multiply - to add equal groups

Formula: *

(4x - 20)/8 = 9y for x

(4x - 20)/8 = 9y for x
Cross multiply:
4x - 20 = 8 * 9y
4x - 20 = 72y
Add 20 to each side to isolate x:
4x - 20 + 20 = 72y + 20
Cancel the 20 on the left side, we get:
4x = 72y + 20
Divide each side by 4:
4x/4 = (72y + 20)/4
Cancel the 4 on the left side:
x = [B](72y + 20)/4[/B]

-2 times the quantity t plus 7

-2 times the quantity t plus 7
The key word here is quantity. In this case, the quantity is t plus 7
t + 7
-2 times the quantity means we multiply -2 times the quantity t + 7
[B]-2(t + 7)
[MEDIA=youtube]nUWLUPfX52k[/MEDIA][/B]

-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]

-g + 3/4a = y for a

-g + 3/4a = y for a
Add g to each side:
-g + g + 3/4a = y + g
Cancel the g terms on the left side:
3/4a = y + g
Cross multiply:
3a = 4(y + g)
Divide each side by 3 to isolate a:
3a/3 = 4(y + g)/3
a = [B]4(y + g)/3[/B]

-g+F/A=h^3 for A

-g+F/A=h^3 for A
Add g to each side:
-g + g+F/A=h^3 + g
Cancel the g's on the left side:
F/A = h^3 + g
Cross multiply:
F = A(h^3 + g)
Divide each side by (h^3 + g)
F/(h^3 + g) = A(h^3 + g)/(h^3 + g)
Cancel (h^3 + g) on the right side:
A = [B]F/(h^3 + g)[/B]

-n = -50

-n = -50
Multiply each side by -1:
-1*-n = -1 * -50
n = [B]50[/B]

-n/8 = 80

-n/8 = 80
Cross multiply:
-n = 80 * 8
-n = 640
Multiply each side by -1:
-1 * -n = -1 * 640
n = [B]-640[/B]

1 multiplied by b squared multiplied by c squared

1 multiplied by b squared multiplied by c squared
b squared means we raise b to the power of 2:
b^2
c squared means we raise c to the power of 2:
c^2
b squared multiplied by c squared
b^2c^2
1 multiplied by b squared multiplied by c squared means we multiply 1 by b^2c^2
1b^2c^2
Multiplying by 1 can be written by [U][I]removing[/I][/U] the 1 since it's an identity multiplication:
[B]b^2c^2[/B]

1/10n = 100

1/10n = 100
Cross multiply:
n = 100 * 10
n = [B]1000[/B]

1/2 of a number decreased by twice a number

1/2 of a number decreased by twice a number
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
[*]1/2 of a number: x/2
[*]Twice a number means we multiply x by 2: 2x
[*]The phrase [I]decreased by[/I] means we subtract
[/LIST]
[B]x/2 - 2x[/B]

1/2 the quantity of x plus y

1/2 the quantity of x plus y
The quantity of x plus y
x + y
1/2 the quantity means we multiply x + y by 1/2:
[B](x + y)/2[/B]

1/2a-10b=c solve for a

1/2a-10b=c solve for a
Multiply each side of the equation by 2:
2/2a - 2(10)b = 2c
Simplify:
a - 20b = 2c
Add 20b to each side:
a - 20b + 20b = 2c + 20b
Cancel the 20b on the left side:
[B]a = 2c + 20b
[/B]
You can also factor out a 2 on the left side for another version of this answer:
[B]a = 2(c + 10b)[/B]

1/3ab^2=6 for a

1/3ab^2=6 for a
Multiply each side by 3:
ab^2 = 18
Divide each side by b^2
a = 18/b^2

1/3x plus twice y

1/3x plus twice y
Twice y means we multiply y by 2:
2y
1/3x plus twice y
[B]1/3x + 2y[/B]

1/a + 1/b = 1/2 for a

1/a + 1/b = 1/2 for a
Subtract 1/b from each side to solve this literal equation:
1/a + 1/b - 1/b = 1/2 - 1/b
Cancel the 1/b on the left side, we get:
1/a = 1/2 - 1/b
Rewrite the right side, using 2b as a common denominator:
1/a = (b - 2)/2b
Cross multiply:
a(b - 2) = 2b
Divide each side by (b - 2)
a = [B]2b/(b - 2)[/B]

1/n + 2/n + 3/n + 4/n = 1

1/n + 2/n + 3/n + 4/n = 1
10/n = 1
Cross multiply:
n * 1 = 10
n = [B]10[/B]

1/n + 2/n + 3/n = 1

1/n + 2/n + 3/n = 1
6/n = 1
Cross multiply:
n * 1 = 6
n = [B]6[/B]

1/n + 3/5 = 1

1/n + 3/5 = 1
Subtract 3/5 from each side where 1 = 5/5
1/n + 3/5 - 3/5 = 5/5 - 3/5
1/n = 2/5
Cross multiply:
5 * 1 = 2 * n
2n = 5
Divide each side by 2:
n = [B]5/2 or 2.5[/B]

1/n^2 = 3/192

1/n^2 = 3/192
Cross multiply:
192 * 1 = 3 * n^2
3n^2 = 192
Divide each side by 3:
3n^2/3 = 192/3
Cancel the 3's on the left side:
n^2 = 64
Take the square root of both sides:
n = [B]8 or -8[/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 the first of 2 consecutive even integers is 8 times the second. Find the integers

10 times the first of 2 consecutive even integers is 8 times the second. Find the integers.
Let the first integer be x. Let the second integer be y. We're given:
[LIST=1]
[*]10x = 8y
[*]We also know a consecutive even integer means we add 2 to x to get y. y = x + 2
[/LIST]
Substitute (1) into (2):
10x = 8(x + 2)
Multiply through:
10x = 8x + 16
To solve for x, [URL='https://www.mathcelebrity.com/1unk.php?num=10x%3D8x%2B16&pl=Solve']we type this equation into our search engine[/URL] and we get:
[B]x = 8[/B]
Since y = x + 2, we plug in x = 8 to get:
y = 8 + 2
[B]y = 10
[/B]
Now, let's check our work. Does x = 8 and y = 10 make equation 1 hold?
10(8) ? 8(10)
80 = 80 <-- Yes!

10 x 12 divided by 9

10 x 12 divided by 9
12/9
1.3333333
Then multiply by 10:
[B]13.33333333[/B]

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]

11 to the power of 6 multiply 11 to the power of 3

11 to the power of 6 multiply 11 to the power of 3
Take this in parts.
[U]Step 1: 11 to the power of 6 means we raise 11 to the 6th power using exponents:[/U]
11^6
[U]Step 2: 11 to the power of 3 means we raise 11 to the 3rd power using exponents:[/U]
11^3
[U]Step 3: Multiply each term together:[/U]
11^6 * 11^3
[U]Step 4: Simplify[/U]
Because we have 2 numbers that are the same, in this case, 11, we can add the exponents together when multiplying:
11^(6 + 3)
[B]11^9
[MEDIA=youtube]gCxVq7LqyHk[/MEDIA][/B]

12 is subtracted from d and the result is tripled

12 is subtracted from d and the result is tripled
12 is subtracted from d:
d - 12
the result is tripled means we multiply d - 12 by 3
[B]3(d - 12)[/B]

12 is subtracted from d and the result is tripled.

12 is subtracted from d and the result is tripled.
12 is subtracted from d:
d - 12
The result is tripled means we multiply d - 12 by 3
[B]3(d - 12)
[MEDIA=youtube]1xqWstiIDP0[/MEDIA][/B]

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 times the sum of x and 9y

13 times the sum of x and 9y
The sum of x and 9y means we add 9y to x:
x + 9y
Now multiply this sum by 13:
[B]13(x + 9y)[/B]

14 increased by twice Carlos’s age. Use the variable c to represent Carlos age

14 increased by twice Carlos’s age. Use the variable c to represent Carlos age
Twice means me multiply a by 2:
2a
14 increased by twice Carlos's age means we add 2a to 14:
[B]14 + 2a[/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]

175 students separated into n classes is 25

175 students separated into n classes is 25
175/n = 25
Cross multiply:
25n = 175
Use our [URL='http://www.mathcelebrity.com/1unk.php?num=25n%3D175&pl=Solve']equation calculator[/URL], we get:
[B]n = 7[/B]

18 multiplied by the quantity of 11 plus r

18 multiplied by the quantity of 11 plus r
The quantity of 11 plus r is written as:
11 + r
18 multiplied by the [I]quantity[/I] means we take 18 and multiply it by the term 11 + r
[B]18(11 + r)
[MEDIA=youtube]2GYjQTjt8qM[/MEDIA][/B]

19 increased by twice Greg’s score use the variable g to represent Greg’s score

19 increased by twice Greg’s score use the variable g to represent Greg’s score
Use g for Greg's score
g
Twice g means we multiply g by 2:
2g
19 increased by means we add 2g to 19
[B]2g + 19
[MEDIA=youtube]E9a_U7z-fHE[/MEDIA][/B]

19 increased by twice Vanessa's age

19 increased by twice Vanessa's age
Let Vanessa's age be a.
Twice means we multiply a by 2:
2a
The phrase [I]increased by[/I] means we add 2a to 19:
[B]19 + 2a[/B]

2 coins are tossed. Find the probability of getting 1 head and 1 tail

2 coins are tossed. Find the probability of getting 1 head and 1 tail
We can either flip HT or TH. Let's review probabilities:
[LIST]
[*]HT = 1/2 * 1/2 = 1/4 <-- We multiply since each event is independent
[*]TH = 1/2 * 1/2 = 1/4 <-- We multiply since each event is independent
[/LIST]
P(1 H, 1 T) = P(HT) + P(TH)
P(1 H, 1 T) = 1/4 + 1/4
P(1 H, 1 T) = 2/4
[URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F4&frac2=3%2F8&pl=Simplify']Using our fraction simplifier[/URL], we can reduce 2/4 to 1/2
P(1 H, 1 T) = [B]1/2[/B]

2 minus 7 times a number

A number is represented by an arbitrary variable, let's call it x.
7 times x means we multiply 7 times x.
7x
2 minus 7x is written:
2 - 7x

2 more than twice the sum of 10 and a number

2 more than twice the sum of 10 and a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The sum of 10 and a number means we add x to 10:
10 + x
Twice the sum means we multiply 10 + x by 2:
2(10 + x)
2 more than twice the sum means we add 2 to 2(10 + x):
[B]2(10 + x) + 2[/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 a number subtracted by x

2 times a number subtracted by x
The phrase [I]a number[/I] means an arbitrary variable, let's call it n.
n
2 times a number means we multiply n by 2:
2n
The phrase [I]subtracted by[/I] means we subtract 2n from x:
[B]x - 2n[/B]

2 times half of a number

A number means an arbitrary variable, let's call it x.
Half of x means we divide x by 2, or multiply by 0.5
x/2
2 times half x is written:
[B]2(x/2)[/B]
If we simplify by cancelling the 2's, we just get x.

2 times itself

2 times itself
Itself means we multiply 2 by 2:
2 * 2
[B]4[/B]

2 times the sum of 7 times a number and 4

2 times the sum of 7 times a number and 4
This is an algebraic expression. Let's take it in 4 parts:
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
[*]7 times a number means we multiply x by 7: 7x
[*]The sum of 7 times a number and 4 means we add 4 to 7x: 7x + 4
[*]Finally, we multiply the sum in #3 by 2
[/LIST]
Build our final algebraic expression:
[B]2(7x + 4)[/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 times the sum of x and 7 plus 10

2 times the sum of x and 7 plus 10
The sum of x and 7 means we add 7 to x
x + 7
2 times the sum means we multiply the quantity x + 7 by 2
2(x + 7)
Plus 10 means we add 10 to the 2(x + 7):
[B]2(x + 7) + 10[/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]

2/5 the cube of a number

2/5 the cube of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The cube of a number means we raise x to the power of 3:
x^3
2/5 of the cube means we multiply x^3 by 2/5:
[B](2x^3)/5[/B]

20% of a number is x. What is 100% of the number? Assume x>0.

20% of a number is x. What is 100% of the number? Assume x>0.
Let the number be n. We're given:
0.2n = x <-- Since 20% = 0.2
To find n, we multiply each side of the equation by 5:
5(0.2)n = 5x
n = [B]5x[/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]

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

250 students have iPhones. This is one third of the population. How many students are there in total

250 students have iPhones. This is one third of the population. How many students are there in total?
Let the population be p. We're given:
1/3p = 250
Cross multiply:
p = 250 * 3
p = [B]750[/B]

26 diminished by twice m

26 diminished by twice m
Twice m means multiply m by 2
2m
26 diminished by twice m means subtract 2m from 26
[B]26 - 2m[/B]

26 students 15 like vanilla 16 like chocolate. 3 do not like either flavour. How many like both vani

26 students 15 like vanilla 16 like chocolate. 3 do not like either flavour. How many like both vanilla and chocolate
Define our people:
[LIST=1]
[*]We have Vanilla Only
[*]Chocolate Only
[*]Both Vanilla and Chocolate
[*]Neither Vanilla Nor Chocolate
[*]Add up 1-4 to get our total
[/LIST]
Total = Vanilla Only + Chocolate Only - Vanilla and Chocolate + Neither
26 = 15 + 16 - V&C + 3
26 = 34 - V&C
Subtract 34 from each side
-V&C = -8
Multiply each side by -1
[B]V&C = 8[/B]

28 less than twice a number

[U]A number means an arbitrary variable, let's call it x.[/U]
[LIST]
[*]x
[/LIST]
[U]Twice a number means multiply by 2[/U]
[LIST]
[*]2x
[/LIST]
[U]28 less than twice a number means we subtract 28[/U]
[LIST]
[*][B]2x - 28[/B]
[/LIST]

2m - n/3 = 5m for n

2m - n/3 = 5m for n
Subtract 2m from each side of the equation:
2m-n/3 - 2m = 5m - 2m
-n/3 = 3m
Multiply each side of the equation by -3 to isolate n:
-3 * -n/3 = -3 * 3m
n = [B]-9m[/B]

2x - b/y = 4c for y

2x - b/y = 4c for y
Subtract 2x from each side:
2x - 2x - b/y = 4c - 2x
Cancel the 2x's on the left side and we get:
-b/y = 4c - 2x
Cross multiply:
-b = y(4c - 2x)
Divide each side by (4c - 2x):
-b/(4c - 2x) = y(4c - 2x)/(4c - 2x)
Cancel the (4c - 2x) on the right side and we get:
[B]y = -b/(4c - 2x) [/B]

2x increased by 3 times a number

2x increased by 3 times a number
The phrase [I]a number[/I] means an arbitary variable, let's call it x.
3 times a number means we multiply x by 3:
3x
The phrase [I]increased by[/I] means we add 3x to 2x:
2x + 3x
Simplifying, we get:
[B]5x[/B]

2x/5 - 9y = 6 for x

2x/5 - 9y = 6 for x
Add 9y to each side to isolate the x term:
2x/5 - 9y + 9y = 9y + 6
Cancel the 9y's on the left side:
2x/5 = 9y + 6
Multiply each side by 5:
2x * 5/5 = 5(9y + 6)
Cancel the 5's on the left side and we get:
2x = 5(9y + 6)
Divide each side by 2 to isolate x:
2x/2 = 5/2(9y + 6)
Cancel the 2's on the left side and we get our final literal equation of:
x = [B]5/2(9y + 6)[/B]

2x/5 - 9y = 6 for x

2x/5 - 9y = 6 for x
Add 9y to each side of the equation:
2x/5 - 9y + 9y = 6 + 9y
Cancel the 9y's on the left side to get:
2x/5 = 6 + 9y
Multiply each side of the equation by 5:
5(2x/5) = 5(6 + 9y)
Cancel the 5's on the left side to get
2x = 5(6 + 9y)
Divide each side of the equation by 2:
2x/2 = 5/2(6 + 9y)
Cancel the 2's on the left side to get:
x = [B]5/2(6 + 9y)[/B]

3 is subtracted from 3/4 of g

3 is subtracted from 3/4 of g
3/4 of g means we multiply g by 3/4:
3g/4
Subtracted from means we subtract 3 from 3g/4
[B]3g/4 - 3[/B]

3 salads, 4 main dishes, and 2 desserts

3 salads, 4 main dishes, and 2 desserts
Total meal combinations are found by multiplying each salad, main dish, and dessert using the fundamental rule of counting.
The fundamental rule of counting states, if there are a ways of doing one thing, b ways of doing another thing, and c ways of doing another thing, than the total combinations of all the ways are found by a * b * c.
With 3 salads, 4 main dishes, and 2 desserts, our total meal combinations are:
3 * 4 * 2 = [B]24 different meal combinations.[/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 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 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 the sum of twice k and 8

3 times the sum of twice k and 8
Twice k means we multiply k by 2:
2k
The sum of twice k and 8:
2k + 8
3 times the sum:
[B]3(2k + 8)[/B]

3 times the sum of x and 9y

3 times the sum of x and 9y
The sum of x and 9y means we add 9y to x:
x + 9y
Now we take this sum, and multiply by 3 to get our final algebraic expression:
3(x + 9y)

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]

3/10 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]

3/4 a number b divided by 5

3/4 a number b divided by 5
3/4 a number b:
3b/4
Divided by 5:
3b/4/5
We multiply top and bottom by 5 to remove the double fraction:
3b*5/4
[B]15b/4[/B]

3/4 of the students went skiing.there are 24 students in the class. How’s many went?

