. How long can a person have the rototiller if the cost must be less than $95?
Setup the inequality:
$19.50 + $7.95x < $95
Subtract 19.50 from both sides:
7.95x < 75.50
Divide each side of the inequality by 7.95 to isolate x
x < 9.5
The next lowest integer is 9. So we take 9 + the first hour of renting to get [B]10 total hours[/B].
Check our work:
$7.95 * 9.5 + $19.50
$71.55 + $19.50 = $91.05

The cost to rent a construction crane is 450 per day plus 150 per hour. What is the maximum number o

The cost to rent a construction crane is 450 per day plus 150 per hour. What is the maximum number of hours the crane can be used each day if the rental cost is not to exceed 1650 per day?
Set up the cost function where h is the number of hours:
C(h) = 150h + 450
We want C(h) <= 1650:
150h + 450 <= 1650
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=150h%2B450%3C%3D1650&pl=Solve']equation/inequality solver[/URL], we get:
[B]h <= 8[/B]

The cube of x is less than 15

The cube of x is less than 15
The cube of x means we raise x to the 3rd power:
x^3
Less than 15 means we setup the following inequality
[B]x^3 < 15[/B]

The difference between 2 numbers is 108. 6 times the smaller is equal to 2 more than the larger. Wh?

The difference between 2 numbers is 108. 6 times the smaller is equal to 2 more than the larger. What are the numbers?
Let the smaller number be x. Let the larger number be y. We're given:
[LIST=1]
[*]y - x = 108
[*]6x = y + 2
[/LIST]
Rearrange (1) by adding x to each side:
[LIST=1]
[*]y = x + 108
[/LIST]
Substitute this into (2):
6x = x + 108 + 2
Combine like terms
6x = x +110
Subtract x from each side:
5x = 110
[URL='https://www.mathcelebrity.com/1unk.php?num=5x%3D110&pl=Solve']Plugging this equation into our search engine[/URL], we get:
x = [B]22[/B]

The difference between a and b is 10

The difference between a and b is 10.
The problem asks for an algebraic expression. Let's take each piece one by one:
[I]Difference between[/I] means we subtract:
a - b
The phrase [I]is [/I]means an equation, so we set a - b equal to 10
[B]a - b = 10[/B]

The difference between A and B is no less than 30

The difference between A and B is no less than 30
The difference between means we subtract.
No less than means greater than or equal to, so we have the following inequality;
[B]A - B >= 30[/B]

the difference between A and B is no less than 30.

the difference between A and B is no less than 30.
The difference between a and b:
a - b
The phrase [I]no less than[/I] means an inequality. You can also say this as [I]greater than or equal to[/I].
[B]a - b >= 30[/B]

The difference between a number and 9 is 27. Find that number

The difference between a number and 9 is 27. Find that number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
The difference between a number and 9
x - 9
The word [I]is[/I] means equal to, so we set x - 9 equal to 27:
x - 9 = 27
To solve this equation for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=x-9%3D27&pl=Solve']type it in our math engine[/URL] and we get:
x = [B]36[/B]

The difference between two positive numbers is 5 and the square of their sum is 169

The difference between two positive numbers is 5 and the square of their sum is 169.
Let the two positive numbers be a and b. We have the following equations:
[LIST=1]
[*]a - b = 5
[*](a + b)^2 = 169
[*]Rearrange (1) by adding b to each side. We have a = b + 5
[/LIST]
Now substitute (3) into (2):
(b + 5 + b)^2 = 169
(2b + 5)^2 = 169
[URL='https://www.mathcelebrity.com/community/forums/calculator-requests.7/create-thread']Run (2b + 5)^2 through our search engine[/URL], and you get:
4b^2 + 20b + 25
Set this equal to 169 above:
4b^2 + 20b + 25 = 169
[URL='https://www.mathcelebrity.com/quadratic.php?num=4b%5E2%2B20b%2B25%3D169&pl=Solve+Quadratic+Equation&hintnum=+0']Run that quadratic equation in our search engine[/URL], and you get:
b = (-9, 4)
But the problem asks for [I]positive[/I] numbers. So [B]b = 4[/B] is one of our solutions.
Substitute b = 4 into equation (1) above, and we get:
a - [I]b[/I] = 5
[URL='https://www.mathcelebrity.com/1unk.php?num=a-4%3D5&pl=Solve']a - 4 = 5[/URL]
[B]a = 9
[/B]
Therefore, we have [B](a, b) = (9, 4)[/B]

The difference in Julies height and 9 is 48 letting j be Julie's height

The difference in Julies height and 9 is 48 letting j be Julie's height
Step 1: If Julie's height is represented with the variable j, then we subtract 9 from j since the phrase [I]difference[/I] means we subtract:
j - 9
Step 2: The word [I]is[/I] means an equation, so we set j - 9 equal to 48 for our final algebraic expression:
[B]j - 9 = 48[/B]

The difference of 100 and x is 57

The difference of 100 and x means we subtract x from 100:
100 - x
Is means equal to, so we set our expression above equal to 57
[B]100 - x = 57
[/B]
If you want to solve for x, use our [URL='http://www.mathcelebrity.com/1unk.php?num=100-x%3D57&pl=Solve']equation calculator[/URL]

The difference of a number and 6 is the same as 5 times the sum of the number and 2. What is the num

The difference of a number and 6 is the same as 5 times the sum of the number and 2. What is the number?
We have two expressions:
[U]Expression 1: [I]The difference of a number and 6[/I][/U]
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The difference of a number and 6 means we subtract 6 from x:
x - 6
[U]Expression 2: [I]5 times the sum of the number and 2[/I][/U]
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The sum of a number and 2 means we add 2 to x:
x + 2
5 times the sum means we multiply x + 2 by 5
5(x + 2)
[U]For the last step, we evaluate the expression [I]is the same as[/I][/U]
This means equal to, so we set x - 6 equal to 5(x + 2)
[B]x - 6 = 5(x + 2)[/B]

The difference of a number times 3 and 6 is equal to 7 . Use the variable w for the unknown n

The difference of a number times 3 and 6 is equal to 7 . Use the variable w for the unknown number.
The phrase a number uses the variable w.
3 times w is written as 3w
The difference of 3w and 6 is written as 3w - 6
Set this equal to 7
[B]3w - 6 = 7
[/B]
This is our algebraic expression. To solve this equation for w, we [URL='http://www.mathcelebrity.com/1unk.php?num=3w-6%3D7&pl=Solve']type the algebraic expression into our search engine[/URL].

The difference of five and five y is the same as eight and two y

The difference of five and five y
5 - 5y
eight and two y
8 + 2y
The phrase [I]is the same as[/I] means equal to. Set 5 - 5y equal to 8 + 2y for our final algebraic expression
[B]5 - 5y = 8 + 2y[/B]
[B][/B]
If the problem asks you to solve for y:
Add 5y to each side:
5 = 8 + 7y
Subtract 8 from each side:
7y = -3
Divide each side by 7:
[B]y= -3/7[/B]

The difference of twice a number and 4 is at least -27

The difference of twice a number and 4 is at least -27.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
Twice a number means multiply the number by 2
2x
[I]and 4[/I] means we add 4 to our expression:
2x + 4
[I]Is at least[/I] means an inequality. In this case, it's greater than or equal to:
[B]2x + 4 >= -27
[/B]
To solve this inequality, [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B4%3E%3D-27&pl=Solve']type it in the search engine[/URL].

The difference of twice a number and 6 is at most 28

The difference of twice a number and 6 is at most 28
This is an algebraic expression. Let's take it in parts:
[LIST=1]
[*]The phrase [I]a number[/I], means an arbitrary variable, let's call it x
[*]Twice this number means we multiply x by 2: 2x
[*][I]The difference of[/I] means subtract, so we subtract 6 to 2x: 2x - 6
[*][I]Is at most [/I]means less than or equal to, so we create an inequality where 2x - 6 is less than or equal to 28, using the <= sign
[/LIST]
[B]2x - 6 <= 28
[/B]
If you wish to solve this inequality, [URL='https://www.mathcelebrity.com/1unk.php?num=2x-6%3C%3D28&pl=Solve']click this link[/URL].

the difference of twice a number and 8 is at most -30

the difference of twice a number and 8 is at most -30.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
Twice this number means we multiply by 2, so we have 2x.
We take the difference of 2x and 8, meaning we subtract 8:
2x - 8
Finally, the phrase [I]at most[/I] means an inequality, also known as less than or equal to:
[B]2x - 8 <= 30 <-- This is our algebraic expression
[/B]
To solve this, we [URL='https://www.mathcelebrity.com/1unk.php?num=2x-8%3C%3D30&pl=Solve']type it into the search engine[/URL] and get x <= 19.

the difference of x and 5 is 2 times of x

the difference of x and 5 is 2 times of x
The difference of x and 5 means we subtract 5 from x
x - 5
The word [I]is[/I] means an equation, so we set x - 5 equal to 2 times x
[B]x - 5 = 2x[/B]

The doctor told Anna that tomorrow she will have to check her blood pressure every 15 minutes from 1

The doctor told Anna that tomorrow she will have to check her blood pressure every 15 minutes from 10:00 AM to 4:00 PM. How many times does she have to take her blood pressure?
10:00 A.M. to 4:00 P.M. is 6 hours.
Each hour is 60 minutes
60 minutes divided by 15 minutes equals 4 blood pressure checks per hour.
4 blood pressure checks per hour * 6 hours = [B]24 blood pressure checks[/B]

The enrollment at High School R has been increasing by 20 students per year. High School R currently

The enrollment at High School R has been increasing by 20 students per year. High School R currently has 200 students. High School T has 400 students and is decreasing 30 students per year. When will the two school have the same enrollment of students?
Set up the Enrollment function E(y) where y is the number of years.
[U]High School R:[/U]
[I]Increasing[/I] means we add
E(y) = 200 + 20y
[U]High School T:[/U]
[I]Decreasing[/I] means we subtract
E(y) = 400 - 30y
When the two schools have the same enrollment, we set the E(y) functions equal to each other
200 + 20y = 400 - 30y
To solve this equation for y, we [URL='https://www.mathcelebrity.com/1unk.php?num=200%2B20y%3D400-30y&pl=Solve']type it in our search engine[/URL] and we get:
y = [B]4[/B]

The first plan has $14 monthly fee and charges an additional $.14 for each minute of calls. The seco

The first plan has $14 monthly fee and charges an additional $.14 for each minute of calls. The second plan had a $21 monthly fee and charges an additional $.10 for each minute of calls. For how many minutes of calls will the cost of the two plans be equal?
Set up the cost equation C(m) for the first plan, where m is the amount of minutes you use
C(m) = 0.14m + 14
Set up the cost equation C(m) for the second plan, where m is the amount of minutes you use
C(m) = 0.10m + 21
Set them equal to each other:
0.14m + 14 = 0.10m + 21
[URL='https://www.mathcelebrity.com/1unk.php?num=0.14m%2B14%3D0.10m%2B21&pl=Solve']Typing this equation into our search engine[/URL], we get:
m = [B]175[/B]

The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered t

The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as Benford's Law. For example, the following distribution represents the first digits in 231 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.
Digit, Probability
1, 0.301
2, 0.176
3, 0.125
4, 0.097
5, 0.079
6, 0.067
7, 0.058
8, 0.051
9, 0.046
[B][U]Fradulent Checks[/U][/B]
Digit, Frequency
1, 36
2, 32
3, 45
4, 20
5, 24
6, 36
7, 15
8, 16
9, 7
Complete parts (a) and (b).
(a) Using the level of significance α = 0.05, test whether the first digits in the allegedly fraudulent checks obey Benford's Law. Do the first digits obey the Benford's Law?

Yes or No Based on the results of part (a), could one think that the employe is guilty of embezzlement? Yes or No Show frequency percentages Digit Fraud Probability Benford Probability 1 0.156 0.301 2 0.139 0.176 3 0.195 0.125 4 0.087 0.097 5 0.104 0.079 6 0.156 0.067 7 0.065 0.058 8 0.069 0.051 9 0.03 0.046 Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277 Critical Value Excel: =CHIINV(0.95,8) = 2.733 Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.

Yes or No Based on the results of part (a), could one think that the employe is guilty of embezzlement? Yes or No Show frequency percentages Digit Fraud Probability Benford Probability 1 0.156 0.301 2 0.139 0.176 3 0.195 0.125 4 0.087 0.097 5 0.104 0.079 6 0.156 0.067 7 0.065 0.058 8 0.069 0.051 9 0.03 0.046 Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277 Critical Value Excel: =CHIINV(0.95,8) = 2.733 Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.

The function f(x) = e^x(x - 3) has a critical point at x =

The function f(x) = e^x(x - 3) has a critical point at x =
The critical point is where the derivative equals 0.
We multiply through for f(x) to get:
f(x) = xe^x - 3e^x
Using the product rule on the first term f'g + fg', we get:
f'(x) = xe^x + e^x - 3e^x
f'(x) = xe^x -2e^x
f'(x) = e^x(x - 2)
We want f'(x) = 0
e^x(x - 2) = 0
When [B]x = 2[/B], then f'(x) = 0

The function f(x) = x^3 - 48x has a local minimum at x = and a local maximum at x = ?