3/4 of the students went skiing.there are 24 students in the class. How’s many went?
We want 3/4 of 24. We [URL='https://www.mathcelebrity.com/fraction.php?frac1=24&frac2=3/4&pl=Multiply']type 3/4 of 24 into our search engine[/URL] and get:
[B]18 students[/B]

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]

30% larger then 1/3 of twice q

30% larger then 1/3 of twice q
Take this algebraic expression in 3 parts:
[LIST=1]
[*]Twice q means multiply q by 2: 2q
[*]1/3 of twice q means we multiply 2q in Step 1 by 1/3: 2q/3
[*]30% larger means we multiply 2q/3 in step 2 by 1.3, since 30% = 0.3: 1.3(2q/3)
[/LIST]
[B]1.3(2q/3)[/B]

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

36 PAGES AND IT 3/8CM THICK, WHAT IS THE THICKNESS OF 1 PAGE?

36 PAGES AND IT 3/8CM THICK, WHAT IS THE THICKNESS OF 1 PAGE?
Set up a proportion in pages to cm:
36 pages /3/8cm = 1 page/x cm
Cross multiply:
36x = 3/8
Divide each side by 36
x = 3/(8 * 36)
x = 1/(8*12)
x = [B]1/96 cm[/B]

3abc^4/12a^3(b^3c^2)^2 * 8ab^-4c/4a^2b

3abc^4/12a^3(b^3c^2)^2 * 8ab^-4c/4a^2b
Expand term 1:
3abc^4/12a^3(b^3c^2)^2
3abc^4/12a^3b^6c^4
Now simplify term 1:
3/12 = 1/4
c^4 terms cancel
Subtract powers from variables since the denominator powers are higher:
b^(6 - 1) = b^5
a^(3 - 1) = a^2
1/4a^2b^5
Now simplify term 2:
8ab^-4c/4a^2b
8/4 = 2
2c/a^(2 - 1)b^(1 - -4)
2c/ab^5
Now multiply simplified term 1 times simplified term 2:
1/4a^2b^5 * 2c/ab^5
(1 * 2c)/(4a^2b^5 * ab^5)
2c/4a^(2 + 1)b^(5 + 5)
2c/4a^3b^10
2/4 = 1/2, so we have:
[B]c/2a^3b^10[/B]

3n/5 = 1

3n/5 = 1
Cross multiply:
3n = 5 * 1
3n = 5
Divide each side by 3:
3n/3 = 5/3
n = 5/3

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]
[B][/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[/B]

4 added to twice a

[U]Twice a means we multiply a by 2[/U]
2a
[U]4 added to that means we add 4[/U]
[B]2a + 4[/B]

4 divided by sin60 degrees

4 divided by sin60 degrees.
We can write as 4/sin(60).
[URL='https://www.mathcelebrity.com/anglebasic.php?entry=60&coff=&pl=sin']Using our trigonometry calculator[/URL], we see sin(60) = sqrt(3)/2.
So we have 4/sqrt(3)/2.
Multiplying by the reciprocal we have:
4*2/sqrt(3)
[B]8/sqrt(3)[/B]

4 times b increased by 9 minus twice y

4 times b increased by 9 minus twice y
Take this algebraic expression in parts:
Step 1: 4 times b means we multiply the variable b by 4:
4b
Step 2: Increased by 9 means we add 9 to 4b:
4b + 9
Step 3: Twice y means we multiply the variable y by 2:
2y
Step 4: The phrase [I]minus[/I] means we subtract 2y from 4b + 9
[B]4b + 9 - 2y[/B]

4 times the difference of 6 times a number and 7

4 times the difference of 6 times a number and 7
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
6 times a number
6x
The difference of 6x and 7 means we subtract 7 from 6x:
6x - 7
Now we multiply this difference by 4:
[B]4(6x - 7)[/B]

4 times the quantity of a number plus 6

4 times the quantity of a number plus 6
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The word [I]plus[/I] means we addd 6 to x
x + 6
The phrase [I]4 times the quantity [/I]means we multiply x + 6 by 4
[B]4(x + 6)[/B]

4 times the sum of 10 and twice x

4 times the sum of 10 and twice x
Twice x means we multiply x by 2:
2x
The sum of 10 and twice x:
10 + 2x
Now multiply this sum by 4:
[B]4(10 + 2x)[/B]

4/5 of the sum of k and 3

4/5 of the sum of k and 3
The sum of k and 3 means we add 3 to k:
k + 3
4/5 of the sum means we multiply 4/5 times the sum k + 3:
[B]4(k + 3)/5[/B]

4d/a - 9 = g for a

4d/a - 9 = g for a
Add 9 to each side:
4d/a - 9 + 9 = g + 9
Cancel the 9's on the left side and we get:
4d/a = g + 9
Cross multiply:
4d = a(g + 9)
Divide each side of the equation by (g + 9) to isolate a:
4d/(g + 9) = a(g + 9)/(g + 9)
Cancel the (g + 9) on the right side, and we get:
a = [B]4d/(g + 9)[/B]

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 diminished by twice the sum of a and b

5 diminished by twice the sum of a and b
Take this algebraic expression in parts:
[LIST]
[*]The sum of a and b: a + b
[*]Twice the sum means we multiply a + b by 2: 2(a + b)
[*]5 diminished by twice the sum means we subtract 2(a + b) from 5
[/LIST]
[B]5 - 2(a + b)[/B]

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 more than twice the cube of a number

5 more than twice the cube of a number.
Take this algebraic expression in pieces.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The cube of a number means we raise it to a power of 3
x^3
Twice the cube of a number means we multiply x^3 by 2
2x^3
5 more than twice the cube of a number means we multiply 2x^3 by 5
5(2x^3)
Simplifying, we get:
10x^3

5 more than twice the cube of a number

5 more than twice the cube of a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
The cube of a number means we raise x to the power of 3:
x^3
Twice the cube means we multiply x^3 by 2
2x^3
Finally, 5 more than twice the cube means we add 5 to 2x^3:
[B]2x^3 + 5[/B]

5 times a number increased by 4 is divided by 6 times the same number

5 times a number increased by 4 is divided by 6 times the same number
Take this algebraic expression in parts.
Part 1: 5 times a number increased by 4
[LIST]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x: x
[*]5 times the number means multiply x by 5: 5x
[*][I]Increased by 4[/I] means we add 4 to 5x: 5x + 4
[/LIST]
Part 2: 6 times the same number
[LIST]
[*]From above, [I]a number[/I] is x: x
[*]6 times the number means we multiply x by 6: 6x
[/LIST]
The phrase [I]is divided by[/I] means we have a quotient, where 5x + 4 is the numerator, and 6x is the denominator.
[B](5x + 4)/6x[/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 g reduced by the square of h

5 times g reduced by the square of h
Take this algebraic expression in pieces:
[LIST=1]
[*]5 times g means we multiply g by 5: 5g
[*]The square of h means we raise h to the 2nd power: h^2
[*]5 times g reduced by the square of h means we subtract h^2 from 5g:
[/LIST]
[B]5g - h^2[/B]

5 times the product of 2 numbers a and b

5 times the product of 2 numbers a and b
The product of 2 numbers a and be means we multiply the variables together:
ab
5 times the product means we multiply ab by 5:
[B]5ab[/B]

5 times the sum of 3 times a number and -5

5 times the sum of 3 times a number and -5
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
3 times a number means we multiply x by 3:
3x
the sum of 3 times a number and -5 means we add -5 to 3x:
3x - 5
5 times the sum means we multiply 3x - 5 by 5:
[B]5(3x - 5)[/B]

5 times the total of 60 and x

5 times the total of 60 and x
The total of 60 and x means we add:
60 + x
5 times the total means we multiply the sum by 5
5(60 + x)

5/12 of a circle what measure in degrees

5/12 of a circle what measure in degrees
A circle measures 360 degrees. We [URL='https://www.mathcelebrity.com/fraction.php?frac1=5%2F12&frac2=360&pl=Multiply']multiply as follows[/URL]:
5/12 * 360 = [B]150 degrees[/B]

5/9v+w=z,for v

5/9v+w=z,for v
Subtract w from each side:
5/9v = z - w
Multiply each side by 9/5
[B]v = 9(z - w)/5[/B]

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]

508 people are there, the daily price is $1.25 for kids and $2.00 for adults. The receipts totaled $

508 people are there, the daily price is $1.25 for kids and $2.00 for adults. The receipts totaled $885.50. How many kids and how many adults were there?
Assumptions:
[LIST]
[*]Let the number of adults be a
[*]Let the number of kids be k
[/LIST]
Given with assumptions:
[LIST=1]
[*]a + k = 508
[*]2a + 1.25k = 885.50 (since cost = price * quantity)
[/LIST]
Rearrange equation (1) by subtracting c from each side to isolate a:
[LIST=1]
[*]a = 508 - k
[*]2a + 1.25k = 885.50
[/LIST]
Substitute equation (1) into equation (2):
2(508 - k) + 1.25k = 885.50
Multiply through:
1016 - 2k + 1.25k = 885.50
1016 - 0.75k = 885.50
To solve for k, we [URL='https://www.mathcelebrity.com/1unk.php?num=1016-0.75k%3D885.50&pl=Solve']type this equation into our search engine[/URL] and we get:
k = [B]174[/B]
Now, to solve for a, we substitute k = 174 into equation 1 above:
a = 508 - 174
a = [B]334[/B]

51 decreased by twice a number

A number is denoted as an arbitrary variable, let's call it x.
Twice a number means we multiply by 2, so 2x.
51 decreased by twice a number means we subtract 2x from 51
[B]51 - 2x[/B]

6 & 1/8 x 1/7

6 & 1/8 x 1/7
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=6%261%2F8&frac2=1%2F7&pl=Multiply']fraction calculator[/URL], we get:
[B]7/8[/B]

6 boys have a mean age of 10 years and 14 girls have a mean age of 5 work out the mean age of the 20

6 boys have a mean age of 10 years and 14 girls have a mean age of 5 work out the mean age of the 20 children
[U]Calculate Sum of boys ages:[/U]
Sum of boys ages/6 = 10
Cross multiply, and we get:
Sum of boys ages = 6 * 10
Sum of boys ages = 60
[U]Calculate Sum of girls ages:[/U]
Sum of girls ages/14 = 5
Cross multiply, and we get:
Sum of girls ages = 14 * 5
Sum of girls ages = 70
Average of 20 children is:
Average of 20 children = (Sum of boys ages + sum of girls ages)/20
Average of 20 children = (60 + 70)/20
Average of 20 children = 130/20
Average of 20 children = [B]6.5 years[/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 numbers have a mean of 4. What is the total of the 6 numbers?

6 numbers have a mean of 4. What is the total of the 6 numbers?
Mean = Sum of numbers / Count of numbers
Plug our Mean of 4 and our count of 6 into this equation:
4 = Sum/Total of Numbers / 6
Cross multiply:
Sum/Total of Numbers = 6 * 4
Sum/Total of Numbers = [B]24[/B]

6 plus twice the sum of a number and 7.

6 plus twice the sum of a number and 7.
The phrase [I]a number[/I] mean an arbitrary variable, let's call it x.
The sum of a number and 7 means we add 7 to the variable x.
x + 7
Twice the sum means we multiply the sum by 2:
2(x + 7)
6 plus means we add 6 to 2(x + 7)
[B]6 + 2(x + 7)[/B]

6 red marbles 9 green marbles and 5 blue marbles two marbles are drawn without replacement what is t

6 red marbles 9 green marbles and 5 blue marbles two marbles are drawn without replacement what is the probability of choosing a green and then a blue marble
First draw:
there are 6 red + 9 green + 5 blue = 20 marbles
We draw 9 possible green out of 20 total marbles = 9/20
Second draw:
We don't replace, so we have 6 red + 8 green + 5 blue = 19 marbles
We draw 5 possible blue of out 19 total marbles = 5/19
Our total probability, since each event is independent, is:
[URL='https://www.mathcelebrity.com/fraction.php?frac1=9%2F20&frac2=5%2F19&pl=Multiply']9/20 * 5/19[/URL] = [B]9/76[/B]

6 times j squared minus twice j squared

6 times j squared minus twice j squared
j squared means we raise the variable j to the power of 2:
j^2
6 times j squared means we multiply j^2 by 6:
6j^2
Twice j squared means we multiply j^2 by 2:
2j^2
The word [I]minus[/I] means we subtract 2j^2 from 6j^2
6j^2 - 2j^2
So if you must simplify, we group like terms and get:
(6 - 2)j^2
[B]4j^2[/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

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 the cube of the sum of x and 8

7 times the cube of the sum of x and 8
Take this algebraic expression in 3 parts:
[LIST=1]
[*]The sum of x and 8 means we add 8 to x: x + 8
[*]The cube of this sum means we raise the sum to the 3rd power: (x + 8)^3
[*]7 times this cubed sum means we multiply (x + 8)^3 by 7:
[/LIST]
[B]7(x + 8)^3[/B]

70 decreased by twice Carlos's age

Let Carlos's age be a. Twice a means we multiply by 2
2a
70 decreased by that amount means we subtract:
[B]70 - 2a[/B]

76 decreased by twice a number. Use the variable n to represent the unknown number

76 decreased by twice a number. Use the variable n to represent the unknown number.
Twice a number (n) means we multiply the unknown number n by 2:
2n
76 decreased by twice a number means we subtract 2n from 76 using the (-) operator
[B]76 - 2n[/B]

8 is subtracted from thrice a number

Thrice a number means we multiply by 3. A number means an arbitrary variable, let's call it x
3x
8 is subtracted from 3x
[B]3x - 8[/B]

8 is subtracted from twice a number

Twice a number:
[LIST]
[*]Choose an arbitrary variable, let's call it x
[*]Twice x means multiply by 2
[*]2x
[/LIST]
8 subtracted from 2x:
[B]2x - 8[/B]

8 less thantriplethedifferenceof2xand6

8 less than triple the difference of 2x and 6
The [I]difference[/I] of 2x and 6 means we [B]subtract[/B] 6 from 2x
2x - 6
[I]Triple[/I] this difference means we [B]multiply by 3[/B]
3(2x - 6)
8 [I]less[/I] means we [B]subtract 8 from this expression
3(2x - 6) - 8[/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 5y and 3

8 times the difference of 5y and 3
The difference of 5y and 3 means we subtract 3 from 5y:
5y - 3
8 times the difference means we multiply (5y - 3) by 8:
[B]8(5y - 3)[/B]

8 times the sum of 5 times a number and 9

8 times the sum of 5 times a number and 9
Take this algebraic expression in parts:
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
5 times a number means:
5x
The sum of this and 9 means we add 9 to 5x:
5x + 9
Now we multiply 8 times this sum:
[B]8(5x + 9)[/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 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 thrice x

9 less than thrice x
Thrice x means multiply x by 3
3x
9 less than that means subtract 9
[B]3x - 9[/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]

a / dc = b for a

a / dc = b for a
Cross multiply:
a = [B]bcd[/B]

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an ho

A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey?
[U]Set up the relationship of still water speed and downstream speed[/U]
Speed down stream = Speed in still water + speed of the current
Speed down stream = x+2
Therefore:
Speed upstream =x - 2
Since distance = rate * time, we rearrange to get time = Distance/rate:
15/(x+ 2) + 15 /(x- 2) = 3
Multiply each side by 1/3 and we get:
5/(x + 2) + 5/(x - 2) = 1
Using a common denominator of (x + 2)(x - 2), we get:
5(x - 2)/(x + 2)(x - 2) + 5(x + 2)/(x + 2)(x - 2)
(5x - 10 + 5x + 10)/5(x - 2)/(x + 2)(x - 2)
10x = (x+2)(x-2)
We multiply through on the right side to get:
10x = x^2 - 4
Subtract 10x from each side:
x^2 - 10x - 4 = 0
This is a quadratic equation. To solve it, [URL='https://www.mathcelebrity.com/quadratic.php?num=x%5E2-10x-4%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type it in our search engine[/URL] and we get:
Speed of the boat in still water =X=5 +- sq. Root of 29 kmph
We only want the positive solution:
x = 5 + sqrt(29)
x = 10.38
[U]Calculate time for upstream journey:[/U]
Time for upstream journey = 15/(10.38 - 2)
Time for upstream journey = 15/(8.38)
Time for upstream journey = [B]1.79[/B]
[U]Calculate time for downstream journey:[/U]
Time for downstream journey = 15/(10.38 + 2)
Time for downstream journey = 15/(12.38)
Time for downstream journey = [B]1.21[/B]

a = v^2/r for r

a = v^2/r for r
Start by cross multiplying to get r out of the denominator:
ar = v^2
Divide each side of the equation by a to isolate r:
ar/a = v^2/a
Cancel the a's on the left side, and we get:
r = [B]v^2/a[/B]

A bag contains 2 red marbles, 3 blue marbles, and 4 green marbles. What is the probability of choosi

A bag contains 2 red marbles, 3 blue marbles, and 4 green marbles. What is the probability of choosing a blue marble, replacing it, drawing a green marble, replacing it, and then drawing a red marble?
Calculate total marbles in the bag:
Total marbles in the bag = Red Marbles + Blue Marbles + Green Marbles
Total marbles in the bag = 2 + 3 + 4
Total marbles in the bag = 9
[U]First choice, blue marble[/U]
P(blue) = Total Blue Marbles / Total Marbles in the bag
P(blue) = 3/9
[URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F9&frac2=3%2F8&pl=Simplify']Using our fraction simplifier[/URL], we see:
P(blue) = 1/3
[U]Second choice, green marble with all the marbles back in the bag after replacement[/U]
P(green) = Total Green Marbles / Total Marbles in the bag
P(green) = 4/9
[U]Third choice, red marble with all the marbles back in the bag after replacement[/U]
P(red) = Total Red Marbles / Total Marbles in the bag
P(red) = 2/9
Since each event is independent, we multiply each probability:
P(blue, green, red) = P(blue) * P(green) * P(red)
P(blue, green, red) = 1/3 * 4/9 * 2/9
P(blue, green, red) = [B]8/243[/B]

A bag contains 3 black, 4 red, 3 yellow, and 2 green marbles. What is the probability of drawing a b

A bag contains 3 black, 4 red, 3 yellow, and 2 green marbles. What is the probability of drawing a black and then a red marble out of the bag without replacing the black marble before drawing the red marble?
The phrase [U][B]without replacement[/B][/U] is a huge clue on this problem.
Take each draw and calculate the probability.
Draw 1: P(Drawing a red)
P(Drawing a red) = Total Red marbles n the jar / Total marbles in the jar
P(Drawing a red) = 4/12
4/12 simplifies to 1/3 using a common factor of 4:
P(Drawing a red) = 1/3
Draw 2: P(Drawing a black)
P(Drawing a black) = Total Black marbles in the jar / Total marbles in the jar
[I]We drew one red marble already. Without replacement means we do not put it back. Therefore, we have 12 - 1 = 11 marbles left in the jar.[/I]
P(Drawing a black) = 3/11
The question asks, what is the the following probability:
P(Drawing a Red, Drawing a Black)
Because each draw is [I][U]independent[/U], [/I]we multiply each draw probability together:
P(Drawing a Red, Black) = P(Drawing a Red) * P(Drawing a Black)
P(Drawing a Red, Black) = 1/3 * 3/11
P(Drawing a Red, Black) = [B]1/11[/B]

A bag contains 6 red balls and 7 green balls. You plan to select 4 balls at random. Determine the pr

A bag contains 6 red balls and 7 green balls. You plan to select 4 balls at random. Determine the probability of selecting 4 green balls.
Assuming draw without replacement of the balls, we have:
[LIST=1]
[*]Selection 1: 7 green out of 13 balls
[*]Selection 2: 6 green out of 12 balls
[*]Selection 3: 5 green out of 11 balls
[*]Selection 4: 4 green out of 10 balls
[/LIST]
Since each draw is independent, we multiply each probability of green:
P(GGGG) = 7/13 * 6/12 * 5/11 * 4/10
P(GGGG) = 840/17,160
P(GGGG) = [B]0.05[/B]

A bag contains 666 red balls, 444 green balls, and 333 blue balls. If we choose a ball, then another

A bag contains 666 red balls, 444 green balls, and 333 blue balls. If we choose a ball, then another ball without putting the first one back in the bag, what is the probability that the first ball will be green and the second will be red?
[U]Calculate total number of balls to start:[/U]
Total Balls = Red Balls + Green Balls + Blue Balls
Total Balls = 666 + 444 + 333
Total Balls = 1,443
[U]Calculate the probability of drawing a green ball on the first pick:[/U]
P(Green) = Green Balls / Total Balls
P(Green) = 444/1443
P(Green) = 0.30769
[U]Calculate the probability of drawing a red ball on the second pick (without replacement):[/U]
Total Balls decrease by 1, since we do not replace. So Total Balls = 1,443 - 1 = 1,442
P(Red) = Red Balls / Total Balls
P(Red) = 666/1442
P(Red) = 0.46186
Now, we want the probability of Green, Red in that order.
Since each event is independent, we multiply the event probabilities
P(Green, Red) = P(Green) * P(Red)
P(Green, Red) = 0.30769 * 0.46186
P(Green, Red) = [B]0.14211[/B]

A bag contains 7 red, 9 white, and 4 blue marbles. Find the probability of picking 3 blue marbles if

A bag contains 7 red, 9 white, and 4 blue marbles. Find the probability of picking 3 blue marbles if each marble is NOT returned to the bag before the next marble is picked.
The problem states we will have no replacement.
[LIST]
[*]First draw probability is 4 blue marbles out of (7 red + 9 white + 4 blue) = 20 marbles (4/20)
[*]Second draw probability is 3 blue marbles out of (7 red + 9 white + 3 blue) = 19 marbles (3/19)
[*]Third draw probability is 2 blue marbles out of (7 red + 9 white + 2 blue) = 18 marbles (2/18)
[/LIST]
Each draw is independent, so we multiply the three draws together:
4/20 * 3/19 * 2/18
24/6840
[B]0.0035[/B]

A ball was dropped from a height of 6 feet and began bouncing. The height of each bounce was three-f

A ball was dropped from a height of 6 feet and began bouncing. The height of each bounce was three-fourths the height of the previous bounce. Find the total vertical distance travelled by the all in ten bounces.
The height of each number bounce (n) is shown as:
h(n) = 6(0.75)^n
We want to find h(10)
h(n) = 6(0.75)^n
Time Height
0 6
1 4.5
2 3.375
3 2.53125
4 1.8984375
5 1.423828125
6 1.067871094
7 0.8009033203
8 0.6006774902
9 0.4505081177
10 0.3378810883
Adding up each bounce from 1-10, we get:
16.98635674
Since vertical distance means both [B]up and down[/B], we multiply this number by 2 to get:
16.98635674 * 2 = 33.97271347
Then we add in the initial bounce of 6 to get:
33.97271347 + 6 = [B]39.97271347 feet[/B]

a baseball player has 9 hits in his first 60 at bats. how many consecutive hits would he need to bri

a baseball player has 9 hits in his first 60 at bats. how many consecutive hits would he need to bring his average up to 0.400?
Let the amount of consecutive hits needed be h. We have:
hits / at bats = Batting Average
Plugging in our numbers, we get:
(9 + h)/60 = 0.400
Cross multiply:
9 + h = 60 * 0.4
9 + h = 24
To solve this equation for h, [URL='https://www.mathcelebrity.com/1unk.php?num=9%2Bh%3D24&pl=Solve']we type it in our search engine[/URL] and we get:
h = [B]15[/B]

A bowl contains 45 oranges. If ? of the oranges are bad; how many are good?

A bowl contains 45 oranges. If ? of the oranges are bad; how many are good?
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=1&frac2=2%2F3&pl=Subtract']fraction operator calculator[/URL], we see that:
1 - 2/3 = 1/3 of the oranges are good.
We want 1/3 of 45. [URL='https://www.mathcelebrity.com/fraction.php?frac1=45&frac2=1/3&pl=Multiply']Typing this expression into our search engine[/URL], we get:
[B]15 good oranges[/B]

A box contains 5 black and 2 white balls. 2 balls are drawn without replacement. Find the probabilit

A box contains 5 black and 2 white balls. 2 balls are drawn without replacement. Find the probability of drawing 2 black balls.
First draw probability of black is:
Total Balls in box = Black balls + white balls
Total Balls in Box = 5 + 2
Total Balls in Box = 7
P(Black) = Black Balls / Total balls in box
P(Black) = 5/7
Second draw probability of black (with no replacement) is:
Total Balls in box = Black balls + white balls
Total Balls in Box = 4 + 2
Total Balls in Box = 6
P(Black) = Black Balls / Total balls in box
P(Black) = 4/6
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F6&frac2=3%2F8&pl=Simplify']fraction simplifier[/URL], we see that 4/6 is:
2/3
Since each event is independent, we can multiply them to find the probability of drawing 2 black balls:
P(Black, Black) = 5/7 * 2/3
[URL='https://www.mathcelebrity.com/fraction.php?frac1=5%2F7&frac2=2%2F3&pl=Multiply']P(Black, Black)[/URL] = 10/21

A box contains 5 plain pencils and 3 pens. A second box contains 2 color pencils and 2 crayons . One

A box contains 5 plain pencils and 3 pens. A second box contains 2 color pencils and 2 crayons. One item from each box is chosen at random. What is the probability that a plain pencil from the first box and a color pencil from the second box are selected
[U]Calculate the probability of a plain pencil in the first box:[/U]
P(plain pencil in the first box) = Total Pencils / Total Objects
P(plain pencil in the first box) = 5 pencils / (5 pencils + 3 pens)
P(plain pencil in the first box) = 5/8
[U]Calculate the probability of a color pencil in the first box:[/U]
P(color in the second box) = Total Pencils / Total Objects
P(color in the second box) = 2 pencils / (2 pencils + 2 crayons)
P(color in the second box) = 2/4
We can simplify this. [URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F4&frac2=3%2F8&pl=Simplify']Type 2/4 into our search engine[/URL] and we get 1/2
Now the problem asks for the probability that a plain pencil from the first box and a color pencil from the second box are selected.
Since each event is independent, we multiply them together to get our answer:
P(plain pencil in the first box, color in the second box) = P(plain pencil in the first box) * P(color in the second box)
P(plain pencil in the first box, color in the second box) = 5/8 * 1/2
P(plain pencil in the first box, color in the second box) = [B]5/16[/B]

A box contains 5 plain pencils and 7 pens. A second box contains 4 color pencils and 4 crayons. One

A box contains 5 plain pencils and 7 pens. A second box contains 4 color pencils and 4 crayons. One item from each box is chosen at random. What is the probability that a plain pencil from the first box and a color pencil from the second box are selected?
Probability of plain pencil from first box:
5/(5 + 7) = 5/12
Probability of color pencil from second box:
4/(4 + 4) = 4/8 = 1/2
Probability of both events together:
Since each event is independent, we multiply probabilities:
5/12 * 1/2 = [B]5/24[/B]

A boy is 10 years older than his brother. In 4 years he will be twice as old as his brother. Find th

A boy is 10 years older than his brother. In 4 years he will be twice as old as his brother. Find the present age of each?
Let the boy's age be b and his brother's age be c. We're given two equations:
[LIST=1]
[*]b = c + 10
[*]b + 4 = 2(c + 4)
[/LIST]
Substitute equation (1) into equation (2):
(c + 10) + 4 = 2(c + 4)
Simplify by multiplying the right side through and grouping like terms:
c + 14 = 2c + 8
[URL='https://www.mathcelebrity.com/1unk.php?num=c%2B14%3D2c%2B8&pl=Solve']Type this equation into our search engine[/URL] and we get:
c = [B]6[/B]
Now plug c = 6 into equation (1):
b = 6 + 10
b = [B]16[/B]

A boy spent one-half of his money for a book and one-third of his money for a pen. The remaining $2.

A boy spent one-half of his money for a book and one-third of his money for a pen. The remaining $2.25 he saved. How much money did he originally have?
Find out what percent of money was spent
Using a common denominator of 6, we have 1/2 + 1/3 = 3/6 + 2/6 = 5/6.
Therefore, 1/6 of his money is left to save. Let the boy's original money be x. We have:
x/6 = 2.25
Cross multiply, we get x = [B]13.50[/B]

A car is bought for $2400 and sold one year later $1440 find the loss as a percentage of the cost pr

A car is bought for $2400 and sold one year later $1440 find the loss as a percentage of the cost price.
(2400 - 1440)/2400
960/2400
0.4
As a percentage, we multiply by 100 to get [B]40%[/B]

A car is purchased for $24,000 . Each year it loses 30% of its value. After how many years will t

A car is purchased for $24,000 . Each year it loses 30% of its value. After how many years will the car be worth $7300 or less? (Use the calculator provided if necessary.) Write the smallest possible whole number answer.
Set up the depreciation equation D(t) where t is the number of years in the life of the car:
D(t) = 24,000/(1.3)^t
The problem asks for D(t)<=7300
24,000/(1.3)^t = 7300
Cross multiply:
7300(1.3)^t = 24,000
Divide each side by 7300
1.3^t = 24000/7300
1.3^t = 3.2877
Take the natural log of both sides:
LN(1.3^t) = LN(3.2877)
Using the natural log identities, we have:
t * LN(1.3) = 1.1902
t * 0.2624 = 1.1902
Divide each side by 0.2624
t = 4.5356
[B]Rounding this up, we have t = 5[/B]

a card is chosen at a random from a deck of 52 cards. it is then replaced and a second card is chose

a card is chosen at a random from a deck of 52 cards. it is then replaced and a second card is chosen. what is the probability of getting a jack and then an eight?
Calculate the probability of drawing a jack from a full deck
There are 4 jacks in a deck of 52 cards
P(J) = 4/52
P(J) = 1/13 <-- We simplify 4/52 by dividing top and bottom of the fraction by 4
Calculate the probability of drawing an eight from a full deck
There are 4 eights in a deck of 52 cards. We[I] replaced[/I] the first card giving us 52 cards to choose from.
P(8) = 4/52
P(8) = 1/13 <-- We simplify 4/52 by dividing top and bottom of the fraction by 4
Since each event is independent, we multiply:
P(J, 8) = P(J) * P(8)
P(J, 8) = 1/13 * 1/13
P(J, 8) = [B]1/169[/B]

A carpet cleaner charges $75 to clean the first 180 sq ft of carpet. There is an additional charge

A carpet cleaner charges $75 to clean the first 180 sq ft of carpet. There is an additional charge of 25¢ per square foot for any footage that exceeds 180 sq ft and $1.30 per step for any carpeting on a staircase. A customers cleaning bill was $253.95. This included the cleaning of a staircase with 14 steps. In addition to the staircase, how many square feet of carpet did the customer have cleaned?
Calculate the cost of the staircase cleaning.
Staircase cost = $1.30 * steps
Staircase cost = $1.30 * 14
Staircase cost = $18.20
Subtract this from the cost of the total cleaning bill of $253.95. We do this to isolate the cost of the carpet.
Carpet cost = $253.95 - $18.20
Carpet cost = $235.75
Now, the remaining carpet cost can be written as:
75 + $0.25(s - 180) = $235.75 <-- were s is the total square foot of carpet cleaned
Multiply through and simplify:
75 + 0.25s - 45 = $235.75
Combine like terms:
0.25s + 30 = 235.75
[URL='https://www.mathcelebrity.com/1unk.php?num=0.25s%2B30%3D235.75&pl=Solve']Type this equation into our search engine[/URL] to solve for s, and we get:
s = [B]823[/B]

A cashier has 44 bills, all of which are $10 or $20 bills. The total value of the money is $730. How

A cashier has 44 bills, all of which are $10 or $20 bills. The total value of the money is $730. How many of each type of bill does the cashier have?
Let a be the amount of $10 bills and b be the amount of $20 bills. We're given two equations:
[LIST=1]
[*]a + b = 44
[*]10a + 20b = 730
[/LIST]
We rearrange equation 1 in terms of a. We subtract b from each side and we get:
[LIST=1]
[*]a = 44 - b
[*]10a + 20b = 730
[/LIST]
Now we substitute equation (1) for a into equation (2):
10(44 - b) + 20b = 730
Multiply through to remove the parentheses:
440 - 10b + 20b = 730
Group like terms:
440 + 10b = 730
Now, to solve for b, we [URL='https://www.mathcelebrity.com/1unk.php?num=440%2B10b%3D730&pl=Solve']type this equation into our search engine[/URL] and we get:
b = [B]29
[/B]
To get a, we take b = 29 and substitute it into equation (1) above:
a = 44 - 29
a = [B]15
[/B]
So we have [B]15 ten-dollar bills[/B] and [B]29 twenty-dollar bills[/B]

A catering service offers 3 appetizers, 6 main courses, and 4 desserts. A customer is to select 2 ap

A catering service offers 3 appetizers, 6 main courses, and 4 desserts. A customer is to select 2 appetizers, 3 main courses, and 3 desserts for a banquet. In how many ways can this be done?
We use the combinations formula, and since each event is independent of the others, we multiply:
2 appetizers, 3 main courses, and 3 desserts = [URL='https://www.mathcelebrity.com/permutation.php?num=3&den=2&pl=Combinations']3C2[/URL] * [URL='https://www.mathcelebrity.com/permutation.php?num=6&den=3&pl=Combinations']6C3[/URL] * [URL='https://www.mathcelebrity.com/permutation.php?num=4&den=3&pl=Combinations']4C3[/URL]
2 appetizers, 3 main courses, and 3 desserts = 3 * 20 * 4
2 appetizers, 3 main courses, and 3 desserts = [B]240[/B]

A catering service offers 4 appetizers, 12 main courses, and 9 desserts. A customer is to select 3 a

A catering service offers 4 appetizers, 12 main courses, and 9 desserts. A customer is to select 3 appetizers, 10 main courses, and 5 desserts for a banquet. In how many ways can this be done?
We use combinations, so we have:
[LIST]
[*][URL='https://www.mathcelebrity.com/permutation.php?num=4&den=3&pl=Combinations']4C3 appetizers[/URL] = 4
[*][URL='https://www.mathcelebrity.com/permutation.php?num=12&den=10&pl=Combinations']12C10 main courses[/URL] = 66
[*][URL='https://www.mathcelebrity.com/permutation.php?num=9&den=5&pl=Combinations']9C5 desserts[/URL] = 126
[/LIST]
We multiply each of these together to get our total combinations:
4 * 66 * 126 = [B]33,264[/B]

A classroom fish tank contains x goldfish. The tank contains 4 times as many guppies as goldfish. En

A classroom fish tank contains x goldfish. The tank contains 4 times as many guppies as goldfish. Enter an equation that represents the total number of guppies, y, in the fish tank.
The phrase [I]4 times as many[/I] means we multiply the goldfish (x) by 4 to get the number of guppies (y):
[B]y = 4x[/B]

A coin is tossed and a die is rolled. Find the probability pf getting a head and a number greater th

A coin is tossed and a die is rolled. Find the probability pf getting a head and a number greater than 4.
Since each event is independent, we multiply the probabilities of each event.
P(H) = 0.5 or 1/2
P(Dice > 4) = P(5) + P(6) = 1/6 + 1/6 = 2/6 = 1/3
P(H) AND P(Dice > 4) = 1/2 * 1/3 = [B]1/6[/B]

A collection of nickels and dime has a total value of $8.50. How many coins are there if there are 3

A collection of nickels and dime has a total value of $8.50. How many coins are there if there are 3 times as many nickels as dimes.
Let n be the number of nickels. Let d be the number of dimes. We're give two equations:
[LIST=1]
[*]n = 3d
[*]0.1d + 0.05n = 8.50
[/LIST]
Plug equation (1) into equation (2) for n:
0.1d + 0.05(3d) = 8.50
Multiply through:
0.1d + 0.15d = 8.50
[URL='https://www.mathcelebrity.com/1unk.php?num=0.1d%2B0.15d%3D8.50&pl=Solve']Type this equation into our search engine[/URL] and we get:
[B]d = 34[/B]
Now, we take d = 34, and plug it back into equation (1) to solve for n:
n = 3(34)
[B]n = 102[/B]

A committee consisting of 4 faculty members and 5 students is to be formed. Every committee position

A committee consisting of 4 faculty members and 5 students is to be formed. Every committee position has the same duties and voting rights. There are 12 faculty members and 15 students eligible to serve on the committee. In how many ways can the committee be formed?
We'll use combinations, so we have:
[LIST]
[*][URL='https://www.mathcelebrity.com/permutation.php?num=12&den=4&pl=Combinations']12 faculty members choose 4 faculty members --> 12 C 4[/URL] = 495
[*][URL='https://www.mathcelebrity.com/permutation.php?num=15&den=5&pl=Combinations']15 students choose 5 students --> 15 C 5[/URL] = 3,003
[/LIST]
To get the total committees, we multiply the total faculty member choices by the total student choices:
Total committees = total faculty members * total students
Total committees = 495 * 3,003
Total committees = [B]1,486,485[/B]

A committee of 6 students are being selected from a class of 10 girls and 8 boys. How many committee

A committee of 6 students are being selected from a class of 10 girls and 8 boys. How many committees are possible if three must be girls and 3 must be boys?
We want combinations. How many ways can we choose 3 boys from 8 boys:
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=3&pl=Combinations']8 choose 3[/URL] = 56
We want combinations. How many ways can we choose 3 girls from 10 girls:
[URL='https://www.mathcelebrity.com/permutation.php?num=10&den=3&pl=Combinations']10 choose 3[/URL] = 120
Our total choices are found by multiplying each event:
Total committees = (8 boys choose 3) * (10 girls choose 3)
Total committees = 56 * 120
Total committees = [B]6,720[/B]

A computer was on sale. The original cost of the computer was $900. It’s on sale for 5/6 the price.

A computer was on sale. The original cost of the computer was $900. It’s on sale for 5/6 the price. How much is the computer now?
We want 5/6 of 900. We [URL='https://www.mathcelebrity.com/fraction.php?frac1=900&frac2=5/6&pl=Multiply']type this in our search engine[/URL] and we get:
[B]750[/B]

a confidence interval for a population mean has a margin of error of 0.081

Margin of error = Interval Size/2
0.081 = Interval Size/2
Cross Multiply:
Interval Size = 0.081 * 2
Interval Size = [B]0.162[/B]

A construction company can remove 2/3 tons of dirt from a construction site each hour. How long wil

A construction company can remove 2/3 tons of dirt from a construction site each hour. How long will it take them to remove 30 tons of dirt from the site?
Let h be the number of hours. We have the following equation:
2/3h = 30
Multiply each side by 3:
2(3)h/3 = 30 * 3
Cancel the 3 on the left side:
2h = 90
[URL='https://www.mathcelebrity.com/1unk.php?num=2h%3D90&pl=Solve']Type 2h = 90 into the search engine[/URL], we get [B]h = 45[/B].

A construction company needs to remove 24 tons of dirt from a construction site. They can remove 3/8

A construction company needs to remove 24 tons of dirt from a construction site. They can remove 3/8 ton s of dirt each hour. How long will I it take to remove the dirt?
Let h be the number of hours it takes, we have:
3/8h = 24
Multiply each side by 8/3
h = 24(8)/3
24/3 = 8, so we have:
h = 8(8)
h = [B]64 hours[/B]

A cookie recipe uses 10 times as much flour as sugar. If the total amount of these ingredients is 8

A cookie recipe uses 10 times as much flour as sugar. If the total amount of these ingredients is 8 1/4 cups, how much flour and how much sugar would it be?
Let f be the number of cups of sugar. And let f be the number of cups of flour. We're given two equations:
[LIST=1]
[*]f = 10s
[*]s + f = 8 & 1/4
[/LIST]
Substitute (1) into (2):
s + 10s = 8 & 1/4
11fs= 33/4 <-- 8 & 1/4 = 33/4
Cross multiply:
44s = 33
Divide each side by 44:
s= 33/44
Divide top and bottom by 11 and we get s [B]= 3/4 or 0.75[/B]
Now substitute this into (1):
f = 10(33/44)
[B]f = 330/44 or 7 & 22/44 or 7.5[/B]

A cubicle is 6 1?2 feet by 8 3?4 feet. What is the area of the cubicle?

A cubicle is 6 1?2 feet by 8 3?4 feet. What is the area of the cubicle?
Area of a cube is length times width:
A = 8 & 3/4 * 6 & 1/2
We need to convert these to improper fractions.
[LIST]
[*]8 & 3/4 [URL='https://www.mathcelebrity.com/fraction.php?frac1=8%263%2F4&frac2=3%2F8&pl=Simplify']converted to an improper fraction[/URL] is 35/4
[*]6 & 1/2 [URL='https://www.mathcelebrity.com/fraction.php?frac1=6%261%2F2&frac2=3%2F8&pl=Simplify']converted to an improper fraction[/URL] is 13/2
[/LIST]
Multiply the improper fractions together:
A = 35/4 * 13/2
[URL='https://www.mathcelebrity.com/fraction.php?frac1=35%2F4&frac2=13%2F2&pl=Multiply']Using our fraction multiplier[/URL], we get:
[B]455/8 sq ft[/B]
If you want to convert this to a mixed fraction, we [URL='https://www.mathcelebrity.com/fraction.php?frac1=455%2F8&frac2=3%2F8&pl=Simplify']type this in our calculator [/URL]and get:
[B]56 & 7/8 sq ft[/B]

A culture of bacteria doubles every hour. If there are 500 bacteria at the beginning, how many bacte

A culture of bacteria doubles every hour. If there are 500 bacteria at the beginning, how many bacteria will there be after 9 hours?
Assumptions and givens;
[LIST]
[*]h is the number of hours.
[*]B(h) is the number of bacteria at time h
[*]B(0) is the starting bacteria amount
[*]Doubling means multiplying by 2, so we have:
[/LIST]
B(h) = B(0) * 2^h
We want h = 9, so we have:
B(9) = 500 * 2^9
B(9) = 500 * 512
B(9) = [B]256,000[/B]

A dress is on sale for $33. This is 3/5 of the regular price. What is the regular price?