The function f(x) = x^3 - 48x has a local minimum at x = and a local maximum at x = ?
f'(x) = 3x^2 - 48
Set this equal to 0:
3x^2 - 48 = 0
Add 48 to each side:
3x^2 = 48
Divide each side by 3:
x^2 = 16
Therefore, x = -4, 4
Test f(4)
f(4) = 4^3 - 48(4)
f(4) = 64 - 192
f(4) = [B]-128 <-- Local minimum[/B]
Test f(-4)
f(-4) = -4^3 - 48(-4)
f(-4) = -64 + 192
f(-4) = [B]128 <-- Local maximum[/B]

The function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the ba

The function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the basis of a ticket price, x. Both the profit and the ticket price are in dollars. What is the maximum profit, and how much should the tickets cost?
Take the [URL='http://www.mathcelebrity.com/dfii.php?term1=-30x%5E2+%2B+360x+%2B+785&fpt=0&ptarget1=0&ptarget2=0&itarget=0%2C1&starget=0%2C1&nsimp=8&pl=1st+Derivative']derivative of the profit function[/URL]:
P'(x) = -60x + 360
We find the maximum when we set the profit derivative equal to 0
-60x + 360 = 0
Subtract 360 from both sides:
-60x = -360
Divide each side by -60
[B]x = 6 <-- This is the ticket price to maximize profit[/B]
Substitute x = 6 into the profit equation:
P(6) = -30(6)^2 + 360(6) + 785
P(6) = -1080 + 2160 + 785
[B]P(6) = 1865[/B]

The graph shows the average length (in inches) of a newborn baby over the course of its first 15 mon

The graph shows the average length (in inches) of a newborn baby over the course of its first 15 months. Interpret the RATE OF CHANGE of the graph.
[IMG]http://www.mathcelebrity.com/community/data/attachments/0/rate-of-change-wp.jpg[/IMG]
Looking at our graph, we have a straight line. For straight lines, rate of change [U][I]equals[/I][/U] slope.
Looking at a few points, we have:
(0, 20), (12, 30)
Using our [URL='https://www.mathcelebrity.com/slope.php?xone=0&yone=20&slope=+2%2F5&xtwo=12&ytwo=30&pl=You+entered+2+points']slope calculator for these 2 points[/URL], we get a slope (rate of change) of:
[B]5/6[/B]

The height of an object t seconds after it is dropped from a height of 300 meters is s(t)=-4.9t^2 +3

The height of an object t seconds after it is dropped from a height of 300 meters is s(t)=-4.9t^2 +300. Find the average velocity of the object during the first 3 seconds? (b) Use the Mean value Theorem to verify that at some time during the first 3 seconds of the fall the instantaneous velocity equals the average velocity. Find that time.
Average Velocity:
[ f(3) - f(0) ] / ( 3 - 0 )
Calculate f(3):
f(3) = -4.9(3^2) + 300
f(3) = -4.9(9) + 300
f(3) = -44.1 + 300
f(3) = 255.9
Calculate f(0):
f(0) = -4.9(0^2) + 300
f(0) = 0 + 300
f(0) = 300
So we have average velocity:
Average velocity = (255.9 - 300)/(3 - 0)
Average velocity = -44.1/3
Average velocity = -[B]14.7
[/B]
Velocity is the first derivative of position
s(t)=-4.9t^2 +300
s'(t) = -9.8t
So we set velocity equal to average velocity:
-9.8t = -14.7
Divide each side by -9.8 to solve for t, we get [B]t = 1.5[/B]

The January record high temperature in Austin, Texas is 92 degrees. The January record low temperatu

The January record high temperature in Austin, Texas is 92 degrees. The January record low temperature is -1 degree. What is the difference between the record high and low temperatures?
Difference = High - Low
Difference = 92 - -1
Difference = 92 + 1. <-- Since minus negative equals positive
Difference = [B]93 degrees[/B]

The larger of 2 numbers is 1 more than 3 times the smaller number

The larger of 2 numbers is 1 more than 3 times the smaller number.
Let the larger number be l. Let the smaller number be s. The algebraic expression is:
3 times the smaller number is written as:
3s
1 more than that means we add 1
3s + 1
Our final algebraic expression uses the word [I]is[/I] meaning an equation. So we set l equal to 3s + 1
[B]l = 3s + 1[/B]

The length of a rectangle is equal to triple the width. Find the length of the rectangle if the peri

The length of a rectangle is equal to triple the width. Find the length of the rectangle if the perimeter is 80 inches.
The perimeter (P) of a rectangle is:
2l + 2w = P
We're given two equations:
[LIST=1]
[*]l = 3w
[*]2l + 2w = 80
[/LIST]
We substitute equation 1 into equation 2 for l:
2(3w) + 2w = 80
6w + 2w = 80
To solve this equation for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=6w%2B2w%3D80&pl=Solve']type it in our search engine[/URL] and we get:
w = 10
To solve for the length (l), we substitute w = 10 into equation 1 above:
l = 3(10)
l = [B]30[/B]

The length of a train car is 50.6 feet. This is 5.8 feet less than 6 times the width. What is the wi

The length of a train car is 50.6 feet. This is 5.8 feet less than 6 times the width. What is the width?
5.8 feet less than 6 times the width is an algebraic expression:
6w - 5.8
We set this equal to the length of 50.6
6w - 5.8 = 50.6
Add 5.8 to each side:
6w - 5.8 + 5.8 = 50.6 + 5.8
Cancel the 5.8 on the left side:
6w = 56.4
Divide each side by 6:
6w/6 = 56.4/6
[URL='http://www.mathcelebrity.com/1unk.php?num=6w-5.8%3D50.6&pl=Solve']Typing this problem into the search engine[/URL], we get [B]w = 9.4[/B].
[MEDIA=youtube]gfM-d_Ej728[/MEDIA]

The length of a wooden frame is 1 foot longer than its width and its area is equal to 12ft˛

The length of a wooden frame is 1 foot longer than its width and its area is equal to 12ft˛
The frame is a rectangle. The area of a rectangle is A = lw. So were given:
[LIST=1]
[*]l = w + 1
[*]lw = 12
[/LIST]
Substitute equation (1) into equation (2) for l:
(w + 1) * w = 12
Multiply through and simplify:
w^2 + w = 12
We have a quadratic equation. To solve for w, we type this equation into our search engine and we get two solutions:
w = 3
w = -4
Since width cannot be negative, we choose the positive result and have:
w = [B]3[/B]
To solve for length, we plug w = 3 into equation (1) above and get:
l = 3 + 1
l = [B]4[/B]

The longest bridge in america is 1700 ft long. Write an inequality that describes the length of ever

The longest bridge in america is 1700 ft long. Write an inequality that describes the length of every bridge.
Let the bridge length be b. Since no bridge will ever be greater than 1700 ft, we have:
[B]b <= 1700[/B]

The mean age of 10 women in an office is 30 years old. The mean age of 10 men in an office is 29 yea

The mean age of 10 women in an office is 30 years old. The mean age of 10 men in an office is 29 years old. What is the mean age (nearest year) of all the people in the office?
Mean is another word for [U]average[/U].
Mean age of women = Sum of all ages women / number of women
We're told mean age of women is 30, so we have:
Sum of all ages women / 10 = 30
Cross multiply, and we get:
Sum of all ages of women = 30 * 10
Sum of all ages of women = 300
Mean age of men = Sum of all ages men / number of men
We're told mean age of men is 29, so we have:
Sum of all ages men / 10 = 29
Cross multiply, and we get:
Sum of all ages of men = 29 * 10
Sum of all ages of men = 290
[U]Calculate mean age (nearest year) of all the people in the office:[/U]
mean age of all the people in the office = Sum of all ages of people in the office (men and women) / Total number of people in the office
mean age of all the people in the office = (300 + 290) / (10 + 10)
mean age of all the people in the office = 590 / 20
mean age of all the people in the office = 29.5
The question asks for nearest year. Since this is a decimal, we round down to either 29 or up to 30.
Because the decimal is greater or equal to 0.5 (halfway), we round [U]up[/U] to [B]30[/B]

The minimum daily requirement of vitamin C for 14 year olds is at least 50 milligrams per day. An av

The minimum daily requirement of vitamin C for 14 year olds is at least 50 milligrams per day. An average sized apple contains 6 milligrams of vitamin C. How many apples would a person have to eat each day to satisfy this requirement?
Let a be the number of apples required. The phrase [I]at least[/I] means greater than or equal to, so we have the inequality:
6a >= 50
To solve this inequality, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=6a%3E%3D50&pl=Show+Interval+Notation']type it in our math engine[/URL] and we get:
[B]a >= 8.3333 apples or rounded up to a full number, we get 9 apples[/B]

The negative of the sum of C and D is equal to the difference of the negative of C and D

The negative of the sum of C and D is equal to the difference of the negative of C and D
The negative of the sum of C and D means -1 times the sum of C and D
-(C + D)
Distribute the negative sign:
-C - D
the difference of the negative of C and D means we subtract D from negative C
-C - D
So this statement is [B]true[/B] since -C - D = -C - D

The phone company charges Rachel 12 cents per minute for her long distance calls. A discount company

The phone company charges Rachel 12 cents per minute for her long distance calls. A discount company called Rachel and offered her long distance service for 1/2 cent per minute, but will charge a $46 monthly fee. How many minutes per month must Rachel talk on the phone to make the discount a better deal?
Minutes Rachel talks = m
Current plan cost = 0.12m
New plan cost = 0.005m + 46
Set new plan equal to current plan:
0.005m + 46 = 0.12m
Solve for [I]m[/I] in the equation 0.005m + 46 = 0.12m
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 0.005m and 0.12m. To do that, we subtract 0.12m from both sides
0.005m + 46 - 0.12m = 0.12m - 0.12m
[SIZE=5][B]Step 2: Cancel 0.12m on the right side:[/B][/SIZE]
-0.115m + 46 = 0
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 46 and 0. To do that, we subtract 46 from both sides
-0.115m + 46 - 46 = 0 - 46
[SIZE=5][B]Step 4: Cancel 46 on the left side:[/B][/SIZE]
-0.115m = -46
[SIZE=5][B]Step 5: Divide each side of the equation by -0.115[/B][/SIZE]
-0.115m/-0.115 = -46/-0.115
m = [B]400
She must talk over 400 minutes for the new plan to be a better deal
[URL='https://www.mathcelebrity.com/1unk.php?num=0.005m%2B46%3D0.12m&pl=Solve']Source[/URL][/B]

The price of a baseball glove is no more than $38.95

The price of a baseball glove is no more than $38.95.
Let p be the price of the baseball glove. The phrase "no more than" means less than or equal to. Our inequality is:
p <= $38.95

The price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she pa

The price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she paid $75. What is the cost of the cheap backpack?
backpack cost = b
Cheap backpack = b - 15
The total of both items equals 75:
b + b - 15 = 75
Solve for [I]b[/I] in the equation b + b - 15 = 75
[SIZE=5][B]Step 1: Group the b terms on the left hand side:[/B][/SIZE]
(1 + 1)b = 2b
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2b - 15 = + 75
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -15 and 75. To do that, we add 15 to both sides
2b - 15 + 15 = 75 + 15
[SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE]
2b = 90
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2b/2 = 90/2
b = 45
Cheap backpack = 45 - 15 = [B]30
[URL='https://www.mathcelebrity.com/1unk.php?num=b%2Bb-15%3D75&pl=Solve']Source[/URL][/B]

the product of 2 less than a number and 7 is 13

the product of 2 less than a number and 7 is 13
Take this algebraic expression in [U]4 parts[/U]:
Part 1 - The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
Part 2 - 2 less than a number means we subtract 2 from x
x - 2
Part 3 - The phrase [I]product[/I] means we multiply x - 2 by 7
7(x - 2)
Part 4 - The phrase [I]is[/I] means an equation, so we set 7(x - 2) equal to 13
[B]7(x - 2) = 13[/B]

The product of 8 and a number k is greater than 4 and no more than 16

Let's take this by pieces.
The product of 8 and a number k is written as: 8k.
Since it's greater than 4, but not more than 16, we include this in the middle of an inequality statement.
4 < 8k <= 16
Notice no more than has an equal sign, it means less than or equal to.
Greater does not include an equal sign.

the product of a number and 15 is not less than 15

the product of a number and 15 is not less than 15
The phrase [I]a number[/I] means an arbitrary variable. Let's call it x.
x
the product of a number and 15 means we multiply x by 15
15x
The phrase [I]not less than[/I] means greater than or equal to. We set 15x greater than prequel to 15
[B]15x >= 15 <-- This is our algebraic expression
[/B]
[U]If the problem asks you to solve for x:[/U]
Divide each side by 15:
15x/15 >= 15/15
[B]x >= 1[/B]

The product of a number b and 3 is no less than 12.