A dress is on sale for $33. This is 3/5 of the regular price. What is the regular price?
Original price is p. We have:
3p/5 = 33
Cross multiply using our [URL='http://www.mathcelebrity.com/prop.php?num1=3p&num2=33&den1=5&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']proportion calculator[/URL], we get [B]p = 55[/B].

A family is renting an apartment. For 2007, the rent is $1376 per month. The monthly rent in 2007

A family is renting an apartment. For 2007, the rent is $1376 per month. The monthly rent in 2007 is 7.5% higher than the monthly rent in 2006. Determine the monthly rent in 2006.
7.5% as a decimal is 0.075
To increase a value by 7.5%, we multiply by 1.075
[U]Calculate Rent Increase[/U]
R(2007) = R(2006) * 1.075
R(2007) = 1376 * 1.075
R(2007) = [B]1,479.20[/B]

a family went to a baseball game. the cost to park the car was $5 AND THE COST PER TICKET WAS $21. W

a family went to a baseball game. the cost to park the car was $5 AND THE COST PER TICKET WAS $21. WRITE A LINEAR FUNCTION IN THE FORM Y=MX+B, FOR THE TOTAL COST OF GOING TO THE BASEBALL GAME,Y, AND THE TOTAL NUMBER PEOPLE IN THE FAMILY,X.
We have:
[B]y = 21x + 5[/B]
Since the cost of each ticket is $21, we multiply this by x, the total number of people in the family.
We add 5 as the cost to park the car, which fits the entire family, and is a one time cost.

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 22. Twice the first number plus the second totals 28. F

A first number plus twice a second number is 22. Twice the first number plus the second totals 28. Find the numbers.
Let the first number be x. Let the second number be y. We're given two equations:
[LIST=1]
[*]x + 2y = 22 <-- Since twice means multiply by 2
[*]2x + y = 28 <-- Since twice means multiply by 2
[/LIST]
We have a set of simultaneous equations. We can solve this three ways
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+22&term2=2x+%2B+y+%3D+28+&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+22&term2=2x+%2B+y+%3D+28&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+22&term2=2x+%2B+y+%3D+28&pl=Cramers+Method']Cramers Rule[/URL]
[/LIST]
No matter which method we use, we get the same answer:
[LIST]
[*][B]x = 11 & 1/3[/B]
[*][B]y = 5 & 1/3[/B]
[/LIST]

A first number plus twice a second number is 3. Twice the first number plus the second totals 24.

A first number plus twice a second number is 3. Twice the first number plus the second totals 24.
Let the first number be x. Let the second number be y. We're given:
[LIST=1]
[*]x + 2y = 3 <-- Because [I]twice[/I] means multiply by 2
[*]2x + y = 24 <-- Because [I]twice[/I] means multiply by 2
[/LIST]
We have a system of equations. We can solve it any one of three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+3&term2=2x+%2B+y+%3D+24&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+3&term2=2x+%2B+y+%3D+24&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=x+%2B+2y+%3D+3&term2=2x+%2B+y+%3D+24&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which way we choose, we get:
[LIST]
[*]x = [B]15[/B]
[*]y = [B]-6[/B]
[/LIST]

A first number plus twice a second number is 6. Twice the first number plus the second totals 15. Fi

A first number plus twice a second number is 6. Twice the first number plus the second totals 15. Find the numbers.
Let the first number be x. Let the second number be y. We're given two equations:
[LIST=1]
[*]x + 2y = 6
[*]2x + y = 15
[/LIST]
Multiply the first equation by -2:
[LIST=1]
[*]-2x - 4y = -12
[*]2x + y = 15
[/LIST]
Now add them
-2x + 2x - 4y + y = -12 + 15
-3y = 3
Divide each side by -3:
y = 3/-3
y =[B] -1[/B]
Plug this back into equation 1:
x + 2(-1) = 6
x - 2 = 6
To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=x-2%3D6&pl=Solve']type this equation into our search engine[/URL] and we get:
x = [B]8[/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 first number plus twice a second number is 7. Twice the first number plus the second totals 23. Fi

A first number plus twice a second number is 7. Twice the first number plus the second totals 23. Find the numbers
Let the first number be a and the second number be b. We have:
[LIST=1]
[*]a + 2b = 7
[*]2a + b = 23
[/LIST]
Rearrange (1) into (3)
(3) a = 7 - 2b
Substitute (3) into (2):
2(7 - 2b) + b = 23
Multiply through:
14 - 4b + b = 23
Combine like terms:
14 - 3b = 23
Subtract 14 from each side:
-3b = 9
Divide each side by -3
[B]b = -3[/B]
Substitute this into (3)
a = 7 - 2b
a = 7 - 2(-3)
a = 7 + 6
[B]a = 13[/B]
[B](a, b) = (13, -3)[/B]

A food truck sells salads for $6.50 each and drinks for $2.00 each. The food trucks revenue from sel

A food truck sells salads for $6.50 each and drinks for $2.00 each. The food trucks revenue from selling a total of 209 salads and drinks in one day was $836.50. How many salads were sold that day?
Let the number of drinks be d. Let the number of salads be s. We're given two equations:
[LIST=1]
[*]2d + 6.50s = 836.50
[*]d + s = 209
[/LIST]
We can use substitution to solve this system of equations quickly. The question asks for the number of salads (s). Therefore, we want all expressions in terms of s. Rearrange Equation 2 by subtracting s from both sides:
d + s - s = 209 - s
Cancel the s's, we get:
d = 209 - s
So we have the following system of equations:
[LIST=1]
[*]2d + 6.50s = 836.50
[*]d = 209 - s
[/LIST]
Substitute equation (2) into equation (1) for d:
2(209 - s) + 6.50s = 836.50
Multiply through to remove the parentheses:
418 - 2s + 6.50s = 836.50
To solve this equation for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=418-2s%2B6.50s%3D836.50&pl=Solve']type it into our search engine and we get[/URL]:
s = [B]93[/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 gasoline tank is leaking at a rate of n gallons in t hours. If the gasoline cost $2 per gallon, wh

A gasoline tank is leaking at a rate of n gallons in t hours. If the gasoline cost $2 per gallon, what is the value of the gasoline that will be lost in m minutes?
n gallons / t hours = n/t gallons per hour are leaking
The value of the gas that leaks each hour is $2, so we have:
2n/t dollar per hour is leaking
Value per minute means we divide by 60:
2n/60t
Dividing top and bottom by 2 to simplify, we have:
n/30t
Given m minutes, we multiply to get:
[B]nm/30t[/B]

a group of students and teachers are going on a field trip. one ninth of the group will fit on 1/3 o

a group of students and teachers are going on a field trip. one ninth of the group will fit on 1/3 of a school bus how many buses are needed to transport the entire group
1/9g = 1/3b
We want to find g, so we multiply each side through by 9
g = 9/3b
Simplify:
g = 3b, so we need [B]3 buses[/B]

A is thrice as much as B

A is thrice as much as B
Thrice means multiply by 3, so we have
[B]A = 3B[/B]

A jar contains 7 red marbles, 8 green marbles, and 6 blue marbles. What is the probability that you

A jar contains 7 red marbles, 8 green marbles, and 6 blue marbles. What is the probability that you draw 4 green marbles in a row if you do not replace the marbles after each draw?
The key phrase in this problem is [I]do not replace[/I].
[U]Draw #1:[/U]
P(Green) = Total Green Marbles in the Jar / Total Marbles in the Jar
Total Green Marbles in the Jar = 8
Total Marbles in the Jar = 7 red + 8 green + 6 blue = 21
P(Green) = 8/21
[U]Draw #2:[/U]
P(Green) = Total Green Marbles in the Jar / Total Marbles in the Jar
Total Green Marbles in the Jar = 8 - 1 = 7
Total Marbles in the Jar = 7 red + 7 green + 6 blue = 20
P(Green) = 7/20
[U]Draw #3:[/U]
P(Green) = Total Green Marbles in the Jar / Total Marbles in the Jar
Total Green Marbles in the Jar = 7 - 1 = 6
Total Marbles in the Jar = 7 red + 6 green + 6 blue = 19
P(Green) = 6/19
[U]Draw #4:[/U]
P(Green) = Total Green Marbles in the Jar / Total Marbles in the Jar
Total Green Marbles in the Jar = 6 - 1 = 5
Total Marbles in the Jar = 7 red + 5 green + 6 blue = 18
P(Green) = 5/18
We want P(Green, Green, Green, Green)
Because each draw is [U][B]independent[/B][/U] of all other draws, we multiply each draw to get the final probability
P(Green, Green, Green, Green) = P(Green on Draw 1) * P(Green on Draw 2) * P(Green on Draw 3) * P(Green on Draw 4) *
P(Green, Green, Green, Green) = 8/21 * 7/20 * 6/19 * 5/18
P(Green, Green, Green, Green) = 1680/143640
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=1680%2F143640&frac2=3%2F8&pl=Simplify']fraction simplifier[/URL], we get:
P(Green, Green, Green, Green) = [B]2/171
[MEDIA=youtube]b2C_D4_d0Ug[/MEDIA][/B]

a jar contains a $5 note, two $10 notes, a $20 note and a $50 note. if 2 notes are taken out by rand

a jar contains a $5 note, two $10 notes, a $20 note and a $50 note. if 2 notes are taken out by random, find the probability that their sum is $15
To get a sum of $15, we'd need to pull the $5 and the $10.
Since both events are indepdenent, we have:
P($5 or 10) or P(whatever is not pulled in the first pull)
First Pull: 2/4 (We can pull either a $10 or a $5, so 2 choices out of 4 bills)
Second Pull: 1/3 <-- since there are only 3 bills and 1 bill to pull
Each pull is independent, so we multiply:
2/4 * 1/3 = 2/12
We can simply this, so [URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F12&frac2=3%2F8&pl=Simplify']we type this fraction in our search engine[/URL] and we get:
[B]1/6[/B]

A kilogram of rice costs $2.05.what is the cost of 30kg of the rice?

A kilogram of rice costs $2.05.what is the cost of 30kg of the rice?
We multiply the price for 1 kilogram by the 30 kilograms total:
$2.05 * 30 = [B]$61.50[/B]

a licence plate that has 3 numbers from 0 to 9 followed by 2 letters

a licence plate that has 3 numbers from 0 to 9 followed by 2 letters
How many license plate combinations can we form?
We multiply as follows:
[LIST]
[*][0-9] = 10 possible digits (D)
[*]A-Z = 26 possible letters (L)
[/LIST]
The problem asks for this:
DDDLL
So we have:
10 * 10 * 10 * 26 * 26 = [B]676,000[/B] plates

A machine shop employee earned $642 last week. She worked 40hours at her regular rate and 9 hours at

A machine shop employee earned $642 last week. She worked 40hours at her regular rate and 9 hours at a time and a half rate. Find her regular hourly rate.
Let the regular hourly rate be h. We're given:
40h + 40(1.5)(h - 40) = 642
Multiply through and simplify:
40h + 60h - 2400 = 642
100h - 2400 = 642
[URL='https://www.mathcelebrity.com/1unk.php?num=100h-2400%3D642&pl=Solve']To solve for h, we type this equation into our search engine[/URL] and we get:
h = [B]30.42[/B]

A medium orange has 70 calories. This is 10 calories less then 1/4 of the calories in a sugar krunch

A medium orange has 70 calories. This is 10 calories less then 1/4 of the calories in a sugar krunchy. How many calories are in a sugar crunchy?
Let s = calories in a sugar crunch. Let o = 70 be the calories in a medium orange. Set up the equation:
o = 1/4s - 10
70 = 1/4s - 10
Add 10 to each side
1/4s = 80
Multiply each side by 4
[B]s = 320[/B]

A number cube is rolled and a coin is tossed. The number cube and the coin are fair. What is the pro

A number cube is rolled and a coin is tossed. The number cube and the coin are fair. What is the probability that the number rolled is greater than 3 and the coin toss is heads? Write your answer as a fraction in simplest form
Let's review the vitals of this question:
[LIST]
[*]The probability of heads on a fair coin is 1/2.
[*]On a fair die, greater than 3 means either 4, 5, or 6. Any die roll face is a 1/6 probability.
[*]So we have a combination of outcomes below:
[/LIST]
Outcomes
[LIST=1]
[*]Heads and 4
[*]Heads and 5
[*]Heads and 6
[/LIST]
For each of the outcomes, we assign a probability. Since the coin flip and die roll are independent, we multiply the probabilities:
[LIST=1]
[*]P(Heads and 4) = 1/2 * 1/6 = 1/12
[*]P(Heads and 5) = 1/2 * 1/6 = 1/12
[*]P(Heads and 6) = 1/2 * 1/6 = 1/12
[/LIST]
Since we want any of those events, we add all three probabilities
1/12 + 1/12 + 1/12 = 3/12
This fraction is not simplified. S[URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F12&frac2=3%2F8&pl=Simplify']o we type this fraction into our search engine, and choose Simplify[/URL].
We get a probability of [B]1/4[/B].
By the way, if you need a decimal answer or percentage answer instead of a fraction, we type in the following phrase into our search engine:
[URL='https://www.mathcelebrity.com/perc.php?num=1&den=4&pcheck=1&num1=+16&pct1=+80&pct2=+35&den1=+90&pct=+82&decimal=+65.236&astart=+12&aend=+20&wp1=20&wp2=30&pl=Calculate']1/4 to decimal[/URL]
Alternative Answers:
[LIST]
[*]For a decimal, we get [B]0.25[/B]
[*]For a percentage, we get [B]25%[/B]
[/LIST]

a number increased by 8 and then tripled

a number increased by 8 and then tripled
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Increased by 8 means we add 8 to x:
x + 8
Then tripled means we multiply the expression x + 8 by 3:
[B]3(x + 8)[/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 K is doubled and then increased by 3

A number K is doubled and then increased by 3
K is doubled means we multiply K by 2:
2K
Increased by 3 means we add:
[B]2K + 3[/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 of bacteria b tripled

a number of bacteria b tripled
The word [I]tripled[/I] means we multiply by 3, so we have:
[B]3b[/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 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 package of soccer accessories costs $25 for cleats, $14 for shin guards , and $12 for a ball. Writ

a package of soccer accessories costs $25 for cleats, $14 for shin guards , and $12 for a ball. Write two equivalent expressions for the total cost of 9 accessory package. Then find the cost.
Let c be the number of cleats, s be the number of shin guards, and b be the number of balls. We have the following cost function for 9 accessory packages:
[B]9(25c + 14s + 12b)[/B]
But if we multiply through, we get an equivalent expression:
[B]225c + 126s + 108b[/B]

A person has $13,000 invested in stock A and stock B. Stock A currently sells for $20 a share and

A person has $13,000 invested in stock A and stock B. Stock A currently sells for $20 a share and stock B sells for $90 a share. If stock B triples in value and stock A goes up 50%, his stock will be worth $33,000. How many shares of each stock does he own?
Set up the given equations, where A is the number of shares for Stock A, and B is the number of shares for Stock B
[LIST=1]
[*]90A + 20B = 13000
[*]3(90A) + 1.5(20B) = 33000 <-- [I]Triple means multiply by 3, and 50% gain means multiply by 1.5[/I]
[/LIST]
Rewrite (2) by multiplying through:
270A + 30B = 33000
Using our simultaneous equations calculator, we get [B]A = 100 and B = 200[/B]. Click the links below to solve using each method:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=90A+%2B+20B+%3D+13000&term2=270A+%2B+30B+%3D+33000&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=90A+%2B+20B+%3D+13000&term2=270A+%2B+30B+%3D+33000&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=90A+%2B+20B+%3D+13000&term2=270A+%2B+30B+%3D+33000&pl=Cramers+Method']Cramers Method[/URL]
[/LIST]
Check our work using equation (1)
90(100) + 20(200) ? 13,000
9000 + 4000 ? 13,000
13000 = 13000

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 Petri dish contains 2000. The number of bacteria triples every 6 hours. How many bacteria will exi

A Petri dish contains 2000. The number of bacteria triples every 6 hours. How many bacteria will exist after 3 days?
Determine the amount of tripling periods:
[LIST]
[*]There are 24 hours in a day.
[*]24 hours in a day * 3 days = 72 hours
[*]72 hours / 6 hours tripling period = 12 tripling periods
[/LIST]
Our bacteria population function B(t) where t is the amount of tripling periods. Tripling means we multiply by 3, so we have:
B(t) = 2000 * 3^t
with t = 12 tripling periods, we have:
B(12) = 2000 * 3^12
B(12) = 2000 * 531441
B(12) = [B]1,062,882,000[/B]

A phone company charges a $30 usage fee $15 per 1GB of data. Write an expression that describes the

A phone company charges a $30 usage fee $15 per 1GB of data. Write an expression that describes the monthly charge and use d to represent data
We multiply gigabyte fee by d and add the usage fee:
[B]15d + 30[/B]

A piece of gym equipment which cost 450 including vat last year is now selling at 500 excluding vat.

A piece of gym equipment which cost 450 including vat last year is now selling at 500 excluding vat. Calculate the percentage increase.
Increase = (New Price - Old Price)/Old Price
Increase = (500-450)/450
50/450 = 0.1111
To get the percentage, multiply by 100
[B]11.11%[/B]

A quarter of the learners in a class have blond hair and two thirds have brown hair. The rest of the

A quarter of the learners in a class have blond hair and two thirds have brown hair. The rest of the learners in the class have black hair. How many learners in the class if 9 of them have blonde hair?
Total learners = Blond + Brown + Black
Total Learners = 1/4 + 2/3 + Black
Total Learners will be 1, the sum of all fractions
1/4 + 2/3 + Black = 1
Using common denominators of 12, we have:
3/12 + 8/12 + Black = 12/12
11/12 + Black = 12/12
Subtract 11/12 from each side:
Black = 1/12
Let t be the total number of people in class. We are given for blondes:
1/4t = 9
Multiply each side by 4
[B]t = 36[/B]
Brown Hair
2/3(36) = 24
Black Hair
1/12(36) = 3

A rational number is such that when you multiply it by 7/3 and subtract 3/2 from the product, you ge

A rational number is such that when you multiply it by 7/3 and subtract 3/2 from the product, you get 92. What is the number?
Let the rational number be x. We're given:
7x/3 - 3/2 = 92
Using a common denominator of 3*2 = 6, we rewrite this as:
14x/6 - 9/6 = 92
(14x - 9)/6 = 92
Cross multiply:
14x - 9 = 92 * 6
14x - 9 = 552
To solve for x, we [URL='https://www.mathcelebrity.com/1unk.php?num=14x-9%3D552&pl=Solve']type this equation into our search engine [/URL]and we get:
x = [B]40.07[/B]

a recipe of 20 bread rolls requires 5 tablespoons of butter. How many tablespoons of butter are need

a recipe of 20 bread rolls requires 5 tablespoons of butter. How many tablespoons of butter are needed for 30 bread rolls?
Set up a proportion of bread rolls per tablespoons of butter where t is the amount of tablespoons of butter needed for 30 bread rolls:
20/5 = 30/t
Cross multiply our proportion:
Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2
20t = 30 * 5
20t = 150
Divide each side of the equation by 20:
20t/20 = 150/20
Cancel the 20's on the left side and we get:
t = [B]7.5[/B]

a rectangle has an area of 238 cm 2 and a perimeter of 62 cm. What are its dimensions?

a rectangle has an area of 238 cm 2 and a perimeter of 62 cm. What are its dimensions?
We know the rectangle has the following formulas:
Area = lw
Perimeter = 2l + 2w
Given an area of 238 and a perimeter of 62, we have:
[LIST=1]
[*]lw = 238
[*]2(l + w) = 62
[/LIST]
Divide each side of (1) by w:
l = 238/w
Substitute this into (2):
2(238/w + w) = 62
Divide each side by 2:
238/w + w = 31
Multiply each side by w:
238w/w + w^2 = 31w
Simplify:
238 + w^2 = 31w
Subtract 31w from each side:
w^2 - 31w + 238 = 0
We have a quadratic. So we run this through our [URL='https://www.mathcelebrity.com/quadratic.php?num=w%5E2-31w%2B238%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation calculator[/URL] and we get:
w = (14, 17)
We take the lower amount as our width and the higher amount as our length:
[B]w = 14
l = 17
[/B]
Check our work for Area:
14(17) = 238 <-- Check
Check our work for Perimeter:
2(17 + 14) ? 62
2(31) ? 62
62 = 62 <-- Check

A rectangular field is to be enclosed with 1120 feet of fencing. If the length of the field is 40 fe

A rectangular field is to be enclosed with 1120 feet of fencing. If the length of the field is 40 feet longer than the width, then how wide is the field?
We're given:
[LIST=1]
[*]l = w + 40
[/LIST]
And we know the perimeter of a rectangle is:
P = 2l + 2w
Substitute (1) into this formula as well as the given perimeter of 1120:
2(w + 40) + 2w = 1120
Multiply through and simplify:
2w + 80 + 2w = 1120
Group like terms:
4w + 80 = 1120
[URL='https://www.mathcelebrity.com/1unk.php?num=4w%2B80%3D1120&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]w = 260[/B]

A rectangular room is 3 times as long as it is wide, and its perimeter is 56 meters.