The product of a number b and 3 is no less than 12.
A number b is just written as b. So we have:
The product of b and 3 is no less than 12.
take this in parts:
[LIST]
[*]The product of b and 3: 3b
[*]The phrase [I]is no less than[/I] means an inequality, so we have greater than or equal to. We set 3b greater than or equal to 12
[/LIST]
[B]3b >= 12[/B]

The product of x and 7 is not greater than 21

The product of x and 7 is not greater than 21
The product of x and 7:
7x
Is not greater than means less than or equal to, so we have our algebraic expression:
7x <= 21
If you want to solve this inequality and interval notation, use our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=7x%3C%3D21&pl=Show+Interval+Notation']calculator[/URL].

The product of x and u is not greater than 21

The product of x and u is not greater than 21
The product of x and u
xu
Not greater than means less than or equal to:
xu <= 21

the quotient of 3 and u is equal to 52 divided by u

the quotient of 3 and u is equal to 52 divided by u
Take this algebraic expression in 3 parts:
[LIST=1]
[*]The quotient of 3 and u means we divide 3 by u: 3/u
[*]52 divided by u means we divide 52 by u: 52/u
[*]The phrase [I]is equal to[/I] means an equation, so we set (1) equal to (2)
[/LIST]
[B]3/u = 52/u[/B]

the quotient of 4 more than a number and 7 is 10

the quotient of 4 more than a number and 7 is 10
Take this algebraic expression in pieces:
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
4 more than a number means we add 4 to x:
x + 4
The quotient of 4 more than a number and 7 means we divide x + 4 by 7
(x + 4)/7
The word [I]is[/I] means an equation, so we set (x + 4)/7 equal to 10
[B](x + 4)/7 = 10[/B]

the quotient of d and 182 is the same as w minus 137

The quotient of d and 182 is the same as w minus 137
Take this algebraic expression in 3 parts:
The quotient of d and 182
d/182
w minus 137
w - 137
The phrase [I]is the same as[/I] means we set d/182 equal to w - 137
[B]d/182 = w - 137[/B]

The quotient of t and 12 is the sum of s and r.

The quotient of t and 12 is the sum of s and r.
Step 1: The quotient of t and 12 is:
t/12
Step 2: The Sum of s and r is
s + r
Step 3: The word [I]is[/I] means equal to, so we set t/12 equal to s + r
[B]t/12 = s + r[/B]

the quotient of x and y is equal to the sum of a and b

the quotient of x and y is equal to the sum of a and b
The quotient of x and y:
x/y
The sum of a and b:
a + b
The phrase [I]is equal to[/I] means an equation, so we set x/y equal to a + b:
[B]x/y = a + b[/B]

The ratio between the sum of a and b and the difference of a and b is equal to 5.

The ratio between the sum of a and b and the difference of a and b is equal to 5.
The sum of a and b:
a + b
The difference of a and b:
a - b
The ratio between the sum of a and b and the difference of a and b
(a + b)/(a - b)
The ratio between the sum of a and b and the difference of a and b is equal to 5.
[B](a + b)/(a - b) = 5[/B]

The ratio of the measures of the 3 angles of a triangle is 1:2:3. What is the measure of the largest

The ratio of the measures of the 3 angles of a triangle is 1:2:3. What is the measure of the largest angle in degrees?
Let the smallest angle be x.
Then we have 3 angles based on the ratio: x, 2x, 3x
We know the sum of the angles of a triangle equals 180. So we have:
x + 2x + 3x = 180
6x = 180
Divide each side by 6:
6x/6 = 180/6
x = 30
The largest angle is 3(30) = [B]90
[MEDIA=youtube]l8Lc6YtK9dg[/MEDIA][/B]

The recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg). One cup of milk con

The recommended daily calcium intake for a 20-year-old is 1,000 milligrams (mg). One cup of milk contains 299 mg of calcium and one cup of juice contains 261 mg of calcium. Which of the following inequalities represents the possible number of cups of milk [I]m[/I] and cups of juice [I]j[/I] a 20-year-old could drink in a day to meet or exceed the recommended daily calcium intake from these drinks alone?
Total calcium = Milk calcium + Juice Calcium
Calculate Milk Calcium:
Milk Calcium = 299m where m is the number of cups of milk
Calculate Juice Calcium:
Juice Calcium = 261j where j is the number of cups of juice
The phrase [I]meet or exceed[/I] means greater than or equal to, so we have an inequality, where Total Calcium is greater than or equal to 1000. So we write our inequality as:
Milk calcium + Juice Calcium >= Total Calcium
[B]299m + 261j >= 1000[/B]

The scale of a map shows that 1/2 inch is equal to 3/4 of a mile. How many inches on a map would be

The scale of a map shows that 1/2 inch is equal to 3/4 of a mile. How many inches on a map would be equal to 3 miles?
Set up a proportion of scale to actual distance
1/2 / 3/4 = x/3
4/3 = x/3
Cross multiply:
3x = 12
Divide each side by 3:
3x/3 = 12/3
x = [B]4 (1/2 inch sections) or 2 inches[/B]

The school yearbook costs $15 per book to produce with an overhead of $5500. The yearbook sells for

The school yearbook costs $15 per book to produce with an overhead of $5500. The yearbook sells for $40. Write a cost and revenue function and determine the break-even point.
[U]Calculate cost function C(b) with b as the number of books:[/U]
C(b) = Cost per book * b + Overhead
[B]C(b) = 15b + 5500[/B]
[U]Calculate Revenue Function R(b) with b as the number of books:[/U]
R(b) = Sales Price per book * b
[B]R(b) = 40b[/B]
[U]Calculate break even function E(b):[/U]
Break-even Point = Revenue - Cost
Break-even Point = R(b) - C(b)
Break-even Point = 40b - 15b - 5500
Break-even Point = 25b - 5500
[U]Calculate break even point:[/U]
Break-even point is where E(b) = 0. So we set 25b - 5500 equal to 0
25b - 5500 = 0
To solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=25b-5500%3D0&pl=Solve']type this equation into our search engine[/URL] and we get:
[B]b = 220[/B]

The science club charges 4.50 per car at their car wash. Write and solve and inequality to find how

The science club charges 4.50 per car at their car wash. Write and solve and inequality to find how many cars they have to wash to earn at least 300
Let x be the number of cars they wash. Set up our inequality. Note, at least 300 means 300 or greater, so we use greater than or equal to.
[U]Inequality:[/U]
[B]4.50x >= 300
[/B]
[U]So solve for x, divide each side by 4[/U]
[B]x >= 66.67[/B]

The square of a number increased by 7 is 23

The square of a number increased by 7 is 23
The phrase [I]a number [/I]means an arbitrary variable, let's call it x.
x
The square of a number means we raise x to the power of 2:
x^2
[I]Increased by[/I] means we add 7 to x^2
x^2 + 7
The word [I]is[/I] means an equation. So we set x^2 + 7 equal to 23:
[B]x^2 + 7 = 23[/B]

The Square of a positive integer is equal to the sum of the integer and 12. Find the integer

The Square of a positive integer is equal to the sum of the integer and 12. Find the integer
Let the integer be x.
[LIST]
[*]The sum of the integer and 12 is written as x + 12.
[*]The square of a positive integer is written as x^2.
[/LIST]
We set these equal to each other:
x^2 = x + 12
Subtract x + 12 from each side:
x^2 - x - 12 = 0
We have a quadratic function. [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-x-12%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']Run it through our search engine[/URL] and we get x = 3 and x = -4.
The problem asks for a positive integer, so we have [B]x = 3[/B]

The square of a positive integer minus twice its consecutive integer is equal to 22. find the intege

The square of a positive integer minus twice its consecutive integer is equal to 22. Find the integers.
Let x = the original positive integer. We have:
[LIST]
[*]Consecutive integer is x + 1
[*]x^2 - 2(x + 1) = 22
[/LIST]
Multiply through:
x^2 - 2x - 2 = 22
Subtract 22 from each side:
x^2 - 2x - 24 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2-2x-24%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get:
x = 6 and x = -4
Since the problem states [U]positive integers[/U], we use:
x = 6 and x + 1 = 7
[B](6, 7)[/B]

the square of the sum of x and y is less than 20

the square of the sum of x and y is less than 20
The sum of x and y means we add y to x:
x + y
the square of the sum of x and y means we raise the term x + y to the 2nd power:
(x + y)^2
The phrase [I]is less than[/I] means an inequality, so we write this as follows:
[B](x + y)^2 < 20[/B]

The sum of 2 and w is less than or equal to 27.

The sum of 2 and w is less than or equal to 27.
Take this algebraic expression in parts:
[LIST]
[*]The sum of 2 and w: 2 + w
[*]The phrase [I]less than or equal to[/I] means an inequality, using the <= sign.
[/LIST]
[B]2 + w <= 27[/B]

The sum of 2 times x and 5 times y is 7

The sum of 2 times x and 5 times y is 7
2 times x:
2x
5 times y:
5y
The sum of 2 times x and 5 times y:
2x + 5y
The word [I]is[/I] means equal to, so we set 2x + 5y equal to 7:
[B]2x + 5y = 7[/B]

the sum of 23 and victor age is 59

the sum of 23 and victor age is 59
Let's Victor's age be a.
The sum of 23 and Victor's age (a) mean we add a to 23:
23 + a
The word [I]is[/I] means an equation, so we set 23 + a equal to 59:
[B]23 + a = 59[/B] <-- This is our algebraic expression
Now if the problem asks you to take it a step further and solve this for a, [URL='https://www.mathcelebrity.com/1unk.php?num=23%2Ba%3D59&pl=Solve']we type this equation into our search engine[/URL] and we get:
[B]a = 36[/B]

The sum of 2x and y is at least 20

The sum of 2x and y is at least 20
The sum of 2x and y:
2x + y
The phrase [I]is at least[/I] means an inequality. We write this as >= or greater than or equal to:
[B]2x + y >= 20[/B]

the sum of 3 and 2x is 10

the sum of 3 and 2x is 10
The sum of 3 and 2x means we add 2x to 3:
3 + 2x
The word [I]is[/I] means an equation, so we set 3 + 2x equal to 10
[B]3 + 2x = 10[/B]

The sum of 3 consecutive integers is greater than 30.

The sum of 3 consecutive integers is greater than 30.
Let the first consecutive integer be n
The second consecutive integer is n + 1
The third consecutive integer is n + 2
The sum is written as:
n + n + 1 + n + 2
Combine like terms:
(n + n + n) + (1 + 2)
3n + 3
The phrase [I]greater than[/I] means an inequality, which we write as:
[B]3n + 3 > 30[/B]

the sum of 4 and x split into 5 equal parts

the sum of 4 and x split into 5 equal parts
The sum of x and 4 means we add 4 to x:
x + 4
Whenever you see the phrase [I]split into[/I], think of divide or divided by:
[B](x + 4)/5[/B]

The sum of 5 and 2x is at most 27

The sum of 5 and 2x is at most 27
The sum of 5 and 2x means we add 2x to 5:
5 + 2x
The phrase [I]at most[/I] means less than or equal to, so we have an inequality where 5 + 2x is less than or equal to 27
[B]5 + 2x <= 27[/B]

the sum of 5 and y is less than or equal to -21

the sum of 5 and y is less than or equal to -21
Take this algebraic expression in parts:
The sum of 5 and y means we add y to 5
5 + y
The phrase [I]less than or equal to[/I] -21 means an inequality. We use the <= sign to relate 5 + y to -21
[B]5 + y <= -21[/B]

The sum of 5 odd consecutive numbers is 145

The sum of 5 odd consecutive numbers is 145.
Let the first odd number be n. We have the other 4 odd numbers denoted as:
[LIST]
[*]n + 2
[*]n + 4
[*]n + 6
[*]n + 8
[/LIST]
Add them all together
n + (n + 2) + (n + 4) + (n + 6) + (n + 8)
The sum of the 5 odd consecutive numbers equals 145
n + (n + 2) + (n + 4) + (n + 6) + (n + 8) = 145
Combine like terms:
5n + 20 = 145
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=5n%2B20%3D145&pl=Solve']equation solver[/URL], we get [B]n = 25[/B]. Using our other 4 consecutive odd numbers above, we get:
[LIST]
[*]27
[*]29
[*]31
[*]33
[/LIST]
Adding the sum up, we get: 25 + 27 + 29 + 31 + 33 = 145.
So our 5 odd consecutive number added to get 145 are [B]{25, 27, 29, 31, 33}[/B].
[MEDIA=youtube]0T2PDuQIIwI[/MEDIA]

The sum of 5x and 2x is at least 70

[I]Is at least [/I]means greater than or equal to:
5x + 2x >= 70
If we combine like terms, we have:
7x >=70
We can further simplify by dividing each side of the inequality by 7
x >=10
If you want the interval notation for that, use the [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3E%3D10&pl=Show+Interval+Notation']interval notation calculator[/URL].

the sum of 6 and 7, plus 5 times a number, is -12

the sum of 6 and 7, plus 5 times a number, is -12
The sum of 6 and 7 means we add the two numbers:
6 + 7
This evaluates to 13
Next, we take 5 times a number. The phrase [I]a number[/I] means an arbitrary variable, let's call it x. So we multiply x by 5:
5x
The first two words say [I]the sum[/I], so we add 13 and 5x
13 + 5x
The word [I]is[/I] means an equation, so we set 13 + 5x equal to -12
[B]13 + 5x = -12[/B] <-- This is our algebraic expression
If the problem asks you to take it a step further and solve for x, then you [URL='https://www.mathcelebrity.com/1unk.php?num=13%2B5x%3D-12&pl=Solve']type this algebraic expression into our search engine[/URL] and you get:
[B]x = -5[/B]

the sum of 7 times y and 3 is equal to 2

the sum of 7 times y and 3 is equal to 2
7 times y:
7y
The sum of 7 times y and 3 means we add 3 to 7y
7y + 3
The phrase [I]is equal to[/I] means an equation, so we set 7y + 3 equal to 2
[B]7y + 3 = 2[/B]

the sum of a and b minus 4 is 12

the sum of a and b minus 4 is 12
the sum of a and b
a + b
the sum of a and b minus 4
a + b - 4
The word [I]is[/I] means equal to, so we set a + b - 4 equal to 12:
a + b - 4 = 12

the sum of a number and 16 is e

A number means an arbitrary variable, let's call it x.
The sum of x and 16 means we add:
x + 16
Is, means equal to, so we set x + 16 = e
x + 16 = e

The sum of a number and 5 all divided by 2 is 17

The sum of a number and 5 all divided by 2 is 17
The phrase [I]a number[/I] means an arbitrary variable, let's call it x
x
The sum of a number and 5:
x + 5
All divided by 2:
(x + 5)/2
The word [I]is[/I] means equal to, so we set (x + 5)/2 equal to 17:
[B](x + 5)/2 = 17[/B]

the sum of a number and its reciprocal is 5/2

the sum of a number and its reciprocal is 5/2
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The reciprocal of the number means 1/x.
The sum means we add them:
x + 1/x
The word [I]is[/I] means an equation, so we set x + 1/x equal to 52
[B]x + 1/x = 52[/B]

The sum of a number and its reciprocal is 72

The sum of a number and its reciprocal is 72
The phrase [I]a number[/I] means an arbitrary variable, let's call it x
x
The reciprocal of the number is written as:
1/x
The sum of a number and its reciprocal means we add:
x + 1/x
The word [I]is[/I] means an equation, so we set x + 1/x equal to 72
[B]x + 1/x = 72[/B]

The sum of a number and its reciprocal is five.

The sum of a number and its reciprocal is five.
The phrase [I]a number[/I] means an arbitrary variable. Let's call it x.
The reciprocal of the number is 1/x.
The sum means we add them together:
x + 1/x
The word [I]is[/I] means an equation, so we set x + 1/x equal to 5
[B]x + 1/x = 5[/B]

the sum of a number and itself is 8

A number means an arbitrary variable, let's call it x.
The sum of a number and itself means adding the number to itself
x + x
Simplified, we have 2x
The word is means equal to, so we have an algebraic expression of:
[B]2x= 8
[/B]
IF you need to solve this equation, divide each side by 2
[B]x = 4[/B]

The sum of a number and twice its reciprocal is 3

The sum of a number and twice its reciprocal is 3
the phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The reciprocal of a number means we take 1 over that number:
1/x
Twice the reciprocal means we multiply 1/x by 2:
2/x
The sum of a number and twice its reciprocal
x + 2/x
The word [I]is[/I] means equal to, so we set x + 2/x equal to 3
[B]x + 2/x = 3[/B]

The sum of a number b and 3 is greater than 4 and no more than 16

The sum of a number b and 3 is greater than 4 and no more than 16
The sum of a number b and 3:
b + 3
Greater than 4 and no more than 16 means we have a combo inequality:
[LIST]
[*]Greater than 4 means we use a > sign
[*]No more than 16 means less than or equal to, so <=
[/LIST]
[B]4 < b + 3 <= 16[/B]

the sum of a number divided by 8 and 3 equals 6

"A Number" means an arbitrary variable, let's call it x.
x divide d by 8 is written as a quotient
x/8
The sum of x/8 and 3 means we add:
x/8 + 3
Finally, equals means we have an equation, so we set our expression above equal to 6
x/8 + 3 = 6

the sum of a number times 3 and 30 is less than 17

the sum of a number times 3 and 30 is less than 17
A number is denoted as an arbitrary variable, let's call it x.
x
Times 3 means we multiply x by 3:
3x
The sum of a number times 3 and 30 means we add 30 to 3x above
3x + 30
Is less than 17 means we have an inequality, so we set 3x + 30 less than 17
3x + 30 < 17
To solve for x and see the interval notation, use [URL='http://www.mathcelebrity.com/1unk.php?num=3x%2B30%3C17&pl=Solve']our calculator[/URL]:

the sum of doubling a number and 100 which totals to 160

the sum of doubling a number and 100 which totals to 160
Take this algebraic expression in pieces:
[LIST=1]
[*]Let the number be n.
[*]Double it, means we multiply n by 2: 2n
[*]The sum of this and 100 means we add 100 to 2n: 2n + 100
[*]The phrase [I]which totals[/I] means we set 2n + 100 equal to 160
[/LIST]
[B]2n + 100 = 160[/B] <-- This is our algebraic expression
If the question asks you to solve for n, then we [URL='https://www.mathcelebrity.com/1unk.php?num=2n%2B100%3D160&pl=Solve']type this equation into our search engine[/URL] and we get:
[B]n = 30[/B]

The sum of five and twice a number is 17

The sum of five and twice a number is 17
[U]The phrase a number means an arbitrary variable, let's call it x[/U]
x
[U]Twice a number means we multiply x by 2:[/U]
2x
[U]The sum of five and twice a number means we add 5 to 2x:[/U]
2x + 5
[U]The phrase [I]is[/I] means an equation, so we set 2x + 5 equal to 17 to get our algebraic expression[/U]
[B]2x + 5 = 17[/B]
[B][/B]
As a bonus, if the problem asks you to solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B5%3D17&pl=Solve']type in this algebraic expression into our math engine[/URL] and we get:
x = 6

the sum of n and twice n is 12

Twice n means we multiply n by 2
2n
The sum of n and twice n means we add
n + 2n
The word [I]is[/I] means equal to, so we set that expression above equal to 12
n + 2n = 12
Combine like terms:
3n = 12
Divide each side of the equation by 3 to isolate n
n = 4
Check our work
Twice n is 2*4 = 8
Add that to n = 4
8 + 4
12

The sum of six times a number and 1 is equal to five times the number. Find the number.

The sum of six times a number and 1 is equal to five times the number. Find the number.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
6 times a number is written as:
6x
the sum of six times a number and 1 is written as:
6x + 1
Five times the number is written as:
5x
The phrase [I]is equal to[/I] means an equation, so we set 6x + 1 equal to 5x:
6x + 1 = 5x
[URL='https://www.mathcelebrity.com/1unk.php?num=6x%2B1%3D5x&pl=Solve']Plugging this into our search engine[/URL], we get:
x = [B]-1[/B]

The sum of the squares of c and d is 25

The sum of the squares of c and d is 25
The square of c means we we raise c to the power of 2:
c^2
The square of d means we we raise d to the power of 2:
d^2
The sum of the squares of c and d means we add d^2 to c^2:
c^2 + d^2
The word [I]is[/I] means equal to, so we set c^2 + d^2 equal to 25:
[B]c^2 + d^2 = 25[/B]

The Sum of three times a number and 18 is -39. Find the number

The Sum of three times a number and 18 is -39. Find the number.
A number means an arbitrary variable, let us call it x.
Three times x:
3x
The sum of this and 18:
3x + 18
Is means equal to, so we set 3x + 18 = -39
3x + 18 = -39
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=3x%2B18%3D-39&pl=Solve']equation solver[/URL], we get [B]x = -19[/B]

The sum of twice an integer and 3 times the next consecutive integer is 48

The sum of twice an integer and 3 times the next consecutive integer is 48
Let the first integer be n
This means the next consecutive integer is n + 1
Twice an integer means we multiply n by 2:
2n
3 times the next consecutive integer means we multiply (n + 1) by 3
3(n + 1)
The sum of these is:
2n + 3(n + 1)
The word [I]is[/I] means equal to, so we set 2n + 3(n + 1) equal to 48:
2n + 3(n + 1) = 48
Solve for [I]n[/I] in the equation 2n + 3(n + 1) = 48
We first need to simplify the expression removing parentheses
Simplify 3(n + 1): Distribute the 3 to each term in (n+1)
3 * n = (3 * 1)n = 3n
3 * 1 = (3 * 1) = 3
Our Total expanded term is 3n + 3
Our updated term to work with is 2n + 3n + 3 = 48
We first need to simplify the expression removing parentheses
Our updated term to work with is 2n + 3n + 3 = 48
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(2 + 3)n = 5n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
5n + 3 = + 48
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 3 and 48. To do that, we subtract 3 from both sides
5n + 3 - 3 = 48 - 3
[SIZE=5][B]Step 4: Cancel 3 on the left side:[/B][/SIZE]
5n = 45
[SIZE=5][B]Step 5: Divide each side of the equation by 5[/B][/SIZE]
5n/5 = 45/5
Cancel the 5's on the left side and we get:
n = [B]9[/B]

The sum of two y and the quantity of three plus y plus twice the quantity two y minus two equals fif

The sum of two y and the quantity of three plus y plus twice the quantity two y minus two equals fifteen
The sum of two y and the quantity of three plus y
2y + (3 + y)
twice the quantity two y minus two
2(2y - 2)
The sum of two y and the quantity of three plus y plus twice the quantity two y minus two
2y + (3 + y) + 2(2y - 2)
Equals 15 to get our algebraic expression
[B]2y + (3 + y) + 2(2y - 2) = 15[/B]
[B][/B]
If the problem asks you to solve for yL
2y + 3 + y + 4y - 4 = 15
Group like terms:
7y - 1 = 15
Add 1 each side:
7y = 16
Divide each side by 7:
y = [B]16/7[/B]

The sum of two y and three is the same as the difference of three y and one

The sum of two y and three
2y + 3
the difference of three y and one
3y - 1
is the same as means equal to. Set 2y + 3 equal to 3y - 1 for our final algebraic expression:
[B]2y + 3 = 3y - 1[/B]
[B][/B]
If the problem asks you to solve for y, subtract 2y from each side:
3 = y - 1
Add 1 to each side:
y = [B]4[/B]

The sum of two-fifths and f is one-half.

The sum of two-fifths and f is one-half.
We write two-fifths as 2/5.
The sum of two-fifths and f is written by adding f to two-fifths using the + sign:
2/5 + f
one-half is written as 1/2
The word [I]is[/I] means equals, so we set up an equation where 2/5 + f equal to 1/2
[B]2/5 + f = 1/2[/B]

The sum of x and 10 equals the sum of 2 times x and 12

The sum of x and 10 equals the sum of 2 times x and 12
The sum of x and 10 means we add 10 to x:
x + 10
2 times x means we multiply x by 2:
2x
the sum of 2 times x and 12 means we add 12 to 2x:
2x + 12
The sum of x and 10 equals the sum of 2 times x and 12:
x + 10 + (2x + 12)
Distribute the parentheses, and we get:
x + 10 + 2x + 12
Group like terms:
(1 + 2)x + 10 + 12
[B]3x + 22[/B]

the sum of x and 96 equals half of x

the sum of x and 96 equals half of x
half of x means we divide x by 2:
x/2
The sum of x and 96:
x + 96
The phrase equals means we set x + 96 equal to x/2:
[B]x + 96 = x/2[/B]

The sum of x and twice y is equal to m.