A rectangular room is 3 times as long as it is wide, and its perimeter is 56 meters.
We're given the following:
[LIST]
[*]l = 3w
[/LIST]
We know the Perimeter (P) of a rectangle is:
P = 2l + 2w
Substituting l = 3w and P = 56 into this equation, we get:
2(3w) + 2w = 56
Multiplying through, we get:
6w + 2w = 56
(6 +2)w = 56
8w = 56
[URL='https://www.mathcelebrity.com/1unk.php?num=8w%3D56&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]w = 7[/B]
Substitute w = 7 into l = 3w, we get:
l = 3(7)
[B]l = 21[/B]

A rectangular room is 3 times as long as it is wide, and its perimeter is 64 meters. Find the dimens

A rectangular room is 3 times as long as it is wide, and its perimeter is 64 meters. Find the dimension of the room.
We're given:
[LIST]
[*]l = 3w
[*]P = 64
[/LIST]
We also know the perimeter of a rectangle is:
2l + 2w = P
We plugin l = 3w and P = 64 into the perimeter equation:
2(3w) + 2w = 64
Multiply through to remove the parentheses:
6w + 2w = 64
To solve this equation for w, we type it in our search engine and we get:
[B]w = 8[/B]
To solve for l, we plug w = 8 into the l = 3w equation above:
l = 3(8)
[B]l = 24[/B]

A restaurant chain sells 15,000 pizzas each month, and 70% of the pizzas are topped with pepperoni.

A restaurant chain sells 15,000 pizzas each month, and 70% of the pizzas are topped with pepperoni. Of these, 2/3 also have peppers. How many pizzas have pepperoni and peppers?
We multiply the pizzas sold by the percentage of pepperoni times the fraction of peppers. Since 70% is 7/10, we have:
Pizzas with pepperoni and peppers = 15,000 * 7/10 * 2/3
7/10 * 2/3 = 14/30. [URL='https://www.mathcelebrity.com/fraction.php?frac1=14%2F30&frac2=3%2F8&pl=Simplify']Using our fraction simplifier calculator[/URL], we can reduce this to 7/15
Pizzas with pepperoni and peppers = 15,000 * 7/15
Pizzas with pepperoni and peppers = [B]7,000[/B]

A restaurant earns $1073 on Friday and $1108 on Saturday. Write and solve an equation to find the am

A restaurant earns $1073 on Friday and $1108 on Saturday. Write and solve an equation to find the amount x (in dollars) the restaurant needs to earn on Sunday to average $1000 per day over the three-day period.
Let Sunday's earnings be s. With 3 days, we divide our sum of earnings for 3 days by 3 to get our 1,000 average, so we have:
(1073 + 1108 + s)/3 = 1000
Cross multiply:
1073 + 1108 + s = 1000 * 3
1073 + 1108 + s = 3000
To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=1073%2B1108%2Bs%3D3000&pl=Solve']type this equation into our search engine[/URL] and we get:
s = [B]819[/B]

A restaurant is going to raise all their prices by 7%. If the current price of an item is p dollars

A restaurant is going to raise all their prices by 7%. If the current price of an item is p dollars, write an expression for the price after the increase.
A 7% increase on price means we multiply the current price of p by 1.07. So our algebraic expression is:
[B]1.07p[/B]

A retired couple invested $8000 in bonds. At the end of one year, they received an interest payment

A retired couple invested $8000 in bonds. At the end of one year, they received an interest payment of $584. What was the simple interest rate of the bonds?
For simple interest, we have:
Balance * interest rate = Interest payment
8000i = 584
Divide each side of the equation by 8000 to isolate i:
8000i/8000 = 584/8000
Cancelling the 8000's on the left side, we get:
i = 0.073
Most times, interest rates are expressed as a percentage.
Percentage interest = Decimal interest * 100%
Percentage interest = 0.073 * 100%
Multiplying by 100 is the same as moving the decimal point two places right:
Percentage interest = [B]7.3%[/B]

A salesperson earns a commission of $364 for selling $2600 in merchandise. Find the commission rate.

A salesperson earns a commission of $364 for selling $2600 in merchandise. Find the commission rate. Write your answer as a percentage.
Commission percentage = Commission Amount / Sales Price
Commission percentage = 364 / 2600
Commission percentage = 0.14
Multiply by 100 to get the percentage:
0.14 * 100 = [B]14%[/B]

A set of 19 scores has a mean of 6.3. A new score of 8 is then included in the data set. What is th

A set of 19 scores has a mean of 6.3. A new score of 8 is then included in the data set. What is the new mean?
We know the mean formula is:
Sum of scores / Number of Scores = Mean
We're given mean = 6.3 and number of scores = 19, so we have:
Sum of scores / 19 = 6.3
Cross multiply:
Sum of scores = 19 * 6.3
Sum of scores = 119.7
Now a new score is added of 8, so we have:
Sum of scores = 119.7 + 8 = 127.7
Number of scores = 19 + 1 = 20
So our new mean is:
Mean = Sum of scores / Number of Scores
Mean = 127.7/20
Mean = [B]6.385[/B]
[COLOR=rgb(0, 0, 0)][SIZE=5][FONT=arial][B][/B][/FONT][/SIZE][/COLOR]

a shop has a sale of 1/5 off all items in stock. if the original price of a dress is £45, what would

a shop has a sale of 1/5 off all items in stock. if the original price of a dress is £45, what would be its sale price?
[URL='https://www.mathcelebrity.com/fraction.php?frac1=45&frac2=1/5&pl=Multiply']1/5 of 45[/URL] = 9
45 - 9 = [B]36[/B]

A softball player had 13 hits in 25 times at bat. What percent of her times at bat resulted in hits?

We take the ratio of hits to at bats:
13/25 = 0.52
To get the percent, we multiply by 100:
100 * 0.52 = 52%

a son is 1/4 of his fathers age. the difference in their ages is 30. what is the fathers age.

a son is 1/4 of his fathers age. the difference in their ages is 30. what is the fathers age.
Declare variables:
[LIST]
[*]Let f be the father's age
[*]Let s be the son's age
[/LIST]
We're given two equations:
[LIST=1]
[*]s = f/4
[*]f - s = 30. [I]The reason why we subtract s from f is the father is older[/I]
[/LIST]
Using substitution, we substitute equaiton (1) into equation (2) for s:
f - f/4 = 30
To remove the denominator/fraction, we multiply both sides of the equation by 4:
4f - 4f/4 = 30 *4
4f - f = 120
3f = 120
To solve for f, we divide each side of the equation by 3:
3f/3 = 120/3
Cancel the 3's on the left side and we get:
f = [B]40[/B]

A spherical water tank holds 11,500ft^3 of water. What is the diameter?

A spherical water tank holds 11,500ft^3 of water. What is the diameter?
The tank holding amount is volume. And the volume of a sphere is:
V = (4pir^3)/3
We know that radius is 1/2 of diameter:
r =d/2
So we rewrite our volume function:
V = 4/3(pi(d/2)^3)
We're given V = 11,500 so we have:
4/3(pi(d/2)^3) = 11500
Multiply each side by 3/4
4/3(3/4)(pi(d/2)^3) = 11,500*3/4
Simplify:
pi(d/2)^3 = 8625
Since pi = 3.1415926359, we divide each side by pi:
(d/2)^3 = 8625/3.1415926359
(d/2)^3 = 2745.42
Take the cube root of each side:
d/2 = 14.0224
Multiply through by 2:
[B]d = 28.005[/B]

A spider has 8 legs. How many legs are there for k spiders?

A spider has 8 legs. How many legs are there for k spiders?
We multiply legs per spider * spiders
[B]8k[/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 stone mason builds 7 houses in 3 days. How many days does it take to build 11 houses?

a stone mason builds 7 houses in 3 days. How many days does it take to build 11 houses?
The build rate of houses per days is proportional. Set up a proportion of [I]houses to days[/I] where d is the number of days it takes to build 11 houses:
7/3 = 11/d
Cross multiply:
Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2
7d = 11 * 3
7d = 33
Divide each side of the equation by 7:
7d/7 = 33/7
d = [B]4.7142857142857[/B]

A store sells small notebooks for $6 and large notebooks for $12. If a student buys 6 notebooks and

A store sells small notebooks for $6 and large notebooks for $12. If a student buys 6 notebooks and spends $60, how many of each did he buy?
Let the amount of small notebooks be s. Let the amount of large notebooks be l. We're given two equations:
[LIST=1]
[*]l + s = 6
[*]12l + 6s = 60
[/LIST]
Multiply equation (1) by -6
[LIST=1]
[*]-6l - 6s = -36
[*]12l + 6s = 60
[/LIST]
Now add equation 1 to equation 2:
12l - 6l + 6s - 6s = 60 - 36
Cancel the 6s terms, and we get:
6l = 24
To solve for l, we [URL='https://www.mathcelebrity.com/1unk.php?num=6l%3D24&pl=Solve']type this equation into our search engine[/URL] and we get:
l = [B]4
[/B]
Now substitute this into equation 1:
4 + s = 6
To solve for s, [URL='https://www.mathcelebrity.com/1unk.php?num=4%2Bs%3D6&pl=Solve']we type this equation into our search engine[/URL] and we get:
s = [B]2[/B]

A survey of 2,000 doctors showed that an average of 3 out of 5 doctors use brand A aspirin. How many

A survey of 2,000 doctors showed that an average of 3 out of 5 doctors use brand A aspirin. How many doctors use brand A aspirin?
We want 3/5 of 2000. We [URL='https://www.mathcelebrity.com/fraction.php?frac1=2000&frac2=3/5&pl=Multiply']type this expression into our search engine[/URL] and we get:
[B]1,200[/B]

A sweater costs $40. That is 5 times as much as a shirt. What is the price of the shirt?

A sweater costs $40. That is 5 times as much as a shirt. What is the price of the shirt?
State this as an equation. Let the price of the shirt be s. 5 times as much means we multiply s by 5:
5s = 40
[URL='https://www.mathcelebrity.com/1unk.php?num=5s%3D40&pl=Solve']Type this equation into the search engine[/URL], we get:
s = [B]8[/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 variable tripled less 40

a variable tripled less 40
[I]A variable[/I] means we pick an arbitrary variable, let's call it x
x
Tripled means we multiply by 3
3x
Less 40 means we subtract 40:
[B]3x - 40[/B]

A varies directly as B and inversely as C.

A varies directly as B and inversely as C.
There exists a constant k such that:
[B]a = kb/c
[/B]
Inversely means we divide by and directly means we multiply by

a varies directly with b and inversely with c

a varies directly with b and inversely with c
Direct variation means we multiply.
Inverse variation means we divide.
There exists a constant k such that:
[B]a = kb/c[/B]

A woman walked for 5 hours, first along a level road, then up a hill, and then she turned around and

A woman walked for 5 hours, first along a level road, then up a hill, and then she turned around and walked back to the starting point along the same path. She walks 4mph on level ground, 3 mph uphill, and 6 mph downhill. Find the distance she walked.
Hint: Think about d = rt, which means that t = d/r. Think about each section of her walk, what is the distance and the rate. You know that the total time is 5 hours, so you know the sum of the times from each section must be 5.
Let Level distance = L and hill distance = H. Add the times it took for each section of the walk:
L/4 + H /3 + H/6 + L/4 = 5
The LCD of this is 12 from our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=4&num2=3&num3=6&pl=LCM']LCD Calculator[/URL]
[U]Multiply each side through by our LCD of 12[/U]
3L + 4H + 2H + 3L = 60
[U]Combine like terms:[/U]
6L + 6H = 60
[U]Divide each side by 3:[/U]
2L + 2H = 20
The woman walked [B]20 miles[/B]

a ^5 x a ^2 without exponents

a ^5 x a ^2 without exponents
When we multiply the same variable or number, we add exponents, so we have:
a^(5 + 2)
a^7
To write a variable raised to an exponent without exponents, we break it up. The formula to do this is:
a^n = a times itself n times
a^7 = [B]a * a * a * a * a * a * a[/B]

a/m - b = c for m

a/m - b = c for m
Add b to both sides:
a/m - b + b = c + b
Cancel b on both sides:
a/m = c + b
Multiply each side by m:
am/m = m(c + b)
Cancel the m's on the left side:
a = m(c + b)
Divide each side by (c + b)
a/(c + b) = m(c + b)/(c + b)
Cancel the (c + b) on the right side, and we get:
m[B] = a/(c + b)[/B]

A=2(l+w) for l

Multiply through:
A = 2l + 2w
To solve for l, subtract 2w from each side:
2l = A - 2w
Divide each side by 2
l = (A - 2w)/2

A=2(l+w) for w

Multiply through using the distributive property, so we have:
A = 2l + 2w
Subtract 2l from each side
2w = A - 2l
Divide each side by w
w = (A - 2l)/2

A=a+b+c+d÷4 for c

A=a+b+c+d÷4 for c
Assume A and a are different variables:
Cross multiply:
a + b + c + d = 4A
Subtract a, b, and d from each side:
a + b + c + d - (a + b + d) = 4A - (a + b + d)
Cancel the a + b + d on the left side
[B]c = 4A - a - b - d[/B]

Aaron buys a bag of cookies that contains 8 chocolate chip cookies, 6 peanut butter cookies,7 sugar

Aaron buys a bag of cookies that contains 8 chocolate chip cookies, 6 peanut butter cookies,7 sugar cookies and 6 oatmeal raisin cookies. What it’s the probability that Aaron randomly selects a peanut butter cookie from the bag, eats it,, then randomly selects another peanut butter cookie?
First draw out of the bag is a peanut butter cookie:
P(PB) = Total Peanut Butter Cookies / Total Cookies
P(PB) = 6/27
Second draw out of the bag is a peanut butter cookie, but we have one less since Aaron ate one:
P(PB) = Total Peanut Butter Cookies - 1 / Total Cookies - 1
P(PB) = (6 - 1)/(27 - 1)
P(PB) = 5/26
Now, since each event is independent, we multiply them to see the probability of choosing a peanut butter cookie, eating it, then reaching in and choosing another peanut butter cookie:
P(PB, PB) = 6/27 * 5/26
[URL='https://www.mathcelebrity.com/fraction.php?frac1=6%2F27&frac2=5%2F26&pl=Multiply']P(PB, PB)[/URL] = [B]5/117[/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]

ab/d + c = e for d

ab/d + c = e for d
I know this is a literal equation because we are asked to solve for a variable [U]in terms of[I] another variable
[/I][/U]
Subtract c from each side to isolate the d term:
ab/d + c - c = e - c
Cancel the c's on the left side and we get:
ab/d = e - c
Cross multiply:
ab = d(e - c)
Divide each side of the equation by (e - c):
ab/(e - c)= d(e - c)/(e - c)
Cancel the (e - c) on the right side, and we get:
d = [B]ab/(e - c)[/B]

ab/d+c=e for d

ab/d+c=e for d
Subtract c from each side:
ab/d+c - c = e - c
ab/d = e - c
Multiply each side by d:
abd/d = d(e - c)
ab = d(e - c)
Divide each side by (e - c):
ab/(e - c) = d(e - c)/(e - c)
d =[B] ab/(e - c)[/B]

About 3/5th of the registered voters participated in 2016 election. California has 25 million regist

About 3/5th of the registered voters participated in 2016 election. California has 25 million registered voters. Find the number of registered voters who participated in 2016 election.
3[URL='https://www.mathcelebrity.com/fraction.php?frac1=25000000&frac2=3/5&pl=Multiply']/5 of 25000000[/URL] = [B]15,000,000[/B]

Absolute Value

Free Absolute Value Calculator - Add, subtract, multiply or divide any two numbers with absolute value signs. Positive Difference.

add 3 and 9, then multiply n by the result

add 3 and 9, then multiply n by the result
Add 3 and 9
3 + 9
12
Then multiply n by the result:
[B]12n[/B]

Add 3 to 6, subtract w from the result, then triple what you have

Add 3 to 6, subtract w from the result, then triple what you have
Add 3 to 6;
3 + 6
Subtract w from the result;
3 + 6 - w
Triple what you have (means multiply by 3):
[B]3(3 + 6 - w)[/B]

Add 5 and 6 and then multiply by 3

Add 5 and 6 and then multiply by 3
Add 5 and 6:
(5 + 6)
Then multiply by 3:
[B]3(5 + 6)
[/B]
If you want to evaluate this term, then we [URL='https://www.mathcelebrity.com/distributive-property.php?a=3&b=5&c=6&pl=Distributive']type it into the math engine[/URL] and we get:
[B]33[/B]

Add 8 and 7, and then multiply by 2.