The sum of x and twice y is equal to m.
Twice y means we multiply y by 2:
2y
The sum of x and twice y:
x + 2y
The phrase [I]is equal to[/I] means an equation, so we set x + 2y equal to m
[B]x + 2y = m[/B]

The sum of x and y is at most 10

The sum of x and y is at most 10
The sum of x and y:
x + y
Is at most 10 means we have an inequality, at most means 10 or less, so less than or equal to
[B]x + y <= 10[/B]

The temperature inside the lab refrigerator is no more than 35 . Use t to represent the temperature

The temperature inside the lab refrigerator is no more than 35 . Use t to represent the temperature (in ) of the refrigerator.
The phrase [I]no more than[/I] means less than or equal to. We have this inequality:
[B]t <= 35[/B]

the total of a and 352 equals a divided by 195

the total of a and 352 equals a divided by 195
Take this algebraic expression in 3 parts:
[LIST=1]
[*]The total of a and 352 means we add 352 to a: a + 352
[*]a divided by 195: a/195
[*]The phrase [I]equals[/I] means we set (1) equal to (2) to get our final algebraic expression:
[/LIST]
[B]a + 352 = a/195[/B]

The volleyball team and the wrestling team at Clarksville High School are having a joint car wash t

The volleyball team and the wrestling team at Clarksville High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets $4 per car. In addition, they have already brought in $81 from past fundraisers. The wrestling team has raised $85 in the past, and they are making $2 per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take?
Set up the earnings equation for the volleyball team where w is the number of cars washed:
E = Price per cars washed * w + past fundraisers
E = 4w + 81
Set up the earnings equation for the wrestling team where w is the number of cars washed:
E = Price per cars washed * w + past fundraisers
E = 2w + 85
If the raised the same amount in total, set both earnings equations equal to each other:
4w + 81 = 2w + 85
Solve for [I]w[/I] in the equation 4w + 81 = 2w + 85
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 4w and 2w. To do that, we subtract 2w from both sides
4w + 81 - 2w = 2w + 85 - 2w
[SIZE=5][B]Step 2: Cancel 2w on the right side:[/B][/SIZE]
2w + 81 = 85
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 81 and 85. To do that, we subtract 81 from both sides
2w + 81 - 81 = 85 - 81
[SIZE=5][B]Step 4: Cancel 81 on the left side:[/B][/SIZE]
2w = 4
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2w/2 = 4/2
w = [B]2 <-- How many cars it will take
[/B]
To get the total earnings, we take either the volleyball or wrestling team's Earnings equation and plug in w = 2:
E = 4(2) + 81
E = 8 + 81
E = [B]89 <-- Total Earnings[/B]

The width of a rectangle is fixed at 4cm. For what lengths will the area be less than 86 cm^2

The width of a rectangle is fixed at 4cm. For what lengths will the area be less than 86 cm^2
The Area (A) of a rectangle is given by:
A = lw
With an area of [I]less than[/I] 86 and a width of 4, we have the following inequality:
4l < 86
To solve for l, we [URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=4l%3C86&pl=Show+Interval+Notation']type this inequality into our search engine[/URL] and we get:
[B]l < 21.5[/B]

There are 15 houses in a neighborhood. Nine of the houses have 6 people in them. The remaining house

There are 15 houses in a neighborhood. Nine of the houses have 6 people in them. The remaining houses have 4 people in them. How many people are in a neighborhood.
9 houses * 6 people per house = 54 people
The remaining houses equal 15 total houses - 9 houses = 6 houses
6 houses remaining times 4 people in each house = 24 people
54 people + 24 people = [B]78 people in the neighborhood[/B]

There are 5 pencil-cases on the desk. Each pencil-case contains at least 10 pencils, but not more th

[SIZE=4]There are 5 pencil-cases on the desk. Each pencil-case contains at least 10 pencils, but not more than 14 pencils. Which of the following could be the total number of pencils in all 5 cases?
A) 35
B) 45
C) 65
D) 75
[U]Determine the minimum amount of pencils (At least means greater than or equal to):[/U]
Minimum Amount of pencils = Cases * Min Quantity
Minimum Amount of pencils = 5 * 10
Minimum Amount of pencils = 50
[SIZE=4][U]Determine the maximum amount of pencils (Not more than means less than or equal to):[/U]
Maximum Amount of pencils = Cases * Min Quantity
Maximum Amount of pencils = 5 * 14
Maximum Amount of pencils = 70[/SIZE]
So our range of pencils (p) is:
50 <= p <= 70
Now take a look at our answer choices. The only answer which fits in this inequality range is [B]C, 65[/B].
[B][/B][/SIZE]

There are 60 rocks placed equally into 3 jars . How many rocks are in each jar?

There are 60 rocks placed equally into 3 jars . How many rocks are in each jar?
60 rocks / 3 jars = [B]20 rocks per jar[/B]

There are 72 boys and 90 girls on the Math team For the next math competition Mr Johnson would like

There are 72 boys and 90 girls on the Math team For the next math competition Mr Johnson would like to arrange all of the students in equal rows with only girls or boys in each row with only girls or boys in each row. What is the greatest number of students that can be put in each row?
To find the maximum number (n) of boys or girls in each row, we want the GCF (Greatest Common Factor) of 72 and 90.
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=72&num2=90&num3=&pl=GCF+and+LCM']Using our GCF calculator for GCF(72,90)[/URL], we get 18.
[LIST]
[*]72 boys divided by 18 = [B]4 rows of boys[/B]
[*]90 girls divided by 18 = [B]5 rows of girls[/B]
[/LIST]

There are 76 milligrams of cholesterol in a 3.2 ounce serving of lobster. How much cholesterol is in

There are 76 milligrams of cholesterol in a 3.2 ounce serving of lobster. How much cholesterol is in a 6 ounce serving?
Let x equal the amount of cholesterol in milligrams for a 6 ounce service. Set up a proportion:
76/3.2 = x/6
Using our [URL='http://www.mathcelebrity.com/prop.php?num1=76&num2=x&den1=3.2&den2=6&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator[/URL] by plugging that expression into the search engine, we get x = 142.5

There are 8 lions, 4 tigers, 5 cheetahs, 6 giraffes, 7 hippos, and 78 monkeys at the City Zoo. If ea

There are 8 lions, 4 tigers, 5 cheetahs, 6 giraffes, 7 hippos, and 78 monkeys at the City Zoo. If each of the 4 zookeepers feeds the same number of animals, how many animals does each zookeeper feed?
Calculate Total Animals:
8 + 4 + 5 + 6 + 7 + 78 = 108
Now divide 108 animals equally into 4 zookeepers
108/4 = [B]27 animals each zookeeper will feed[/B]

Think of a number. Double the number. Subtract 6 from the result and divide the answer by 2. The quo

Think of a number. Double the number. Subtract 6 from the result and divide the answer by 2. The quotient will be 20. What is the number
Let's call our number n.
Double the number means we multiply n by 2:
2n
Subtract 6 from the result means we subtract 6 from 2n:
2n - 6
Divide the answer by 2:
(2n - 6)/2
We can simplify this as n - 3
The quotient will be 20. This means the simplified term above is set equal to 20:
[B]n - 3 = 20 [/B] <-- This is our algebraic expression
If you want to take it a step further, and solve for n in the algebraic expression above, we [URL='https://www.mathcelebrity.com/1unk.php?num=n-3%3D20&pl=Solve']type this expression into our calculator[/URL], and get:
n = 23

Thirty is half of the sum of 4 and a number

Thirty is half of the sum of 4 and a number.
The phrase [I]a number[/I] represents an arbitrary variable, let's call it x.
The sum of 4 and a number:
4 + x
Half of this sum means we divide by 2:
(4 + x)/2
Set this equal to 30:
[B](4 + x)/2 = 30[/B] <-- This is our algebraic expression

Three more than 2x is greater than or equal to 1 and less than or equal to 11

This is a double inequality. Let's take it by pieces:
Three more than 2x is denoted as 2x + 3. We add since we see the phrase, [I]more than[/I].
Because it's greater than or equal to 1, we have:
1 <= 2x + 3
Finally, that same phrase is [U]also[/U] less than or equal to 11.
2x + 3 <= 11.
Piecing these two inequalities together, we have:
1 <= 2x + 3 <= 11.

Three more than 2x is greater than or equal to 1 and less than or equal to 11

This is a two-part inequality. Let's take it piece by piece.
Three more than 2x means we add.
2x + 3
It's greater than or equal to 1, denoted below:
1 <= 2x + 3
It's also less than or equal to 11, denoted below
2x + 3 <= 11
Piece these two inequalities together:
1 <= 2x + 3 <= 11

Three x is five less than twice x

Twice x means we multiply x by 2:
2x
five less than twice x
2x - 5
Three x
3x
The word [I]is[/I] means equal to. Set 2x - 5 equal to 3x for our algebraic expression:
[B]2x - 5 = 3x
[/B]
If the problem asks you to solve for x, subtract 2x from each side
[B]x = -5[/B]

Time and Distance

Let h be the number of hours that pass when Charlie starts. We have the following equations:
[LIST]
[*]Charlie: D = 40h + 9
[*]Danny: D = 55h
[/LIST]
Set them equal to each other:
40h + 9 = 55h
Subtract 40h from both sides:
15h = 9
h = 3/5
[B]3/5 of an hour = 3(60)/5 = 36 minutes[/B]

Time and Distance

Thank you so much
[QUOTE="math_celebrity, post: 1003, member: 1"]Let h be the number of hours that pass when Charlie starts. We have the following equations:
[LIST]
[*]Charlie: D = 40h + 9
[*]Danny: D = 55h
[/LIST]
Set them equal to each other:
40h + 9 = 55h
Subtract 40h from both sides:
15h = 9
h = 3/5
[B]3/5 of an hour = 3(60)/5 = 36 minutes[/B][/QUOTE]

To ship a package with UPS, the cost will be $7 for the first pound and $0.20 for each additional po

To ship a package with UPS, the cost will be $7 for the first pound and $0.20 for each additional pound. To ship a package with FedEx, the cost will be $5 for the first pound and $0.30 for each additional pound. How many pounds will it take for UPS and FedEx to cost the same? If you needed to ship a package that weighs 8 lbs, which shipping company would you choose and how much would you pay?
[U]UPS: Set up the cost function C(p) where p is the number of pounds:[/U]
C(p) = Number of pounds over 1 * cost per pounds + first pound
C(p) = 0.2(p - 1) + 7
[U]FedEx: Set up the cost function C(p) where p is the number of pounds:[/U]
C(p) = Number of pounds over 1 * cost per pounds + first pound
C(p) = 0.3(p - 1) + 5
[U]When will the costs equal each other? Set the cost functions equal to each other:[/U]
0.2(p - 1) + 7 = 0.3(p - 1) + 5
0.2p - 0.2 + 7 = 0.3p - 0.3 + 5
0.2p + 6.8 = 0.3p + 4.7
To solve this equation for p, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.2p%2B6.8%3D0.3p%2B4.7&pl=Solve']type it in our search engine[/URL] and we get:
p = [B]21
So at 21 pounds, both UPS and FedEx costs are equal
[/B]
Now, find out which shipping company has a better rate at 8 pounds:
[U]UPS:[/U]
C(8) = 0.2(8 - 1) + 7
C(8) = 0.2(7) + 7
C(8) = 1.4 + 7
C(8) = 8.4
[U]FedEx:[/U]
C(8) = 0.3(8 - 1) + 5
C(8) = 0.3(7) + 5
C(8) = 2.1 + 5
C(8) = [B]7.1[/B]
[B]Therefore, FedEx is the better cost at 8 pounds since the cost is lower[/B]
[B][/B]

Today is my birthday! Four-fifths of my current age is greater than three-quarters of my age one yea

Today is my birthday! Four-fifths of my current age is greater than three-quarters of my age one year from now. Given that my age is an integer number of years, what is the smallest my age could be?
Let my current age be a. We're given:
4/5a > 3/4(a + 1)
Multiply through on the right side:
4a/5 > 3a/4 + 3/4
Let's remove fractions by multiply through by 5:
5(4a/5) > 5(3a/4) + 5(3/4)
4a > 15a/4 + 15/4
Now let's remove the other fractions by multiply through by 4:
4(4a) > 4(15a/4) + 4(15/4)
16a > 15a + 15
[URL='https://www.mathcelebrity.com/1unk.php?num=16a%3E15a%2B15&pl=Solve']Typing this inequality into our search engine[/URL], we get:
a > 15
This means the smallest [I]integer age[/I] which the problem asks for is:
15 + 1 = [B]16[/B]

Tom has 72 muffins,which he need to box up into dozens. How many boxes does he need

Tom has 72 muffins,which he need to box up into dozens. How many boxes does he need?
A dozen equals 12. So we have:
72 muffins / 12 per dozen in a box = [B]6 boxes[/B]

Tom has a collection 21 CDs and Nita has a collection of 14 CDs. Tom is adding 3 cds a month to his

Tom has a collection 21 CDs and Nita has a collection of 14 CDs. Tom is adding 3 cds a month to his collection while Nita is adding 4 CDs a month to her collection. Find the number of months after which they will have the same number of CDs?
Set up growth equations for the CDs where c = number of cds after m months
Tom: c = 21 + 3m
Nita: c = 14 + 4m
Set the c equations equal to each other
21 + 3m = 14 + 4m
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=21%2B3m%3D14%2B4m&pl=Solve']equation calculator[/URL], we get [B]m = 7[/B]

Tom is the deli manager at a grocery store. He needs to schedule employees to staff the deli departm

Tom is the deli manager at a grocery store. He needs to schedule employees to staff the deli department at least 260 person-hours per week. Tom has one part-time employeewho works 20 hours per week. Each full-time employee works 40 hours per week. Write an inequality to determine n, the number of full-time employees Tom must schedule, so that his employees will work at least 260 person-hours per week.
Set up the inequality:
[LIST]
[*]Add the part-timer's hours of 20
[*]Full time hours is 40 times n employees
[*]At least means greater than or equal to, so we use the >= sign
[/LIST]
[B]40n + 20 >= 260[/B]

Transitive Property of Equality

Free Transitive Property of Equality Calculator - Demonstrates the Transitive property of equality using a number.
Numerical Properties

Translate the sentence into an inequality. Twice y is less than 21.

Translate the sentence into an inequality. Twice y is less than 21.
Twice y
2y
Is less than 21 means we have an inequality:
[B]2y < 21[/B]

Translate this sentence into an equation. 43 is the sum of 17 and Gregs age. Use the variable g to

Translate this sentence into an equation. 43 is the sum of 17 and Gregs age. Use the variable g to represent Gregs age.
The sum of 17 and Greg's age:
g + 17
The word [I]is[/I] means equal to, so we set g + 17 equal to 43
[B]g + 17 = 43[/B] <-- This is our algebraic expression
If you want to solve this equation for g, use our [URL='http://www.mathcelebrity.com/1unk.php?num=g%2B17%3D43&pl=Solve']equation calculator[/URL].
[B]g = 26[/B]

Translate this sentence into an equation. 48 is the difference of Ritas age and 11 . Use the variabl

Translate this sentence into an equation. 48 is the difference of Ritas age and 11 . Use the variable r to represent Ritas age.
The difference of Rita's age and 11 is written:
r - 11
The phrase [I]is[/I] means equal to, so we set r - 11 equal to 48
r - 11 = 48

Translate this sentence into an equation. 49 is the difference of Diegos age and 17. Use the variabl

Translate this sentence into an equation. 49 is the difference of Diegos age and 17. Use the variable d to represent Diegos age.
The difference means we subtract, so we have d as Diego's age minus 17
d - 17
The word "is" means an equation, so we set d - 17 equal to 49
[B]d - 17 = 49[/B]

Translate to an inequality. The cost is smaller than $94,000

Translate to an inequality. The cost is smaller than $94,000.
Let the cost be c. We have:
[B]c<94,000[/B]

Triangle Inequality

Free Triangle Inequality Calculator - This calculator displays 2 scenarios

1) Enter 3 sides of a triangle, and it will determine if the side lengths satisfy the properties of the triangle inequality and form a triangle

2) Enter 2 sides of a triangle, and this will determine an acceptable range for the length of the 3rd side of a triangle so that the 3rd side respects the Triangle Inequality.

1) Enter 3 sides of a triangle, and it will determine if the side lengths satisfy the properties of the triangle inequality and form a triangle

2) Enter 2 sides of a triangle, and this will determine an acceptable range for the length of the 3rd side of a triangle so that the 3rd side respects the Triangle Inequality.

triangle sum theorem

The triangle sum theorem states the sum of the three angles in a triangle equals 180 degrees.
So if you're given two angles and need too find the 3rd angle, add the 2 known angles up, and subtract them from 180 to get the 3rd angle measure.

triple the value of c plus 3 is 84

Triple the value of c means we multiply c by 3
3c
Plus 3 means we add 3
3c + 3
Is, means equal to, so we set our expression equal to 84
[B]3c +3 = 84
[/B]
If you want to solve that equation, use our [URL='http://www.mathcelebrity.com/1unk.php?num=3c%2B3%3D84&pl=Solve']equation solver[/URL]:
c = 27

Twenty-five is nine more than four times a number

The phrase [I]a number[/I] means an arbitrary variable. We can pick any letter a-z except for i and e.
Let's choose x.
Four times a number:
4x
nine more than four times a numbrer
4x + 9
The phrase [I]is[/I] means equal to. We set 4x + 9 equal to 25 as our algebraic expression:
[B]4x + 9 = 25
[/B]
If the problem asks you to solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=4x%2B9%3D25&pl=Solve']type it in our math solver[/URL] and get:
x = [B]4[/B]

Twenty-five is the same as ten added to twice a number

The phrase [I]a number[/I] means an arbitrary variable. We can pick any letter a-z except for i and e.
Let's choose x.
Twice a number means we multiply x by 2:
2x
ten added to twice a number
2x + 10
The phrase [I]is the same as [/I]means equal to. Set 25 equal to 2x + 10 to get our algebraic expression
[B]25 = 2x + 10
[/B]
If the problem asks you to solve for x, [URL='https://www.mathcelebrity.com/1unk.php?num=25%3D2x%2B10&pl=Solve']type it in our math solver [/URL]and get
x = [B]7.5[/B]

Twice a number decreased by eight is zero

The phrase [I]a number[/I] means an arbitrary variable. We can pick any letter a-z except for i and e.
Let's choose x.
Twice a number:
2x
decreased by eight
2x - 8
[I]is [/I]means equal to. Set 2x - 8 equal to zero for our algebraic expression:
[B]2x - 8 = 0
[/B]
If the problem asks you to solve for x, add 8 to each side:
2x = 8
Divide each side by 2:
x= [B]4[/B]

twice the difference between x and 28 is 3 times a number

twice the difference between x and 28 is 3 times a number
The difference between x and 28:
x - 28
Twice the difference means we multiply x - 28 by 2:
2(x - 28)
The phrase [I]a number[/I] means an arbitrary variable, let's call it x
x
3 times a number:
3x
The word [I]is[/I] means equal to, so we set 2(x - 28) equal to 3x:
[B]2(x - 28) = 3x[/B]

twice the difference of a number and 3 is equal to 3 times the sum of a number and 2

twice the difference of a number and 3 is equal to 3 times the sum of a number and 2.
We've got 2 algebraic expressions here. Let's take them in parts.
Left side algebraic expression: twice the difference of a number and 3
[LIST]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
[*]The word [I]difference[/I] means we subtract 3 from the variable x
[*]x - 3
[*]Twice this difference means we multiply (x - 3) by 2
[*]2(x - 3)
[/LIST]
Right side algebraic expression: 3 times the sum of a number and 2
[LIST]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
[*]The word [I]sum[/I] means we add 2 to the variable x
[*]x + 2
[*]3 times the sum means we multiply (x + 2) by 3
[*]3(x + 2)
[/LIST]
Now, we have both algebraic expressions, the problem says [I]is equal to[/I]
This means we have an equation, where we set the left side algebraic expression equal to the right side algebraic expression using the equal sign (=) to get our answer
[B]2(x - 3) = 3(x + 2)[/B]

twice the difference of a number and 55 is equal to 3 times the sum of a number and 8

twice the difference of a number and 55 is equal to 3 times the sum of a number and 8
Take this algebraic expression in pieces.
Left side: The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The difference of this number and 55 means we subtract 55 from x
x - 55
Twice the difference means we multiply x - 55 by 2
2(x - 55)
Right side: The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The sum of a number and 8 means we add 8 to x
x + 8
3 times the sum means we multiply x + 8 by 3
3(x + 8)
Now that we have the left and right side of the expressions, we see the phrase [I]is equal to[/I]. This means an equation, so we set the left side equal to the right side:
[B]2(x - 55) = 3(x + 8)[/B]

Twice the quantity of seven plus x is the same as the difference of x and seven

seven plus x
7 + x
Twice the quantity of seven plus x
2(7 + x)
Difference of x and seven
x - 7
The phrase [I]is the same as[/I] means equal to. This is our algebraic expression:
[B]2(7 + x) = x - 7
[/B]
If the problem asks you to solve for x, distribute 2 on the left side:
14 + 2x = x - 7
Subtract x from the right side
14 + x = -7
Subtract 14 from each side
[B]x = -21[/B]

twice the square root of a number increased by 5 is 23

twice the square root of a number increased by 5 is 23
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
The square root of a number means we raise x to the 1/2 power:
sqrt(x)
the square root of a number increased by 5 means we add 5 to sqrt(x):
sqrt(x) + 5
twice the square root of a number increased by 5 means we multiply sqrt(x) + 5 by 2:
2(sqrt(x) + 5)
The phrase [I]is 23[/I] means we set 2(sqrt(x) + 5) equal to 23:
[B]2(sqrt(x) + 5) = 23[/B]

twice the sum of a and b is thrice c

twice the sum of a and b is thrice c
The sum of a and b:
a + b
twice the sum of a and b means we multiply the sum of a and b by 2:
2(a + b)
Thrice c means we multiply c by 3:
3c
The word [I]is[/I] means equal to, so we set 2(a + b) equal to 3c:
[B]2(a + b) = 3c
[MEDIA=youtube]G_D4b8Jv89Q[/MEDIA][/B]

Twice the sum of a number and 6 is equal to three times the difference of the number and 3. Find the

[SIZE=6]Twice the sum of a number and 6 is equal to three times the difference of the number and 3. Find the number.
The phrase [/SIZE][I][SIZE=7]a number[/SIZE][/I][SIZE=6] means an arbitrary variable, let's call it x.
The sum of a number and 6 means we add 6 to x:
x + 6
Twice the sum of a number and 6 means we multiply x + 6 by 2:
2(x + 6)
the difference of the number and 3 means we subtract 3 from x
x - 3
three times the difference of the number and 3 means we multiply x - 3 by 3:
3(x- 3)
The word [I]is[/I] means we set 2(x + 6) equal to 3(x - 3)
2(x + 6) = 3(x - 3)
Use the distributive property to multiply through:
2x + 12 = 3x - 9
Subtract 2x from each side:
2x - 2x + 12 = 3x - 2x - 9
x - 9 = 12
Add 9 to each side:
x - 9 + 9 = 12 + 9
x = [B]21[/B]
[B][/B]
[B][MEDIA=youtube]CeZl_oZnSiw[/MEDIA][/B][/SIZE]

Twice x increased by the cube of y equals z

Twice x increased by the cube of y equals z
[LIST]
[*]Twice x means we multiply x by 2: 2x
[*]Increased this by the cube of y which is y^3. So we have 2x + y^3
[*]Now, we set this entire expression equal to z: 2x + y^3 = z
[/LIST]

Two consecutive even integers that equal 126

Two consecutive even integers that equal 126
Let the first integer equal x. So the next even integer must be x + 2.
The sum which is equal to 126 is written as x + (x + 2) = 126
Simplify:
2x + 2 = 126
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2x%2B2%3D126&pl=Solve']equation calculator,[/URL] we get:
x = 62
This means the next consecutive even integer is 62 = 2 = 64.
So our two even consecutive integers with a sum of 126 are [B](62, 64)[/B]

two thirds of a number is no more than -10

two thirds of a number is no more than -10
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Two thirds of a number mean we multiply x by 2/3:
2x/3
The phrase [I]no more than[/I] -10 means less than or equal to -10, so we have an inequality:
[B]2x/3 <= -10[/B]

Tyrese’s sister is 41 inches tall. A ride at the amusement park states that riders must be at least

Tyrese’s sister is 41 inches tall. A ride at the amusement park states that riders must be at least 52 inches tall to ride. Which statements describe how much taller Tyrese’s sister must be to ride?
Let h be the required additional height.
The phrase [I]at least[/I] means an inequality, using the >= sign, so we have:
h + 41 >= 52
If we want another way to express this, we [URL='https://www.mathcelebrity.com/1unk.php?num=h%2B41%3E%3D52&pl=Solve']type this inequality into our math engine[/URL] and we get:
[B]h >= 11[/B]

u and 201 more equals q

201 more means we add:
u + 201
We set that expression equal to q
u + 201 = q

u cubed equals nine

u cubed equals nine
u cubed means we raise u to the 3rd power:
u^3
We set this equal to 9:
[B]u^3 = 9[/B]

v equals 66 decreased by d

66 decreased by d means we subtract:
66 - d
v equals means we set our entire expression equal to v
[B]66 - d = v[/B]

v is equal to the product of 7 and the sum of u and 6

v is equal to the product of 7 and the sum of u and 6
[LIST]
[*]Sum of u and 6: u + 6
[*]the product of 7 and the sum of u and 6: 7(u + 6)
[*]We set this expression equal to v:
[/LIST]
[B]v = 7(u + 6)[/B]

Vanessa is packing her bags for her vacation. She has 5 unique action figures, but only 3 fit in her

Vanessa is packing her bags for her vacation. She has 5 unique action figures, but only 3 fit in her bag. How many different groups of 3 action figures can she take?
The key word here is [U]different[/U]. This means combinations.
We use our [URL='http://www.mathcelebrity.com/permutation.php?num=5&den=3&pl=Combinations']combinations calculator[/URL] to find 5 C 3 which equals [B]10[/B].

Vectors

Free Vectors Calculator - Given 2 vectors A and B, this calculates:

* Length (magnitude) of A = ||A||

* Length (magnitude) of B = ||B||

* Sum of A and B = A + B (addition)

* Difference of A and B = A - B (subtraction)

* Dot Product of vectors A and B = A x B

A ÷ B (division)

* Distance between A and B = AB

* Angle between A and B = θ

* Unit Vector U of A.

* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).

* Cauchy-Schwarz Inequality

* The orthogonal projection of A on to B, proj_{B}A and and the vector component of A orthogonal to B → A - proj_{B}A

Also calculates the horizontal component and vertical component of a 2-D vector.

* Length (magnitude) of A = ||A||

* Length (magnitude) of B = ||B||

* Sum of A and B = A + B (addition)

* Difference of A and B = A - B (subtraction)

* Dot Product of vectors A and B = A x B

A ÷ B (division)

* Distance between A and B = AB

* Angle between A and B = θ

* Unit Vector U of A.

* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).

* Cauchy-Schwarz Inequality

* The orthogonal projection of A on to B, proj

Also calculates the horizontal component and vertical component of a 2-D vector.

Victoria is 4 years older than her neighbor. The sum of their ages is no more than 14 years.