Add 8 and 7, and then multiply by 2.
Add 8 and 7:
8 + 7
Then multiply by 2:
2(8 + 7)
If you want to evaluate this order of operations, then [URL='https://www.mathcelebrity.com/distributive-property.php?a=2&b=8&c=7&pl=Distributive']type it in our search engine[/URL] to get:
[B]30[/B]

add c and b, multiply the result by a, then double what you have

add c and b, multiply the result by a, then double what you have
Take this algebraic expression in pieces:
[LIST]
[*]add c and b: c + b
[*]Multiply the result by a: a(c + b)
[*]Double what you have means take the last step result, and multiply it by 2:
[/LIST]
[B]2a(c + b)[/B]

add c to b, subtract d from the result, then double what you have

add c to b, subtract d from the result, then double what you have
Add c to b:
b + c
Subtract d from the result:
b + c - d
Double what you have means multiply the entire expression by 2:
[B]2(b + c - d)[/B]

add c to d, multiply a by the result, then divide what you have by b

add c to d, multiply a by the result, then divide what you have by b
Add c to d:
d + c
Multiply a by the result:
a(d + c)
then divide what you have by b:
[B]a(d + c)/b[/B]

add c to d, then multiply 9 by the result

add c to d, then multiply 9 by the result
Add c to d
c + d
Multiply 9 by the result:
[B]9(c + d)[/B]

add f and g, then triple the result

add f and g, then triple the result
Add f and g
f + g
Triple the result means we multiply f + g by 3
[B]3(f + g)[/B]

add p to 7, add the result to 10, then multiply 4 by what you have

add p to 7, add the result to 10, then multiply 4 by what you have
Add p to 7:
p + 7
Add the result to 10:
p + 7 + 10
p + 17 <-- combine like terms
Then multiply 4 by what you have:
[B]4(p + 17)[/B]

Add q and t, subtract s from the result, then multiply by r

Add q and t, subtract s from the result, then multiply by r
Take this algebraic expression in parts:
[LIST]
[*]Add q and t: q + t
[*]Subtract s from the result: q + t - s
[*]Multiply by r means we multiply the entire expression by r:
[/LIST]
[B]r(q + t - s)[/B]

add r and q, divide the result by s, then triple what you have

add r and q, divide the result by s, then triple what you have
Add r and q:
r + q
Divide the result by s. The result above is r + q, so we have:
(r + q)/s
Triple what you have means we multiply the expression above by 3:
[B]3(r + q)/s[/B]

add r to 3, triple the result, then divide s by what you have

add r to 3, triple the result, then divide s by what you have
Take this algebraic expression in parts:
[LIST=1]
[*]Add r to 3: 3 + r
[*]Triple the result means multiply the result above by 3: 3(3 + r)
[*]Then divide s by what you have. [B]s/3(3 + r)[/B]
[/LIST]

add s and t, multiply the result by u, then add r to what you have

add s and t, multiply the result by u, then add r to what you have.
Take this algebraic expression in 3 parts:
[LIST=1]
[*]Add s and t: s + t
[*]Multiply the result by u means me multiply (s + t) times u: u(s + t)
[*]Then add r to what you have. [I]what you have means the result in #2.[/I]
[/LIST]
[B]u(s + t) + r[/B]

add s to r, double the result

add s to r, double the result
Add s to r:
r + s
Double the result means multiply r + s by 2:
[B]2(r + s)[/B]

add s to v, multiply the result by u, then multiply t by what you have

add s to v, multiply the result by u, then multiply t by what you have
Take this algebraic expression in parts:
[LIST]
[*]Add s to v: v + s
[*]Multiply the result by u: u(v + s)
[*]Then multiply t by what you have:
[/LIST]
[B]tu(v + s)[/B]

add t and r and double the result

add t and r and double the result
Add t and r:
t + r
Double the result means multiply by 2:
[B]2(t + r)[/B]

add u and t divide s by the result then triple what you have

add u and t divide s by the result then triple what you have
Take this algebraic expression in parts:
[LIST]
[*]Add u and t: u + t
[*]Divide s by the result: s/(u + t)
[*]Triple what you have means we you multiply s/(u + t) by 3
[/LIST]
[B]3s/(u + t)[/B]

Add u and w, triple the result, then add what you have to v

Add u and w, triple the result, then add what you have to v
Add u and w
u + w
Triple the result means multiply the sum of u and w by 3:
3(u + w)
Then add what you have to v:
[B]v + 3(u + w)[/B]

add w to u, triple the result, then add v to what you have

add w to u, triple the result, then add v to what you have
Take this algebraic expression in parts:
[LIST]
[*]add w to u: w + u
[*]triple the result means we multiply w + u by 3: 3(w + u)
[*]Then add v to what you have
[/LIST]
[B]3(w + u) + v[/B]

add x and 3; then multiply by y

Add x and 3
x + 3
Then multiply by y (they mean the total)
[B]y(x + 3)[/B]

After John worked at a job for 10 years, his salary doubled. If he started at $x, his salary after 1

After John worked at a job for 10 years, his salary doubled. If he started at $x, his salary after 10 years is _____.
Doubled means we multiply by 2, so we have a new salary in 10 years of:
[B]2x[/B]

Alberto’s salary was $1500 greater than 5 times Nick’s salary. Write an equation stating Alberto’s a

Alberto’s salary was $1500 greater than 5 times Nick’s salary. Write an equation stating Alberto’s and Nick’s salaries in terms of x and y.
Let x be Alberto's salary. Let y be Nick's salary. We have:
Let's break this down:
[LIST=1]
[*]5 times Nick's salary (y), means we multiply the variable y by 5
[*]$1500 greater means we add $1500 to 5y
[/LIST]
[B]x = 5y - 1500[/B]

algebraic expression for the sum of x and double the value of y

algebraic expression for the sum of x and double the value of y
Double the value of y means we multiply y by 2:
2y
The sum of x and 2y means we add 2y to x:
[B]x + 2y[/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 spent $60 at the grocery store. Of this amount, he spent $51 on fruit. What percentage of the to

Ali spent $60 at the grocery store. Of this amount, he spent $51 on fruit. What percentage of the total did he spend on fruit?
51/60 = 0.85
Multiply 0.85 by 100 to get the percentage
0.85 * 100 = [B]85%[/B]

An angle is 30 degrees less than 5 times it's complement. Find the angle.

An angle is 30 degrees less than 5 times it's complement. Find the angle.
Let the angle be a. The complement of a is 90 - a. We're given the following equation:
a = 5(90 - a) - 30 <-- Less means we subtract
Multiplying though, we get:
a = 450 - 5a - 30
a = 420 - 5a
[URL='https://www.mathcelebrity.com/1unk.php?num=a%3D420-5a&pl=Solve']Typing this equation into our search engine[/URL], we get:
a =[B] 70[/B]

An experienced accountant can balance the books twice as fast as a new accountant. Working together

An experienced accountant can balance the books twice as fast as a new accountant. Working together it takes the accountants 10 hours. How long would it take the experienced accountant working alone?
Person A: x/2 job per hour
Person B: 1/x job per hour
Set up our equation:
1/x + 1/(2x) = 1/10
Multiply the first fraction by 2/2 to get common denominators;
2/(2x) + 1/(2x) = 1/10
Combine like terms
3/2x = 1/10
Cross multiply:
30 = 2x
Divide each side by 2:
[B]x = 15[/B]

An irregular pentagon is a five sided figure. The two longest sides of a pentagon are each three tim

An irregular pentagon is a five sided figure. The two longest sides of a pentagon are each three times as long as the shortest side. The remaining two sides are each 8m longer than the shortest side. The perimeter of the Pentagon is 79m. Find the length of each side of the Pentagon.
Let long sides be l. Let short sides be s. Let medium sides be m. We have 3 equations:
[LIST=1]
[*]2l + 2m + s = 79
[*]m = s + 8
[*]l = 3s
[/LIST]
Substitute (2) and (3) into (1):
2(3s) + 2(s + 8) + s = 79
Multiply through and simplify:
6s + 2s + 16 + s = 79
9s + 16 = 79
[URL='https://www.mathcelebrity.com/1unk.php?num=9s%2B16%3D79&pl=Solve']Using our equation calculator[/URL], we get [B]s = 7[/B].
This means from Equation (2):
m = 7 + 8
[B]m = 15
[/B]
And from equation (3):
l = 3(7)
[B]l = 21[/B]

An isosceles triangles non-congruent angle is 16 more than twice the congruent ones. What is the mea

An isosceles triangles non-congruent angle is 16 more than twice the congruent ones. What is the measure of all 3 angles?
Let the congruent angles measurement be c. And the non-congruent angle measurement be n. We're given:
[LIST=1]
[*]n = 2c + 16 <-- Twice means we multiply by 2, and more than means we add 16
[*]2c + n = 180 <-- Since the sum of angles in an isosceles triangle is 180
[/LIST]
Substitute (1) into (2):
2c + (2c + 16) = 180
Group like terms:
4c + 16 = 180
[URL='https://www.mathcelebrity.com/1unk.php?num=4c%2B16%3D180&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]c = 41[/B]
Substituting this value into Equation 1, we get
n = 2(41) + 16
n = 82 + 16
[B]n = 98[/B]

An item cost $370 before tax, and the sales tax is 25.90 what is the percentage?

An item cost $370 before tax, and the sales tax is 25.90 what is the percentage?
Sales Tax = Tax Amount/Original Bill
Sales Tax = 25.90/370
Sales Tax = 0.07
Multiply by 100 to convert to a percent, we have[B] 7%[/B]

An orchard has 378 orange trees. The number of rows exceeds the number of trees per row by 3. How ma

An orchard has 378 orange trees. The number of rows exceeds the number of trees per row by 3. How many trees are there in each row?
We have r rows and t trees per row. We're give two equations:
[LIST=1]
[*]rt = 378
[*]r = t + 3
[/LIST]
Substitute equation (2) into equation (1) for r:
(t + 3)t = 378
Multiply through:
t^2 + 3t = 378
We have a quadratic equation. To solve this equation, we [URL='https://www.mathcelebrity.com/quadratic.php?num=t%5E2%2B3t%3D378&pl=Solve+Quadratic+Equation&hintnum=+0']type it in our search engine [/URL]and we get:
t = 18 and t = -21
Since t cannot be negative, we get trees per row (t):
[B]t = 18[/B]

Ande has 8 pints of milk. If he drinks 1/4 of a pint of milk each day, how long will the 8 pints of

Ande has 8 pints of milk. If he drinks 1/4 of a pint of milk each day, how long will the 8 pints of milk last him?
Milk Days = Total Pints of Milk / pints drank per day
Milk Days = 8 / 1/4
Dividing by a fraction is the same as multiplying by it's reciprocal.
The [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F4&frac2=3%2F8&pl=Reciprocal']reciprocal of[/URL] 1/4 is 4/1, so we have:
8 * 4/1 = [B]32 days[/B]

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]

Anna made $60 babysitting. She spent 4/5 of the money on new shoes. How much money does she have lef

Anna made $60 babysitting. She spent 4/5 of the money on new shoes. How much money does she have left?
[U]Calculate shoe spend[/U]
[URL='https://www.mathcelebrity.com/fraction.php?frac1=60&frac2=4/5&pl=Multiply']4/5 of 60[/URL] = 48
[U]Calculate leftover money[/U]
Leftover money = Babysitting money - shoe spend
Leftover money = 60 - 48
Leftover money = [B]12[/B]

Anna painted 1/6 of a wall, Eric painted 1/5 of the wall, and Meadow painted 1/4 of the wall. There

Anna painted 1/6 of a wall, Eric painted 1/5 of the wall, and Meadow painted 1/4 of the wall. There are now 3910 square feet left to paint. How many square feet did Anna paint?
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=4&num2=5&num3=6&pl=LCM']Using 60 as a common denominator through least common multiple[/URL], we get:
1/6 = 10/60
1/5 = 12/60
1/4 = 15/60
10/60 + 12/60 + 15/60 = 37/60
Remaining part of the wall is 60/60 - 37[B]/[/B]60 = 23/60
3910/23 = 170 for each 1/60 of a wall
Anna painted 1/6 or 10/60 of the wall. So we multiply 170 * 10 = [B]1,700 square feet[/B]

April, May and June have 90 sweets between them. May has three-quarters of the number of sweets that

April, May and June have 90 sweets between them. May has three-quarters of the number of sweets that June has. April has two-thirds of the number of sweets that May has. How many sweets does June have?
Let the April sweets be a.
Let the May sweets be m.
Let the June sweets be j.
We're given the following equations:
[LIST=1]
[*]m = 3j/4
[*]a = 2m/3
[*]a + j + m = 90
[/LIST]
Cross multiply #2;
3a = 2m
Dividing each side by 2, we get;
m = 3a/2
Since m = 3j/4 from equation #1, we have:
3j/4 = 3a/2
Cross multiply:
6j = 12a
Divide each side by 12:
a = j/2
So we have:
[LIST=1]
[*]m = 3j/4
[*]a = j/2
[*]a + j + m = 90
[/LIST]
Now substitute equation 1 and 2 into equation 3:
j/2 + j + 3j/4 = 90
Multiply each side by 4 to eliminate fractions:
2j + 4j + 3j = 360
To solve this equation for j, we [URL='https://www.mathcelebrity.com/1unk.php?num=2j%2B4j%2B3j%3D360&pl=Solve']type it in our search engine[/URL] and we get:
j = [B]40[/B]

Assuming a standard 52-card deck, what's the probability of dealing three eights in a row when the c

Assuming a standard 52-card deck, what's the probability of dealing three eights in a row when the cards are returned and the deck is shuffled between each draw?
There are four (8's) in a standard 52 card deck. The probability of drawing an 8 is:
4/52
[URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F52&frac2=3%2F8&pl=Simplify']Using our fraction simplifier[/URL], we get:
1/13
Now, with each draw, we replace the deck. So each draw of an 8 has a 1/13 probability. And since each of the three draws is independent, we multiply each probability:
1/13 * 1/13 * 1/13 = [B]1/2197 or 0.00045516613[/B]

At a carnival, the price of an adult ticket is $6 while a child ticket is $4. On a certain day, 30 m

At a carnival, the price of an adult ticket is $6 while a child ticket is $4. On a certain day, 30 more child tickets than adult tickets were sold. If a total of $6360 was collected from the total ticket sale that day, how many child tickets were sold?
Let the number of adult tickets be a. Let the number of child tickets be c. We're given two equations:
[LIST=1]
[*]c = a + 30
[*]6a + 4c = 6360
[/LIST]
Substitute equation (1) into equation (2):
6a + 4(a + 30) = 6360
Multiply through to remove parentheses:
6a + 4a + 120 = 6360
T[URL='https://www.mathcelebrity.com/1unk.php?num=6a%2B4a%2B120%3D6360&pl=Solve']ype this equation into our search engine[/URL] to solve for a and we get:
a = 624
Now substitute a = 624 back into equation (1) to solve for c:
c = 124 + 30
c = [B]154[/B]

at a party, there are 72 people. The ratio of men to ladies to kids is 4 to 3 to 2.

at a party, there are 72 people. The ratio of men to ladies to kids is 4 to 3 to 2.
[LIST]
[*]How many men at the party?
[*]How many ladies at the party?
[*]How many kids at the party?
[/LIST]
Our total ratio denominator is 4 + 3 + 2 = 9. To find the number of each type of person, we take their ratio divided by their ratio numerator times 72 people at the party
[U]Calculate ratios:[/U]
[LIST]
[*]Men: [URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F9&frac2=72&pl=Multiply']4/9 * 72[/URL] = [B]32[/B]
[*]Ladies: [URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F9&frac2=72&pl=Multiply']3/9 * 72[/URL] = [B]24[/B]
[*]Kids: [URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F9&frac2=72&pl=Multiply']2/9 * 72[/URL] = [B]16[/B]
[/LIST]
[U]Check our work:[/U]
Men + Ladies + Kids = 32 + 24 + 16
Men + Ladies + Kids = 72 <-- This checks out!

At Falling Creek Middle School, they noticed that 3 out of every 4 buses were on time. If there are

At Falling Creek Middle School, they noticed that 3 out of every 4 buses were on time. If there are a total of 32 buses that drop off at this school, how many buses will NOT be on time.
If 3/4 are on time, we have 1 - 3/4 are not on time.
1 = 4/4
4/4 - 3/4 = 1/4 are not on time
We multiply 1/4(32) = [B]8 buses will NOT be on time[/B].

At the 2002 Winter Olympics, Austria won 2 gold medals. This was 1/8 of the total medals Austria won

At the 2002 Winter Olympics, Austria won 2 gold medals. This was 1/8 of the total medals Austria won. How many did Austria win?
Let t be the total medals Austria won. We have:
2 = x/8
Cross multiply, we get:
x = 2 * 8
x = [B]16[/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 what simple interest rate will 4500$ amount to 8000$ in 5 years?

At what simple interest rate will 4500$ amount to 8000$ in 5 years?
Simple Interest is written as 1 + it.
With t = 5, we have:
4500(1 + 5i) = 8000
Divide each side by 4500
1 + 5i = 1.77777778
Subtract 1 from each side:
5i = 0.77777778
Divide each side by 5
i = 0.15555
As a percentage we multiply by 100 to get [B]15.5%[/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 decreased by twice a

b is decreased by twice a
Twice a means we multiply a by 2:
2a
b decreased by twice a means we subtract 2a from b:
[B]b - 2a[/B]

B+c =10/a for a

B+c =10/a for a
Cross multiply:
a(B + c) = 10
Divide each side by a
[B]a = 10/(B + c)[/B]

b/3d - h = 343 for b

b/3d - h = 343 for b
A literal equation means we solve for one variable in terms of another variable or variables
Add h to each side to isolate the b term:
b/3d - h + h = 343 + h
Cancel the h's on the left side, we get:
b/3d = 343 + h
Cross multiply:
b = [B]3d(343 + h)[/B]

Balls numbered 1 to 10 are placed in a bag. Two of the balls are drawn out at random. Find the proba

Balls numbered 1 to 10 are placed in a bag. Two of the balls are drawn out at random. Find the probability that the numbers on the balls are consecutive.
Build our sample set:
[LIST]
[*](1, 2)
[*](2, 3)
[*](3, 4)
[*](4, 5)
[*](5, 6)
[*](6, 7)
[*](7, 8)
[*](8, 9)
[*](9, 10)
[/LIST]
Each of these 9 possibilities has a probability of:
1/10 * 1/9
This is because we draw without replacement. To start, the bag has 10 balls. On the second draw, it only has 9. We multiply each event because each draw is independent.
We have 9 possibilities, so we have:
9 * 1/10 * 1/9
Cancelling, the 9's, we have [B]1/10[/B]

Base Conversion Operations

Free Base Conversion Operations Calculator - This calculator allows you to add, subtract, multiply, and divide two numbers with different bases.

Basic Math Operations

Free Basic Math Operations Calculator - Given 2 numbers, this performs the following arithmetic operations:

* Addition (Adding) (+)

* Subtraction (Subtracting) (-)

* Multiplication (Multiplying) (x)

* Long division (Dividing) with a remainder (÷)

* Long division to decimal places (÷)

* Partial Sums (Shortcut Sums)

* Short Division

* Duplication and Mediation

* Addition (Adding) (+)

* Subtraction (Subtracting) (-)

* Multiplication (Multiplying) (x)

* Long division (Dividing) with a remainder (÷)

* Long division to decimal places (÷)

* Partial Sums (Shortcut Sums)

* Short Division

* Duplication and Mediation

Because of the promotion, attendance exceeded normal by 275 percent. If the normal attendance was 66

Because of the promotion, attendance exceeded normal by 275 percent. If the normal attendance was 6600 people, how many people attended?
Find the exceeded multiplier:
Exceeded Multiplier = Exceed percent / 100
[URL='https://www.mathcelebrity.com/perc.php?num=+5&den=+8&num1=+16&pct1=+80&pct2=+90&den1=+80&pct=275&pcheck=4&decimal=+65.236&astart=+12&aend=+20&wp1=20&wp2=30&pl=Calculate']275% as a decimal[/URL] = 2.75
We multiply as follows:
Exceeded Attendance = Normal Attendance * Exceeded Multiplier
Exceeded Attendance = 6600 * 2.75
Exceeded Attendance = [B]18,150[/B]

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].

Bob bought 10 note books and 4 pens for 18$. Bill bought 6 notebooks and 4 pens for 12$. Find the pr

Bob bought 10 note books and 4 pens for 18$. Bill bought 6 notebooks and 4 pens for 12$. Find the price of one note book and one pen.
Let the price of each notebook be n. Let the price of each pen be p. We're given two equations:
[LIST=1]
[*]10n + 4p = 18
[*]6n + 4p = 12
[/LIST]
Since we have matching coefficients for p, we subtract equation 1 from equation 2:
(10 - 6)n + (4 - 4)p = 18 - 12
Simplifying and cancelling, we get:
4n = 6
[URL='https://www.mathcelebrity.com/1unk.php?num=4n%3D6&pl=Solve']Type this equation into our search engine[/URL] and we get:
[B]n = 1.5[/B]
Now, substitute this value for n into equation (2):
6(1.5) + 4p = 12
Multiply through:
9 + 4p = 12
[URL='https://www.mathcelebrity.com/1unk.php?num=9%2B4p%3D12&pl=Solve']Type this equation into our search engine[/URL] and we get:
[B]p = 0.75[/B]

Bob finished reading his book in x days. Each day, he read 4 pages. His book has 28 pages

Bob finished reading his book in x days. Each day, he read 4 pages. His book has 28 pages
Our equation for this is found by multiplying pages per day times number of days;
4x = 28
To solve for x, [URL='https://www.mathcelebrity.com/1unk.php?num=4x%3D28&pl=Solve']we type the equation into our search engine[/URL] and we get:
x = [B]7[/B]

Brendan bought an aquarium originally priced at $50 but on sale for 50% off. After 12% sales tax, wh

Brendan bought an aquarium originally priced at $50 but on sale for 50% off. After 12% sales tax, what was the total cost?
50% off of 50 means they pay half, or 1/2(50) = 25.
Now, this gets taxed at 12%. So we multiply 25 * 1.12
Total Cost = 25(1.12)
Total Cost = [B]$28[/B]

Brian's age is 3/4 of Marcus'. The sum of their ages is 14. How old are Marcus and Brian?