Victoria is 4 years older than her neighbor. The sum of their ages is no more than 14 years.
Let Victoria's age be v. And her neighbor's age be n. We're given:
[LIST=1]
[*]v = n + 4
[*]v + n <=14 <-- no more than means less than or equal to
[/LIST]
Substitute Equation (1) into Inequality (2):
(n + 4) + n <= 14
Combine like terms:
2n + 4 <= 14
[URL='https://www.mathcelebrity.com/1unk.php?num=2n%2B4%3C%3D14&pl=Solve']Typing this inequality into our search engine[/URL], we get:
n <= 5
Substituting this into inequality (2):
v + 5 <= 14
[URL='https://www.mathcelebrity.com/1unk.php?num=v%2B5%3C%3D14&pl=Solve']Typing this inequality into our search engine[/URL], we get:
[B]v <= 9[/B]

Video store movie rental plans. Plan A 25 membership fee plus 1.25 for movie. Plan B 40 for unlimite

Video store movie rental plans. Plan A 25 membership fee plus 1.25 for movie. Plan B 40 for unlimited rentals. What number of movies rentals is plan B less than plan A?
Let x equal the number of movies rented and C the cost for rentals
Plan A: C = 1.25x + 25
Plan B: C = 40
Set up the inequality:
1.25x + 25 > 40
Subtract 25 from each side:
1.25x > 15
Divide each side of the inequality by 1.25
x > 12
So [B]13[/B] rentals or more make Plan B less than Plan A.

Water flows from tank A to tank B at the rate of 2 litres per minute.

Water flows from tank A to tank B at the rate of 2 litres per minute. Initially tank A has 200 litres in it and tank B has 100 Litres in it. Water drains from tank B at 0.5 litres per minute.
After how many minutes are there equal volumes of water in the 2 tanks?
Write an equation and solve it.

Water flows from tank A to tank B at the rate of 2 litres per minute.

[QUOTE="Jahn, post: 78, member: 5"]Water flows from tank A to tank B at the rate of 2 litres per minute. Initially tank A has 200 litres in it and tank B has 100 Litres in it. Water drains from tank B at 0.5 litres per minute.
After how many minutes are there equal volumes of water in the 2 tanks?
Write an equation and solve it.[/QUOTE]
Tank A: V = 200 - 2x
Tank B: V = 100 - 0.5x
Where x is the number of minutes passed.
Set them equal to each other
200 - 2x = 100 - 0.5x
Subtract 100 from each side:
100 - 2x = -0.5x
Add 2x to each side:
1.5x = 100
Divide each side of the equation by x:
x = 66.66666667

What pair of consecutive integers gives the following: 7 times the smaller is less than 6 times the

What pair of consecutive integers gives the following: 7 times the smaller is less than 6 times the larger?
Let x and y be consecutive integers, where y = x + 1
We have 7x < 6y as our inequality.
Substituting x, y = x + 1, we have:
7x < 6(x + 1)
7x < 6x + 6
Subtracting x from each side, we have:
x < 6, so y = 6 + 1 = 7
(x, y) = (6, 7)

WHAT SHOULD BE SUBTRACTED FROM -9876 TO OBTAIN -9512

WHAT SHOULD BE SUBTRACTED FROM -9876 TO OBTAIN -9512
We set up an arbitrary number x.
Subtracted from is written as
-9876 - x
The phrase [I]to obtain[/I] means an equation, so we set -9876 - x equal to -9512
-9876 - x = -9512
To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=-9876-x%3D-9512&pl=Solve']type this equation into our search engine[/URL] and we get:
x = [B]364[/B]

When 20 is subtracted from 3 times a certain number, the result is 43

A certain number means an arbitrary variable, let's call it x
x
3 times x
3x
20 is subtracted from 3 time x
3x - 20
The result is means equal to, so we set 3x - 20 equal to 43 for our algebraic expression
[B]3x - 20 = 43
[/B]
If you need to solve this, use our [URL='http://www.mathcelebrity.com/1unk.php?num=3x-20%3D43&pl=Solve']equation calculator[/URL]:
[B]x = 21[/B]

When 39 is added to a number, the result is 40 times the number. Find the number. Let n be the unkn

When 39 is added to a number, the result is 40 times the number. Find the number. Let n be the unknown number. Write the translated equation below.
[LIST=1]
[*]39 added to a number is written as n + 39
[*]40 times the number is written as 40n
[*]The result is means we have an equation, so set (1) equal to (2)
[/LIST]
n+ 39 = 40n
Running [URL='http://www.mathcelebrity.com/1unk.php?num=n%2B39%3D40n&pl=Solve']n + 39 = 40n through the search engine[/URL], we get[B] n = 1[/B].

When 4 times a number is increased by 40, the answer is the same as when 100 is decreased by the num

When 4 times a number is increased by 40, the answer is the same as when 100 is decreased by the number. Find the number
The phrase [I]a number, [/I]means an arbitrary variable, let's call it "x".
4 times a number, increased by 40, means we multiply 4 times x, and then add 40
4x + 40
100 decreased by the number means we subtract x from 100
100 - x
The problem tells us both of these expressions are the same, so we set them equal to each other:
4x + 40 = 100 - x
Add x to each side:
4x + x + 40 = 100 - x + x
The x's cancel on the right side, so we have:
5x + 40 = 100
[URL='https://www.mathcelebrity.com/1unk.php?num=5x%2B40%3D100&pl=Solve']Typing this equation into the search engine[/URL], we get [B]x = 12[/B].

When 4 times a number is increased by 40, the answer is the same as when 100 is decreased by the num

When 4 times a number is increased by 40, the answer is the same as when 100 is decreased by the number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
4 times a number means we multiply x by 4:
4x
Increased by 40 means we add 40 to 4x:
4x + 40
100 decreased by the number means we subtract x from 100:
100 - x
The phrase [I]is the same as[/I] means equal to, so we set 4x + 40 equal to 100 - x
4x + 40 = 100 - x
Solve for [I]x[/I] in the equation 4x + 40 = 100 - x
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 4x and -x. To do that, we add x to both sides
4x + 40 + x = -x + 100 + x
[SIZE=5][B]Step 2: Cancel -x on the right side:[/B][/SIZE]
5x + 40 = 100
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 40 and 100. To do that, we subtract 40 from both sides
5x + 40 - 40 = 100 - 40
[SIZE=5][B]Step 4: Cancel 40 on the left side:[/B][/SIZE]
5x = 60
[SIZE=5][B]Step 5: Divide each side of the equation by 5[/B][/SIZE]
5x/5 = 60/5
x = [B]12[/B]
Check our work for x = 12:
4(12) + 40 ? 100 - 12
48 + 40 ? 100 - 12
88 = 88

When 54 is subtracted from the square of a number, the result is 3 times the number.

When 54 is subtracted from the square of a number, the result is 3 times the number.
This is an algebraic expression. Let's take it in parts.
The phrase [I]a number[/I] means an arbitrary variable, let's call it "x".
x
Square the number, means raise it to the 2nd power:
x^2
Subtract 54:
x^2 - 54
The phrase [I]the result[/I] means an equation, so we set x^2 - 54 equal to 3
[B]x^2 - 54 = 3[/B]

When a dog noticed a fox, they were 60 meters apart. The dog immediately started to chase the fox at

When a dog noticed a fox, they were 60 meters apart. The dog immediately started to chase the fox at a speed of 750 meters per minute. The fox started to run away at a speed of 720 meters per minute. How soon will the dog catch the fox?
The dog sits a position p.
Distance = Rate x Time
The dogs distance in minutes is D = 720t
The fox sits at position p + 60
Distance = Rate x Time
The fox's distance in minutes is D = 750t - 60 <-- Subtract 60 since the fox is already ahead 60 meters.
We want to know when their distance (location) is the same. So we set both distance equations equal to each other:
720t = 750t - 60
[URL='https://www.mathcelebrity.com/1unk.php?num=720t%3D750t-60&pl=Solve']Using our equation calculator[/URL], we get [B]t = 2[/B].
Let's check our work:
Dog's distance is 720(2) = 1440
Fox's distance is 750(2) - 60 = 1,440

When twice a number is reduced by 15 you get 95 what is the number

When twice a number is reduced by 15 you get 95 what is the number?
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
[I]Twice[/I] x means we multiply x by 2
2x
[I]Reduced by[/I] 15 means we subtract 15
2x - 15
[I]You get[/I] means equal to, as in an equation. Set 2x - 15 = 95
2x - 15 = 95 <-- This is our algebraic expression.
[URL='https://www.mathcelebrity.com/1unk.php?num=2x-15%3D95&pl=Solve']Type 2x - 15 = 95 into the search engine[/URL] and we get [B]x = 55[/B].

Which of the following is NOT TRUE about the distribution for averages?

Which of the following is NOT TRUE about the distribution for averages?
a. The mean, median, and mode are equal.
b. The area under the curve is one.
c. The curve never touches the x-axis.
d. The curve is skewed to the right.
Answer is d, the curve is skewed to the right
For a normal distribution:
[LIST]
[*] The area under the curve for a standard normal distribution equals 1
[*] Mean media mode are equal
[*] Never touches the x-axis since in theory, all events have some probability of occuring
[/LIST]

Which phrase is represented by the equation p equals kv

Which phrase is represented by the equation p equals kv
[B]p varies directly as v[/B]

Write a system of equations to describe the situation below, solve using any method, and fill in the

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. Hugo is going to send some flowers to his wife. Somerville Florist charges $2 per rose, plus $39 for the vase. Dwaynes Flowers, in contrast, charges $3 per rose and $10 for the vase. If Hugo orders the bouquet with a certain number of roses, the cost will be the same with either flower shop. What would the total cost be? How many roses would there be?
Let r be the number of roses and C(r) be the cost function. The vase is a one-time cost.
Somerville Florist:
C(r) = 2r + 39
Dwaynes Flowers
C(r) = 3r + 10
Set them equal to each other:
2r + 39 = 3r + 10
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=2r%2B39%3D3r%2B10&pl=Solve']equation calculator[/URL], we get:
[B]r = 29[/B]

Write the sentence as an equation. 19 is equal to c less than 321

Write the sentence as an equation. 19 is equal to c less than 321
c less than 321:
321 - c
The phrase [I]is equal to[/I] means an equation, so we set 321 - c equal to 19:
[B]321 - c = 19[/B]

X bisects WY. XY=32 and WY=2x. Find x and WY

X bisects WY. XY=32 and WY=2x. Find x and WY\
Bisects means split into two equal parts. So we have:
XY = 32
WX = XY
If XY = 32, then:
WY = 2 * 32 =[B] 64[/B]
So x = [B]32[/B]

X is at least as large as 4

X is at least as large as 4.
This is an algebraic expression, where the phrase [I]at least as large as[/I] means greater than or equal to:
[B]x >=4[/B]

X is the speed limit is a maximum 65 mph

X is the speed limit is a maximum 65 mph
A maximum of means less than or equal to. Or, no more than. So we have the inequality:
[B]X <= 65[/B]

X minus 5 plus x equals 79

X minus 5 plus x equals 79
x minus 5
x - 5
plus x
x - 5 + x
equals 79
x - 5 + x = 79
Group like terms:
(x + x) - 5 = 79
[B]2x - 5 = 79[/B]

X plus 9 is equal to 3 times x minus 4

X plus 9 is equal to 3 times x minus 4
x plus 9:
x + 9
3 times x minus 4:
3x - 4
The phrase [I]is equal to[/I] means an equation, so we set x + 9 equal to 3x - 4:
[B]x + 9 = 3x - 4[/B]

X to the 9th is less than or equal to 38

X to the 9th is less than or equal to 38:
x to the 9th means 9th power:
x^9
We set this less than or equal 38:
[B]x^9 <= 38[/B]

x tripled less two is 5

x tripled less two is 5
x tripled means we multiply x by 3
3x
Less two means we subtract 2 from 3x
3x - 2
[I]Is[/I] means equal to, so we set 3x - 2 equal to 5
[B]3x - 2 = 5[/B]
[B][/B]
To solve this equation, use our [URL='http://www.mathcelebrity.com/1unk.php?num=3x-2%3D5&pl=Solve']equation solver[/URL].

y is the sum of twice a number and 3

y is the sum of twice a number and 3
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
twice a number means we multiply x by 2:
2x
the sum of twice a number and 3:
2x + 3
The word [I]is[/I] means equal to, so we set 2x + 3 equal to y
[B]y = 2x + 3[/B]

y minus 10 is equal to the product of y and 8

y minus 10 is equal to the product of y and 8.
Take this algebraic expression in 3 parts:
Part 1: y minus 10
Subtract 10 from the variable y
y - 10
Part 2: The product of y and 8
We multiply 8 by the variable y
8y
Part 3: The phrase [I]is equal to[/I] means an equation, so we set y - 10 equal to 8y
[B]y - 10 = 8y[/B]

Yolanda wants to rent a boat and spend less than $41. The boat costs $8 per hour, and Yolanda has a

Yolanda wants to rent a boat and spend less than $41. The boat costs $8 per hour, and Yolanda has a discount coupon for $7 off. What are the possible numbers of hours Yolanda could rent the boat?
A few things to build this problem:
[LIST=1]
[*]Discount subtracts from our total
[*]Cost = Hourly rate * hours
[*]Less than means an inequality using the < sign
[/LIST]
Our inequality is:
8h - 7 < 41
To solve this inequality for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=8h-7%3C41&pl=Solve']type it in our math engine[/URL] and we get:
h < [B]6[/B]

You and two friends share 7 cookies equally. How many cookies do you each get?

You and two friends share 7 cookies equally. How many cookies do you each get?
7/3 = 2 with a remainder of 1/3
So everybody gets 2 whole cookies, and they split the last cookie into 1/3.
[B]2 & 1/3[/B]

You and your friend are playing a number-guessing game. You ask your friend to think of a positive n

You and your friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add three. If your friend's final answer is 53, what was the original number chosen?
Let n be our original number.
Square the number means we raise n to the power of 2:
n^2
Multiply the result by 2:
2n^2
And then add three:
2n^2 + 3
If the friend's final answer is 53, this means we set 2n^2 + 3 equal to 53:
2n^2 + 3 = 53
To solve for n, we subtract 3 from each side, to isolate the n term:
2n^2 + 3 - 3 = 53 - 3
Cancel the 3's on the left side, and we get:
2n^2 = 50
Divide each side of the equation by 2:
2n^2/2 = 50/2
Cancel the 2's, we get:
n^2 = 25
Take the square root of 25
n = +-sqrt(25)
n = +-5
We are told the number is positive, so we discard the negative square root and get:
n = [B]5[/B]

You and your friend are saving for a vacation. You start with the same amount and save for the same

You and your friend are saving for a vacation. You start with the same amount and save for the same number of weeks. You save 75 per week, and your friend saves 50 per week. When vacation time comes, you have 950, and your friend has 800. How much did you start with, and for how many weeks did you save?
[U]Let w be the number of weeks. Set up two equations where s is the starting amount:[/U]
(1) s + 75w =950
(2) s + 50w = 800
[U]Rearrange (1) into (3) to solve for s by subtracting 75w[/U]
(3) s = 950 - 75w
[U]Rearrange (2) into (4) to solve for s by subtracting 50w[/U]
(4) s = 800 - 50w
[U]Set (3) and (4) equal to each other so solve for w[/U]
950 - 75w = 800 - 50w
[U]Add 75w to each side, and subtract 950 from each side:[/U]
25w = 150
[U]Divide each side by w[/U]
[B]w = 6[/B]
Now plug w = 6 into (3)
s = 950 - 75(6)
s = 950 - 450
[B]s = 500[/B]

You are comparing the costs of producing shoes at two different manufacturers. Company 1 charges $5

You are comparing the costs of producing shoes at two different manufacturers. Company 1 charges $5 per pair of shoes plus a $650 flat fee. Company 2 charges $4 per pair of shoes plus a $700 flat fee. How many pairs of shoes are produced when the total costs for both companies are equal?
Let s be the number of shoes. We have two equations:
(1) C = 5s + 650
(2) C = 4s + 700
Set the costs equal to each other
5s + 650 = 4s + 700
Subtract 4s from each side
s + 650 = 700
Subtract 650 from each side
[B]s =50[/B]

You are offered two different sales jobs. The first company offers a straight commission of 6% of th

You are offered two different sales jobs. The first company offers a straight commission of 6% of the sales. The second company offers a salary of $330 per week plus 2% of the sales. How much would you have to sell in a week in order for the straight commission offer to be at least as good?
Let s be the sales and C be the weekly commission for each sales job. We have the following equations:
[LIST=1]
[*]C = 0.06s
[*]C = 330 + 0.02s
[/LIST]
Set them equal to each other:
0.06s = 330 + 0.02s
Subtract 0.02s from each side:
0.04s = 330
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=0.04s%3D330&pl=Solve']equation solver[/URL], we get [B]s = 8,250[/B]

You can get 2 different moving companies to help you move. The first one charges $150 up front then

You can get 2 different moving companies to help you move. The first one charges $150 up front then $38 an hour. The second one charges $230 then $30 an hour, at what exact time will Both companies cost the same
[U]Company 1: We set up the cost equation C(h) where h is the number of hours[/U]
C(h) = Hourly Rate * h + up front charge
C(h) = 38h + 150
[U]Company 2: We set up the cost equation C(h) where h is the number of hours[/U]
C(h) = Hourly Rate * h + up front charge
C(h) = 30h + 230
The question asks for h when both cost equations C(h) are equal. So we set both C(h) equations equal to other:
38h + 150 = 30h + 230
To solve for h, we [URL='https://www.mathcelebrity.com/1unk.php?num=38h%2B150%3D30h%2B230&pl=Solve']type this equation into our search engine [/URL]and we get:
h = [B]10[/B]

You can pay a daily entrance fee of $3 or purchase a membership for the 12 week summer season for $8

You can pay a daily entrance fee of $3 or purchase a membership for the 12 week summer season for $82 and pay only $1 per day to swim. How many days would you have to swim to make the membership worthwhile?
Set up cost equations:
Daily entrance fee:
3d where d is the number of days of membership
Membership fee
82 + 1d
Set them equal to each other
82 + 1d = 3d
Subtract d from each side:
2d = 82
Divide each side by 2
[B]d = 41[/B]

You can spend at most $35. If you buy 5 tickets, how much can you spend on each ticket

You can spend at most $35. If you buy 5 tickets, how much can you spend on each ticket
We're given the number of tickets as 5.
We know cost = price * quantity
Let p = price
The phrase [B]at most[/B] means less than or equal to, so we have:
5p <= 35
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=5p%3C%3D35&pl=Show+Interval+Notation']Plugging this inequality into our search engine[/URL], we have:
[B]p <= 7[/B]

You deposit $150 into an account that yields 2% interest compounded quarterly. How much money will

You deposit $150 into an account that yields 2% interest compounded quarterly. How much money will you have after 5 years?
2% per year compounded quarterly equals 2/4 = 0.5% per quarter. 5 years * 4 quarter per year = 20 quarters of compounding.
Using our [URL='https://www.mathcelebrity.com/compoundint.php?bal=150&nval=20&int=2&pl=Quarterly']balance calculator[/URL], we get [B]$165.73[/B] in the account after 20 years.

You have $10.00 to spend on tacos. Each taco costs $0.50. Write and solve an inequality that explain

You have $10.00 to spend on tacos. Each taco costs $0.50. Write and solve an inequality that explains how many tacos you can buy.
Let's start with t as the number of tacos.
We know that cost = price * quantity, so we have the following inequality for our taco spend:
[B]0.5t <= 10
[/B]
Divide each side of the inequality by 0.5 to isolate t:
0.5t/0.5 <= 10/0.5
Cancel the 0.5 on the left side and we get:
t <= [B]20
[MEDIA=youtube]yy51EsGi1nM[/MEDIA][/B]

You have $140 in a savings account and save $10 per week. Your friend has $95 in a savings account a

You have $140 in a savings account and save $10 per week. Your friend has $95 in a savings account and saves $19 per week. How many weeks will it take for you and your friend to have the same balance?
[U]Set up the savings account S(w) for you where w is the number of weeks[/U]
S(w) = 140 + 10w
[U]Set up the savings account S(w) for your friend where w is the number of weeks[/U]
S(w) = 95 + 19w
The problem asks for the number of weeks (w) when the balances are the same. So set both equations equal to each other:
140 + 10w = 95 + 19w
To solve this equation for w, [URL='https://www.mathcelebrity.com/1unk.php?num=140%2B10w%3D95%2B19w&pl=Solve']we type it in the search engine[/URL] and get:
w = [B]5[/B]

You have $20 to spend on taxi fare. The ride costs $5 plus $2.50 per kilometer. Write the inequality

You have $20 to spend on taxi fare. The ride costs $5 plus $2.50 per kilometer. Write the inequality.
Let k be the number of kilometers. We want our total to be $20 [I]or less. [/I]We have the following inequality:
[B]2.50k + 5 <= 20[/B]

You have $6.50 to make copies. It cost $0.45. Write and solve an equality that represents the number

You have $6.50 to make copies. It cost $0.45. Write and solve an equality that represents the number of copies
Hoow many exact copies can you make? Let the number of copies be c. We have:
0.45c = 6.50
[URL='https://www.mathcelebrity.com/1unk.php?num=0.45c%3D6.50&pl=Solve']Type this equation into our search engine[/URL] and we get:
c = 14.444
We round down and say we can make 14 copies.
[B]c = 14[/B]
Now, if the problem asks you for an [I]inequality[/I], we want to see how many copies we can make without exceeding our $6.50 spend. So it's less than or equal to:
[B]c <= 14[/B]

You have $80. Jeans cost $29 and shirts cost $12. Mom told you to buy one pair of jeans and use the

You have $80. Jeans cost $29 and shirts cost $12. Mom told you to buy one pair of jeans and use the rest of the money to buy shirts. Find the inequality.
Let j be the number of jeans. Let s be the number of shirts. We are given:
[LIST]
[*]Mom told you to buy one pair of jeans. So we have $80 to start with - $29 for 1 pair of jeans = $51 left over
[/LIST]
Now, since shirts cost $12 each, and our total number of shirts we can buy is s, our inequality is [B]12s <= 51[/B].
We want to find the s that makes this inequality true.
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=12s%3C%3D51&pl=Show+Interval+Notation']Run this statement through our calculator[/URL], and we get s <= 4.25. But, we need s to be an integer, so we have s <= 4.

You have to pay 29 a month until you reach 850 how many months will that take

You have to pay 29 a month until you reach 850 how many months will that take.
Let m be the number of months. We set up the inequality:
29m > = 850 <-- We want to know when we meet or exceed 850, so we use greater than or equal to
[URL='https://www.mathcelebrity.com/interval-notation-calculator.php?num=29m%3E%3D850&pl=Show+Interval+Notation']Type this inequality into our search engine[/URL], and we get:
m >= 29.31
We round up to the next integer month, to get [B]m = 30[/B].

you must be 65 or older to join inequality

you must be 65 or older to join inequality
Let a be the age. 65 or older means greater than or equal to 65:
[B]a >=65[/B]

You need to hire a catering company to serve meals to guests at a wedding reception. Company A charg

You need to hire a catering company to serve meals to guests at a wedding reception. Company A charges $500 plus $20 per guest. Company B charges $800 plus $16 per guest. For how many guests are the total costs the same at both companies?
Set up the Cost equations for both companies where g is the number of guests:
[LIST]
[*]C(a) = 20g + 500
[*]C(b) = 16g + 800
[/LIST]
Set each equation equal to each other and use our [URL='http://www.mathcelebrity.com/1unk.php?num=20g%2B500%3D16g%2B800&pl=Solve']equation solver[/URL] to get:
[B]g = 75[/B]

Your brother has $2000 saved for a vacation. His airplane ticket is $637. Write and solve an inequal

Your brother has $2000 saved for a vacation. His airplane ticket is $637. Write and solve an inequality to find how much he can spend for everything else.
Let x be the amount your brother can spend. Subtracting the cost of the plane ticket from savings, we have:
x <= 2000 - 637
[B]x <= 1,363[/B]

Your clothes washer stopped working during the spin cycle and you need to get a person in to fix the

Your clothes washer stopped working during the spin cycle and you need to get a person in to fix the washer. Company A costs $20 for the visit and $15 for every hour the person is there to fix the problem. Company B costs $40 for the visit and $5 for every hour the person is there to fix the problem. When would Company B be cheaper than Company A?
Set up the cost functions:
[LIST]
[*]Company A: C(h) = 15h + 20
[*]Company B: C(h) = 5h + 40
[/LIST]
Set them equal to each other:
15h + 20 = 5h + 40
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=15h%2B20%3D5h%2B40&pl=Solve']equation solver[/URL], we get h = 2.
With [B]h = 3[/B] and beyond, Company B becomes cheaper than Company A.

Your grade must be at least 60 to pass this class

Your grade must be at least 60 to pass this class
Assumptions and givens:
[LIST]
[*]The phrase [I]at least[/I] means greater than or equal to.
[*]Let g be your grade
[/LIST]
We have:
[B]g >= 60[/B]

Zalika thinks of a number. She subtracts 6 then multiplies the result by 5. The answer is the same a

Zalika thinks of a number. She subtracts 6 then multiplies the result by 5. The answer is the same as subtracting 5 from the number then multiplying by 4.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x. We're given two expressions in relation to this number (x):
[U]She subtracts 6 then multiplies the result by 5[/U]
[LIST]
[*]Subtract 6: x - 6
[*]Multiply the result by 5: 5(x - 6)
[/LIST]
[U]She subtracts 5 from the number then multiplying by 4[/U]
[LIST]
[*]Subtract 6: x - 5
[*]Multiply the result by 5: 4(x - 5)
[/LIST]
Finally, the expression [I]is the same as[/I] means an equation, so we set the first expression equal to the second expression to make the following equation:
5(x - 6) = 4(x - 5)
Now, let's solve the equation for x. To do this, we [URL='https://www.mathcelebrity.com/1unk.php?num=5%28x-6%29%3D4%28x-5%29&pl=Solve']type this equation into our search engine [/URL]and we get:
x = [B]10[/B]

“The shortest person in a class is 55 inches tall and tallest person in that class is 75 inches tall

The shortest person in a class is 55 inches tall and tallest person in that class is 75 inches tall. write an absolute value equation that requires the minimum and maximum height. Use X to represent heights.
We write our inequality as:
[B]55 <= X <= 75[/B]