Brian's age is 3/4 of Marcus'. The sum of their ages is 14. How old are Marcus and Brian?
Let Marcus's age be m.
Then Brian's age = 3/4m
The sum is:
m + 3m/4 = 14
Combine like terms
7m/4 = 14
Cross multiply:
7m = 56
[URL='http://www.mathcelebrity.com/1unk.php?num=7m%3D56&pl=Solve']Plugging this into the search engine[/URL], we get m = 8.
So Brian's age = 3(8)/4 = 24/4 = 6

Bridget can grow 6 flowers with every seed packet. With 4 seed packets, how many total flowers can B

Bridget can grow 6 flowers with every seed packet. With 4 seed packets, how many total flowers can Bridget have in her garden?
Set up a proportion of flowers to seed packets where f is the number of flowers for 4 seed packets. We have:
6/1 = f/4
Cross multiply:
f(1) = 24
f = 24

c increased by a factor of 20

c increased by a factor of 20
This means we multiply c by 20:
[B]20c[/B]

C varies directly as d use k as the constant of variation

C varies directly as d use k as the constant of variation
Direct variation means we multiply below:
[B]C = kd[/B]

C varies directly as the cube of a and inversely as the 4th power of B

C varies directly as the cube of a and inversely as the 4th power of B
The cube of a means we raise a to the 3rd power:
a^3
The 4th power of B means we raise b to the 4th power:
b^4
Varies directly means there exists a constant k such that:
C = ka^3
Also, varies inversely means we divide by the 4th power of B
C = [B]ka^3/b^4[/B]
Varies [I]directly [/I]as means we multiply by the constant k.
Varies [I]inversely [/I]means we divide k by the term which has inverse variation.
[MEDIA=youtube]fSsG1OB3qdk[/MEDIA]

c/a=db/r for a

c/a=db/r for a
Cross multiply the proportion:
cr = adb
Divide each side of the equation by db to isolate a:
cr/db = adb/db
Cancel the db terms on the left side and we get:
a = [B]cr/db[/B]

calculate cos(x) given tan(x)=8/15

calculate cos(x) given tan(x)=8/15
tan(x) = sin(x)/cos(x)
sin(x)/cos(x) = 8/15
Cross multiply:
15sin(x) = 8cos(x)
Divide each side by 8
[B]cos(x) = 15sin(x)/8[/B]

Caleb earns points on his credit card that he can use towards future purchases.

Let f = dollars spent on flights, h dollars spent on hotels, and p dollars spent on all other purchases.
[U]Set up our equations:[/U]
(1) 4f + 2h + p = 14660
(2) f + h + p = 9480
(3) f = 2h + 140
[U]First, substitute (3) into (2)[/U]
(2h + 140) + h + p = 9480
3h + p + 140 = 9480
3h + p = 9340
[U]Subtract 3h to isolate p to form equation (4)[/U]
(4) p = 9340 - 3h
[U]Take (3) and (4), and substitute into (1)[/U]
4(2h + 140) + 2h + (9340 - h) = 14660
[U]Multiply through[/U]
8h + 560 + 2h + 9340 - 3h = 14660
[U]Combine h terms and constants[/U]
(8 + 2 - 3)h + (560 + 9340) = 14660
7h + 9900 = 14660
[U]Subtract 9900 from both sides:[/U]
7h = 4760
[U]Divide each side by 7[/U]
[B]h = 680[/B]
[U]Substitute h = 680 into equation (3)[/U]
f = 2(680) + 140
f = 1360 + 140
[B]f = 1,500[/B]
[U]
Substitute h = 680 and f = 1500 into equation (2)[/U]
1500 + 680 + p = 9480
p + 2180 = 9480
[U]Subtract 2180 from each side:[/U]
[B]p = 7,300[/B]

Carlos was asked to write an equivalent equation to 2x/5 = 1 - x. he wrote it as 2x = 1 - 5x. do you

Carlos was asked to write an equivalent equation to 2x/5 = 1 - x. he wrote it as 2x = 1 - 5x. do you agree with his conclusion? explain your answer for x
Cross multiply
2x/5 = 1 - x
2x = 5(1 - x)
2x = 5 - 5x
I disagree with his conclusion. He forgot to multiply the 5 through to [B]both terms[/B]

Charlie buys a 40 pound bag of cat food. His cat eats a 1/2 pound of food per day.

Charlie buys a 40 pound bag of cat food. His cat eats a 1/2 pound of food per day.
Set up an equation:
1/2x = 40 where x is the number of days
Multiply through by 2
[B]x = 80[/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]

Committees of 4 men 5 women form a group of 11 men and 11 women.

Committees of 4 men 5 women form a group of 11 men and 11 women.
We want combinations.
4 men from 11 men is the combination 11C4. [URL='https://www.mathcelebrity.com/permutation.php?num=11&den=4&pl=Combinations']Using our combinations calculator[/URL], we get:
11C4 = 330
5 women from 11 women is the combination 11C5. [URL='https://www.mathcelebrity.com/permutation.php?num=11&den=5&pl=Combinations']Using our combinations calculator[/URL], we get:
11C5 = 462
We multiply the committee of men times the committee of women:
11C4 * 11C5 = 330 * 432
11C4 * 11C5 = [B]142,560[/B]

Consider the following 15 numbers 1, 2, y - 4, 4, 5, x, 6, 7, 8, y, 9, 10, 12, 3x, 20 - The mean o

Consider the following 15 numbers
1, 2, y - 4, 4, 5, x, 6, 7, 8, y, 9, 10, 12, 3x, 20
- The mean of the last 10 numbers is TWICE the mean of the first 10 numbers
- The sum of the last 2 numbers is FIVE times the sum of the first 3 numbers
(i) Calculate the values of x and y
We're given two equations:
[LIST=1]
[*](x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10 = 2(1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/10
[*]3x - 20 = 5(1 + 2 + y - 4)
[/LIST]
Let's evaluate and simplify:
[LIST=1]
[*](x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10 = (1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/5
[*]3x - 20 = 5(y - 1)
[/LIST]
Simplify some more:
[URL='https://www.mathcelebrity.com/polynomial.php?num=x%2B6%2B7%2B8%2By%2B9%2B10%2B12%2B3x%2B20&pl=Evaluate'](x + 6 + 7 + 8 + y + 9 + 10 + 12 + 3x + 20)/10[/URL] = (4x + y + 72)/10
[URL='https://www.mathcelebrity.com/polynomial.php?num=1%2B2%2By-4%2B4%2B5%2Bx%2B6%2B7%2B8%2By&pl=Evaluate'](1 + 2 + y - 4 + 4 + 5 + x + 6 + 7 + 8 + y)/5[/URL] = (2y + x + 29)/5
5(y - 1) = 5y - 5
So we're left with:
[LIST=1]
[*](4x + y + 72)/10 = (2y + x + 29)/5
[*]3x - 20 = 5y - 5
[/LIST]
Cross multiply equations in 1, we have:
5(4x + y + 72) = 10(2y + x + 29)
20x + 5y + 360 = 20y + 10x + 290
We have:
[LIST=1]
[*]20x + 5y + 360 = 20y + 10x + 290
[*]3x - 20 = 5y - 5
[/LIST]
Combining like terms:
[LIST=1]
[*]10x - 15y = -70
[*]3x - 5y = 15
[/LIST]
Now we have a system of equations which we can solve any of three ways:
[LIST]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10x+-+15y+%3D+-70&term2=3x+-+5y+%3D+15&pl=Substitution']Substitution Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10x+-+15y+%3D+-70&term2=3x+-+5y+%3D+15&pl=Elimination']Elimination Method[/URL]
[*][URL='https://www.mathcelebrity.com/simultaneous-equations.php?term1=10x+-+15y+%3D+-70&term2=3x+-+5y+%3D+15&pl=Cramers+Method']Cramer's Rule[/URL]
[/LIST]
No matter which method we choose, we get the same answer:
(x, y) = [B](-115, -72)[/B]

cot(?)=12 and ? is in Quadrant I, what is sin(?)?

cot(?)=12 and ? is in Quadrant I, what is sin(?)?
cot(?) = cos(?)/sin(?)
12 = cos(?)/sin(?)
Cross multiply:
12sin(?) = cos(?)
Divide each side by 12:
sin(?) = [B]12cos(?)[/B]

Currently 16 out of every 25 American adults drink coffee every day. In a town with a population of

Currently 16 out of every 25 American adults drink coffee every day. In a town with a population of 7900 adults, how many of these adults would you expect to drink coffee ever
We'd multiply 16/25 times 7900:
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=7900&frac2=16/25&pl=Multiply']fraction multiplication calculator by type 16/25 of 7900[/URL], we get:
[B]5056[/B]

cx+b/d=y for b

cx+b/d=y for b
Subtract cx from each side to isolate b/d:
cx - cx + b/d = y - cx
Cancel the cx terms on each side:
b/d = y - cx
Cross multiply:
b = [B]d(y - cx)[/B]

Dale has a bookcase with d shelves. There are 14 books on each shelf. Using d, write an expression f

Dale has a bookcase with d shelves. There are 14 books on each shelf. Using d, write an expression for the total number of books.
We multiply the number of shelves by the number of books per shelf.
[B]14d[/B]

Dan bought 7 new baseball trading cards to add to his collection. The next day his dog ate half of h

Dan bought 7 new baseball trading cards to add to his collection. The next day his dog ate half of his collection. There are now only 26 cards left. How many cards did Dan start with?
Let the starting amount of cards be s. We're given:
[LIST]
[*]Dan bought 7 new cards: s + 7
[*]The dog ate half of his collection. This means he's left with half, or (s + 7)/2
[*]Now, he's got 26 cards left. So we set up the following equation:
[/LIST]
(s + 7)/2 = 26
Cross multiply:
s + 7 = 26 * 2
s + 7 = 52
To solve for s, [URL='https://www.mathcelebrity.com/1unk.php?num=s%2B7%3D52&pl=Solve']we plug this equation into our search engine[/URL] and we get:
s = [B]45[/B]

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

Dan 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 30 cards left. How many cards did Dan start with?
Let the original collection count of cards be b.
So we have (b + 8)/2 = 30
Cross multiply:
b + 8 = 30 * 2
b + 8 = 60
[URL='http://www.mathcelebrity.com/1unk.php?num=b%2B8%3D60&pl=Solve']Use the equation calculator[/URL]
[B]b = 52 cards[/B]

Danny's mom ate 1/6 of an ice cream cake. Danny and his sister want to split the remainder of it. Ho

Danny's mom ate 1/6 of an ice cream cake. Danny and his sister want to split the remainder of it. How much of the cake would each get?
If Danny's mom ate 1/6 of the cake, then we have:
1 - 1/6 of the cake left.
We [URL='https://www.mathcelebrity.com/fraction.php?frac1=1&frac2=1%2F6&pl=Subtract']use our fraction subtraction calculator[/URL] for 1 - 1/6 to get:
5/6
If Danny and his sister split the remainder, then we divide 5/6 by 2. It's also the same as multiplying 5/6 by 1/2:
We [URL='https://www.mathcelebrity.com/fraction.php?frac1=5%2F6&frac2=1%2F2&pl=Multiply']use our fraction multiplication calculator[/URL] to get:
[B]5/12 for Danny and his sister[/B]

David has b dollars in his bank account; Claire has three times as much money as David. The sum of t

David has b dollars in his bank account; Claire has three times as much money as David. The sum of their money is $240. How much money does Claire have?
David has b
Claire has 3b since three times as much means we multiply b by 3
The sum of their money is found by adding David's bank balance to Claire's bank balance to get the equation:
3b + b = 240
To solve for b, [URL='https://www.mathcelebrity.com/1unk.php?num=3b%2Bb%3D240&pl=Solve']we type this equation into our search engine[/URL] and we get:
b = 60
So David has 60 dollars in his bank account.
Therefore, Claire has:
3(60) = [B]180[/B]

David obtained 5 out of 20 votes in the election. What percentage of the votes did david receive?

David obtained 5 out of 20 votes in the election. What percentage of the votes did david receive?
5/20 is the fraction.
You can simplify by dividing top and bottom by 5 to get 1/4
As a decimal, this is 0.25
To get a percentage, multiply the decimal by 100
100 * 0.25 = 25%
You can also use our [URL='http://www.mathcelebrity.com/perc.php?num=+5&den=+20&pcheck=1&num1=6500&pct1=70&pct2=70&den1=80&idpct1=10&hltype=1&idpct2=90&pct=82&decimal=+65.236&astart=12&aend=20&wp1=20&wp2=30&pof1=40&pof2=20&pl=Calculate']decimal-percentage-fraction converter[/URL]

decrease a number by 7 and multiply by 6.

decrease a number by 7 and multiply by 6.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Decrease a number by 7:
x - 7
Multiply by 6
[B]6(x - 7)[/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]

Derek must choose a 4 digit PIN. Each Digit can be chosen from 0 to 9. Derek does not want to reuse

Derek must choose a 4 digit PIN. Each Digit can be chosen from 0 to 9. Derek does not want to reuse any digits. He also only wants an even number that begins with 5. How many possible PINS could he choose from?
[LIST=1]
[*]First digit must begin with 5. So we have 1 choice
[*]We subtract 1 possible digit from digit 3 to have 8 - 1 = 7 possible digits
[*]This digit can be anything other than 5 and the even number in the next step. So we have 0-9 is 10 digits - 2 = 8 possible digits
[*]Last digit must end in 0, 2, 4, 6, 8 to be even. So we have 5 choices
[/LIST]
Our total choices from digits 1-4 are found by multiplying each possible digit choice:
1 * 7 * 8 * 5 = [B]280 possible PINS[/B]

Determine the formula of the given statement by following the procedures. Choose any number then add

Determine the formula of the given statement by following the procedures. Choose any number then add 2. Multiply your answer to 3 and minus 2
For the phrase [I]choose any number[/I] we can use an arbitrary variable, let's call it x.
Add 2:
x + 2
Multiply your answer to 3:
3(x + 2)
And minus 2 which means we subtract:
[B]3(x + 2) - 2[/B]

Diego is jogging at a rate of 5mi/h. A function relates how far Deigo jogs to his rate of speed.

Let d be distance and h be hours in time. Set up our function.
[LIST]
[*]f(h) = d
[*][B]f(h) = 5h[/B]
[/LIST]
Read this out, it says, for every hour Diego jogs, multiply that by 5 to get the distance he jogs.

Dina is twice as old as Andrea. The sum of their age is 72. Find their present ages.

Dina is twice as old as Andrea. The sum of their age is 72. Find their present ages.
Let d be Dina's age. Let a be Andrea's age. We're given:
[LIST=1]
[*]d = 2a <-- Twice means multiply by 2
[*]a + d = 72
[/LIST]
Substitute equation (1) into equation (2):
a + 2a = 72
[URL='https://www.mathcelebrity.com/1unk.php?num=a%2B2a%3D72&pl=Solve']Type this equation into our search engine[/URL] and we get:
[B]a = 24[/B]
Substitute a = 24 into equation (1):
d = 2(24)
[B]d = 48
So Andrea is 24 years old and Dina is 48 years old[/B]

Dina is twice the age of Anton. If Anton is 12, how will you represent the age of Dina?

Dina is twice the age of Anton. If Anton is 12, how will you represent the age of Dina?
Twice means multiply by 2, so we have:
Dina = 2 * Anton's age
Dina = 2 * 12
Dina = [B]24[/B]

Divide 73 into two parts whose product is 402

Divide 73 into two parts whose product is 40
Our first part is x
Our second part is 73 - x
The product of the two parts is:
x(73 - x) = 40
Multiplying through, we get:
-x^2 + 73x = 402
Subtract 40 from each side, we get:
-x^2 + 73x - 402 = 0
This is a quadratic equation. To solve this, we type it in our search engine, choose "solve Quadratic", and we get:
[LIST=1]
[*]x = [B]6[/B]
[*]x = [B]67[/B]
[/LIST]

divide a by 8, triple the result, then add 7

divide a by 8, triple the result, then add 7
[LIST]
[*]Divide a by 8: a/8
[*]Triple the result means multiply by 3: 3a/8
[*]Then add 7
[/LIST]
[B]3a/8 + 7[/B]

Divide a by b, double the result, then multiply c by what you have

Divide a by b, double the result, then multiply c by what you have
Take this algebraic expression in parts:
[LIST]
[*]Divide a by b: a/b
[*]Double the result means multiply by 2: 2a/b
[*]Then multiply c by what you have:
[/LIST]
[B]2ac/b[/B]

divide a by c, triple the result, then subtract what you have from b

divide a by c, triple the result, then subtract what you have from b
Let's take this algebraic expression in parts:
[LIST=1]
[*]Divide a by c: a/c
[*]Triple the result. This means we multiply a/c by 3: 3a/c
[*]Then subtract what you have (the result) from b: b - 3a/c
[/LIST]
[B]b - 3a/c[/B]

divide the sum of the square of a and b by thrice c

divide the sum of the square of a and b by thrice c
Sum of the squares of a and b is found as follows:
[LIST]
[*]a squared means we raise a to the power of 2: a^2
[*]b squared means we raise b to the power of 2: b^2
[*]Sum of the squares means we add both terms: a^2 + b^2
[*]Thrice c means we multiply c by 3: 3c
[/LIST]
Divide means we have a quotient:
[B](a^2 + b^2)/3c[/B]

divide u by s multiply the result by v

divide u by s multiply the result by v
Divide u by s:
u/s
Multiply the result by v:
[B]uv/s[/B]

double 10, add the result to 9, then add y

double 10, add the result to 9, then add y
Double 10 means multiply 10 by 2:
10 * 2
20
Add the result to 9, means we add 20 to 9:
20 + 9
29
Then we add y:
[B]29 + y[/B]

double 6 , divide the result by y ,then raise what you have to the 10th power

double 6 , divide the result by y ,then raise what you have to the 10th power
Take this in pieces:
Double 6 means multiply 6 by 2 --> 6(2) = 12
Divide the result by y:
12/y
Then raise what you have to the 10th power:
[B](12/y)^10[/B]

double n, multiply answer by 3

double n, multiply answer by 3
Double n means multiply n by 2
2n
Multiply the answer by 3:
3(2n) = [B]6n[/B]

double the quotient of 4 and 7

double the quotient of 4 and 7
The quotient fo 4 and 7:
4/7
Double means multiply by this expression by 2:
[B]2(4/7)[/B]
If you need to evaluate and simplify this, it's:
[B]8/7[/B]

double v, add u, then divide t by what you have

double v, add u, then divide t by what you have
Double v means we multiply the variable v by 2:
2v
Add u:
2v + u
We build a fraction, with t as the numerator, and 2v + u as the denominator
[B]t/(2v + u)[/B]

double v, raise the result to the 6th power, then multiply what you have by w

double v, raise the result to the 6th power, then multiply what you have by w
Double v means multiply v by 2:
2v
Raise the result to the 6th power, means we use an exponent of 6 on 2v:
(2v)^6
Then multiply what you have by w, means take the result above, and multiply by w:
[B]w(2v)^6[/B]

Each tree in an orchard containing 2,650 trees requires 210 grams of fertiliizer. At $1.25 per kilog

Each tree in an orchard containing 2,650 trees requires 210 grams of fertiliizer. At $1.25 per kilogram of fertilizer, how much does it cost to fertilize the orchard?
[U]Calculate the total fertilizer needed:[/U]
Total fertilizer needed = Number of trees * grams of fertilizer per tree
Total fertilizer needed = 2650 * 210
[URL='https://www.mathcelebrity.com/longdiv.php?num1=2650&num2=210&pl=Multiply']Total fertilizer needed[/URL] = 556500 grams
[U]1 kilogram = 1000 grams, so we convert our 556500 grams to kilograms:[/U]
kilograms of fertilizer = grams of fertilizer / 1000
kilograms of fertilizer = 556500/1000
kilograms of fertilizer = 556.5
[U]Calculate fertilizer cost:[/U]
Fertilizer cost = kilograms of fertilizer * cost per kilogram
Fertilizer cost = 556.5 * 1.25
Fertilizer cost = [B]695.63[/B]

Eight gallons were poured into a gas tank that was 1/4 full. Now the tank is 3/4 full. How many gall

Eight gallons were poured into a gas tank that was 1/4 full. Now the tank is 3/4 full. How many gallons does a full tank hold?
3/4 - 1/4 = 2/4
8 gallons = 2/4
2/4 = 1/2
8 gallons = 1/2
Multiply by 2:
8 * 2 = 2/2 = [B]16 gallons[/B] = 1 full tank

Eight workers dug 3/8 of a tunnel in 10 days. If they need to finish the remaining 5/8 of the tunnel

Eight workers dug 3/8 of a tunnel in 10 days. If they need to finish the remaining 5/8 of the tunnel in 3 & 1/3 days, how many more workers must they hire?
Eight workers * 10 days = 80 days for one worker
If 3/8 of a tunnel took 80 days for one worker, then each 1/8 of a tunnel takes a single worker:
80/3 = 26 & 2/3 days
Multiply by 5 for the remaining 5/8 and we get:
133 & 1/3
We need 133 & 1/3 / 3 & 1/3 = 40 workers needed
Additional workers = Workers Needed - Original Workers
Additional workers = 40 - 8
Additional workers = [B]32[/B]

Expand Master and Build Polynomial Equations

Free Expand Master and Build Polynomial Equations Calculator - This calculator is the __ultimate__ expansion tool to multiply polynomials. It expands algebraic expressions listed below using all 26 variables (a-z) as well as negative powers to handle polynomial multiplication. Includes multiple variable expressions as well as outside multipliers.

Also produces a polynomial equation from a given set of roots (polynomial zeros). * Binomial Expansions c(a + b)^{x}

* Polynomial Expansions c(d + e + f)^{x}

* FOIL Expansions (a + b)(c + d)

* Multiple Parentheses Multiplications c(a + b)(d + e)(f + g)(h + i)

Also produces a polynomial equation from a given set of roots (polynomial zeros). * Binomial Expansions c(a + b)

* Polynomial Expansions c(d + e + f)

* FOIL Expansions (a + b)(c + d)

* Multiple Parentheses Multiplications c(a + b)(d + e)(f + g)(h + i)

Explain the steps you would take to find an equation for the line perpendicular to 4x - 5y = 20 and

Explain the steps you would take to find an equation for the line perpendicular to 4x - 5y = 20 and sharing the same y-intercept
Get this in slope-intercept form by adding 5y to each side:
4x - 5y + 5y = 5y + 20
Cancel the 5y's on the left side and we get:
5y + 20 = 4x
Subtract 20 from each side
5y + 20 - 20 = 4x - 20
Cancel the 20's on the left side and we get:
5y = 4x - 20
Divide each side by 5:
5y/5 = 4x/5 - 4
y = 4x/5 - 4
So we have a slope of 4/5
to find our y-intercept, we set x = 0:
y = 4(0)/5 - 4
y = 0 - 4
y = -4
If we want a line perpendicular to the line above, our slope will be the negative reciprocal:
The reciprocal of 4/5 is found by flipping the fraction making the numerator the denominator and the denominator the numerator:
m = 5/4
Next, we multiply this by -1:
-5/4
So our slope-intercept of the perpendicular line with the same y-intercept is:
[B]y = -5x/4 - 4[/B]

Explain why 1/2 and 3/6 are equivalent

Explain why 1/2 and 3/6 are equivalent.
Multiply any number by 1, and we get the same number.
Multiply 1/2 by 3/3 which is 1
(1 * 3)/(2 *3) = 3/6

ey/n + k = t for y

ey/n + k = t for y
Let's take this literal equation in pieces:
Subtract k from each side:
ey/n + k - k = t - k
Cancel the k's on the left side, we have:
ey/n = t - k
Now multiply each side by n:
ney/n = n(t - k)
Cancel the n's on the left side, we have:
ey = n(t - k)
Divide each side by e:
ey/e = n(t - k)/e
Cancel the e's on the left side, we have:
[B]y = n(t - k)/e[/B]

f - g = 1/4b for b

f - g = 1/4b for b
Multiply each side of the equation by 4 to remove the 1/4 and isolate b:
4(f - g) = 4/4b
4/4 = 1, so we have:
b = [B]4(f - g)[/B]
[I]the key to this problem was multiplying by the reciprocal of the constant[/I]

F varies directly as g and inversely as r^2

F varies directly as g and inversely as r^2
[U]Givens and assumptions[/U]
[LIST]
[*]We take a constant of variation called k.
[*][I]Varies directly means we multiply our variable term by k[/I]
[*][I]Varies inversely means we divide k by our variable term[/I]
[/LIST]
The phrase varies directly or varies inversely means we have a constant k such that:
[B]F = kg/r^2[/B]

f varies jointly with u and h and inversely with the square of y.

f varies jointly with u and h and inversely with the square of y.
Variation means we have a constant k.
Varies jointly with u and h means we multiply k by hu
Varies inversely with the square of y means we divide by y^2
[B]f = khu/y^2[/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]

f+g/e=r for g

f+g/e=r for g
Subtract f from each side
g/e = r - f
Multiply each side by e
[B]g = e(r - f)[/B]

F/B=(M-N*L)/D for L

F/B=(M-N*L)/D for L
Cross multiply:
DF/B = M - N*L
Subtract M from each side:
DF/B - M = -N*L
Divide each side by -N
[B]L = -DF/BN[/B]

Faith is 1/5 her mother's age. Their combined ages are 30. How old is faith?

Faith is 1/5 her mother's age. Their combined ages are 30. How old is faith?
Let Faith's age be f. Let her mother's age be m. We're given:
[LIST=1]
[*]f = m/5
[*]f + m = 30
[/LIST]
Rearrange (1) by cross-multiplying:
m = 5f
Substitute this into equation (2):
f + 5f = 30
[URL='https://www.mathcelebrity.com/1unk.php?num=f%2B5f%3D30&pl=Solve']Type this equation into our search engine[/URL] and we get:
f = [B]5[/B]

Farah rolls a fair dice and flips a fair coin. What is the probability of obtaining a 5 and a head?

Farah rolls a fair dice and flips a fair coin. What is the probability of obtaining a 5 and a head? Give your answer in its simplest form.
Probability of a 5 is 1/6
Probability of a head is 1/2
Since each event is independent, we get the total probability by multiplying both together:
P(5,H) = 1/6 * 1/2
P(5,H) = [B]1/12[/B]

Find 2 consecutive numbers such that the sum of twice the smaller number and 3 times the larger numb

Find 2 consecutive numbers such that the sum of twice the smaller number and 3 times the larger number is 73.
Let x be the smaller number and y be the larger number. We are given:
2x + 3y = 73
Since the numbers are consecutive, we know that y = x + 1. Substitute this into our given equation:
2x + 3(x + 1) = 73
Multiply through:
2x + 3x + 3 = 73
Group like terms:
5x + 3 = 73
[URL='https://www.mathcelebrity.com/1unk.php?num=5x%2B3%3D73&pl=Solve']Type 5x + 3 = 73 into the search engine[/URL], and we get [B]x = 14[/B].
Our larger number is 14 + 1 = [B]15
[/B]
Therefore, our consecutive numbers are[B] (14, 15)[/B]

Find 3 consecutive integers such that the sum of twice the smallest and 3 times the largest is 126

Find 3 consecutive integers such that the sum of twice the smallest and 3 times the largest is 126.
Let the first integer be n, the second integer be n + 1, and the third integer be n + 2. We have:
Sum of the smallest and 3 times the largest is 126:
n + 3(n + 2) = 126
Multiply through:
n + 3n + 6 = 126
Group like terms:
4n + 6 = 126
[URL='https://www.mathcelebrity.com/1unk.php?num=4n%2B6%3D126&pl=Solve']Type 4n + 6 = 126 into our calculator[/URL], we get n = 30. Which means the next two integers are 31 and 32.
[B]{30, 31, 32}[/B]

Find r in P(7, r)

Find r in P(7, r)
Recall the permutations formula:
7! / (7-r!) = 840.
We [URL='https://www.mathcelebrity.com/factorial.php?num=7!&pl=Calculate+factorial']run 7! through our search engine[/URL] and we get:
[URL='https://www.mathcelebrity.com/factorial.php?num=7!&pl=Calculate+factorial']7![/URL] = 5040
5040 / (7 - r)! = 840
Cross multiply, and we get:
5040/840 = 7 - r!
6 = (7 - r)!
Since 6 = 3*2*! = 3!, we have;
3! = (7 - r)!
3 = 7 - r
To solve for r, we [URL='https://www.mathcelebrity.com/1unk.php?num=3%3D7-r&pl=Solve']type this equation into our search engine[/URL] and we get:
r = [B]4[/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 the two square roots of 81

find the two square roots of 81
When we multiply 9 * 9, we get 81
When we multiply -9 * -9, we get 81
So our two square roots of 81 are:
[LIST]
[*][B]-9, 9[/B]
[/LIST]

Find y if the line through (1,y) and (4,5) has a slope of 3

Find y if the line through (1,y) and (4,5) has a slope of 3.
Slope formula is:
m = (y2 - y1)/(x2 - x1)
With m = 3, we have:
3 = (5 - y)/(4 - 1)
3 = (5 - y)/3
Cross multiply:
5 - y = 9
Subtract 5 from each side
-y = 4
Multiply each side by -1
[B]y = -4[/B]

Finn has 8 toy cars. Dirk has t times as many toy cars as Finn

Finn has 8 toy cars. Dirk has t times as many toy cars as Finn
The phrase [I]times as many [/I]means we multiply:
[B]8t[/B]

flip 7 coins How many total outcomes are there

flip 7 coins How many total outcomes are there
A flip of a coin has 2 outcomes, heads or tails. Since each outcome is independent of the other outcomes, we multiply each flip by 2 outcomes:
Total outcomes = 2 * 2 * 2 * 2 * 2 * 2 * 2
Total outcomes = 2^7
Total outcomes = [B]128[/B]

for every 10 white cars a dealer sells he sells 7 silver, 6 blue, 5 red, 4 yellow, 3 green, 2 black,

for every 10 white cars a dealer sells he sells 7 silver, 6 blue, 5 red, 4 yellow, 3 green, 2 black, 2 purple and 1 brown car. If he sells 120 cars how many blue cars?
[U]Take this in blocks, so each block has:[/U]
10 white + 7 silver + 6 blue + 5 red + 4 yellow + 3 green + 2 black + 2 purple + 1 brown = 40 cars
[U]Calculate the number of blocks:[/U]
120 cars / 40 cars = 3 blocks.
[U]For 120 cars sold, it takes 3 blocks, which means we multiply:[/U]
6 blue cars per block * 3 blocks = [B]18 blue cars[/B]

Four coins are flipped. What is the probability of the coins all landing on heads

Four coins are flipped. What is the probability of the coins all landing on heads
The probability of one head is 1/2. Since all 4 flips are independent, we multiply each flip probability:
P(HHHH) = 1/2 * 1/2 * 1/2 * 1/2
P(HHHH) = [B]1/16[/B]

Four fifths of 300

Four fifths of 300.
Multiply 4/5 by 300:
4(300)/5
1200/5
[URL='https://www.mathcelebrity.com/fraction.php?frac1=1200%2F5&frac2=3%2F8&pl=Simplify']Simplifying this fraction[/URL], we get:
240

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]

Fractions and Mixed Numbers

Free Fractions and Mixed Numbers Calculator - Given (improper fractions, proper fraction, mixed numbers, or whole numbers), this performs the following operations:

* Addition (Adding)

* Subtraction (Subtracting)

* Positive Difference (Absolute Value of the Difference)

* Multiplication (Multiplying)

* Division (Dividing: complex fraction division is included)

* Compare Fractions

* Simplifying of proper and improper fractions as well as mixed numbers. Fractions will be reduced down as far as possible (Reducing Fractions).

* Reciprocal of a Fraction

* Find all fractions between two fractions

* reduce a fraction

* Addition (Adding)

* Subtraction (Subtracting)

* Positive Difference (Absolute Value of the Difference)

* Multiplication (Multiplying)

* Division (Dividing: complex fraction division is included)

* Compare Fractions

* Simplifying of proper and improper fractions as well as mixed numbers. Fractions will be reduced down as far as possible (Reducing Fractions).

* Reciprocal of a Fraction

* Find all fractions between two fractions

* reduce a fraction

From a group of 10 men and 8 women, how many ways can 2 men and 3 women be chosen for 5 positions

From a group of 10 men and 8 women, how many ways can 2 men and 3 women be chosen for 5 positions?
We use combinations. Since men and women are independent, we multiply each result:
We want 10 men choose 2 men multiplied by 8 women choose 3 women.
[URL='https://www.mathcelebrity.com/permutation.php?num=10&den=2&pl=Combinations']Type 10C2 into our search engine[/URL] and we get 45
[URL='https://www.mathcelebrity.com/permutation.php?num=8&den=3&pl=Combinations']Type 8C3 into our search engine[/URL] and we get 56
Multiply both together:
45 * 56 = [B]2,520 ways[/B]

FV-O/T=A for o

FV-O/T=A for o
Add O/T to each side:
FV-O/T + O/T = A + O/T
We have:
A + O/T = FV
Subtract A from each side:
A - A + O/T = FV + A
Cancelling the A's, e have:
O/T = FV - A
Cross multiply the T:
[B]O = T(FV - A)[/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

Ok. To go further in this equation. It reads:
...How many did he originally take to the event? Multiply the answer by 3 and reverse the digits. This will give you the answer for ACH in the coordinates.
Does that make sense to reverse 303?
:-/
Thank you for your help!!

Geocache puzzle help

Ok. To go further in this equation. It reads:
...How many did he originally take to the event? Multiply the answer by 3 and reverse the digits. This will give you the answer for ACH in the coordinates.
Does that make sense to reverse 303?
:-/
Thank you for your help!!

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 me post the whole equation paragraph:
Brainteaser # 1: Answer for ACH
A fellow geocacher decided that he would try to sell some hand-made walking sticks at the local geocaching picnic event. In the first hour, he sold one-half of his sticks, plus one-half of a stick. The next hour, he sold one-third of his remaining sticks plus one-third of a stick. In the third hour, he sold one-fourth of what he had left, plus three-fourths of a stick. The last hour, he sold one-fifth of the remaining sticks, plus one-fifth of a stick. He did not cut up any sticks to make these sales. He returned home with 19 sticks. How many did he originally take to the event? Multiply the answer by 3 and reverse the digits. This will give you the answer for ACH in the coordinates. Make sure to multiply and reverse the digits.
What would the answer be?

Geocache puzzle help

Multiply the answer by 3: 101 * 3 = 303
Reverse the digits:
303 reversed is a palindrome, so it's still [B]303[/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 g(a)=a² - 2a - 1 and f(x)=x² - 2x, Find: a) f(a+2)-f(a)/2 b) g(a+h)-g(a)/h

Given g(a)=a² - 2a - 1 and f(x)=x² - 2x:
Find:
a) f(a+2) - f(a)/2
b) g(a+h) - g(a)/h
a) f(a + 2) = (a + 2)^2 - 2(a + 2)
f(a + 2) = a^2 + 2a + 4 - 2a - 4
Simplify and combine like terms:
the 2a and 4's cancel, so we have:
f(a + 2) = a^2
f(a)/2 = (a^2 - 2a)/2
Subtract one from the other, we get:
a^2 - a^2/2 - a
[B]a) a^2/2 - a
------------------------[/B]
b) g(a + h) = (a + h)^2 - 2(a + h) - 1
g(a + h) = a^2 +2ah + h^2 - 2a - 2h - 1
g(a)/2 = (a^2 - 2a - 1)/h
g(a)/2 = (a^2 - 2a - 1)/h
Subtract one from the other:
g(a+h) - g(a)/h
a^2 +2ah + h^2 - 2a - 2h - 1 - (a^2 - 2a - 1)/h
Multiply through by h
[B]a^2h + 2ah^2 + h^3 - 2ah - 2h^2 - h - a^2 + 2a + 1[/B]

Given that E[Y]=2 and Var [Y] =3, find E[(2Y + 1)^2]

Given that E[Y]=2 and Var [Y] =3, find E[(2Y + 1)^2]
Multiply through
E[(2Y + 1)^2] = E[4y^2 + 4y + 1]
We can take the expected value of each term
E[4y^2] + E[4y] + E[1]
For the first term, we have:
4E[Y^2]
We define the Var[Y] = E[Y^2] - (E[Y])^2
Rearrange this term, we have E[Y^2] = Var[Y] + (E[Y])^2
E[Y^2] = 3+ 2^2
E[Y^2] = 3+ 4
E[Y^2] = 7
So our first term is 4(7) = 28
For the second term using expected value rules of separating out a constant, we have
4E[Y] = 4(2) = 8
For the third term, we have:
E[1] = 1
Adding up our three terms, we have:
E[4y^2] + E[4y] + E[1] = 28 + 8 + 1
E[4y^2] + E[4y] + E[1] = [B]37[/B]

Given w(x) = 3x + 8, find w(2b + 6).

Given w(x) = 3x + 8, find w(2b + 6).
Plug the value of 2b + 6 in for x
w(2b + 6) = 3(2b + 6) + 8
Multiply through:
w(2b + 6) = 6b + 18 + 8
Group like terms:
w(2b + 6) = [B]6b + 26[/B]

Given y= 4/3x what is the constant of proportionality

Given y= 4/3x what is the constant of proportionality
Direct variation means the constant of proportionality is y/x.
Cross multiplying, we get:
y/x = [B]4/3[/B]

Gregg has 8 cards.Half red,half black. He picks 2 cards from the deck.What is the probability both o

Gregg has 8 cards.Half red,half black. He picks 2 cards from the deck.What is the probability both of them are red?
Half means 4 cards are red and 4 cards are black.
The first draw probability of red is:
4 total red cards out of 8 total cards = 4/8.
[URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F8&frac2=3%2F8&pl=Simplify']Simplified, this is[/URL] 1/2
The second draw is 3 total red cards out of 7 remaining cards. Since 1 red was drawn (4 - 1) = 3 reds left and 1 card was drawn (8 -1) = left
3/7
Since each draw is independent, we multiply the probabilities:
1/2 * 3/7 = [B]3/14[/B]

gy=-g/v+w for g

gy=-g/v+w for g
Multiply each side of the equation by v to eliminate fractions:
gvy = -g + vw
Add g to each side:
gvy + g = -g + g + vw
Cancel the g's on the right side and we geT:
gvy + g = vw
Factor out g on the left side:
g(vy + 1) = vw
Divide each side of the equation by (vy + 1):
g(vy + 1)/(vy + 1) = vw/(vy + 1)
Cancel the (vy + 1) on the left side and we geT:
g = [B]vw/(vy + 1)[/B]

Hayden bought 48 new trading cards. Three-fourths of the new cards are baseball cards. How many base

Hayden bought 48 new trading cards. Three-fourths of the new cards are baseball cards. How many baseball cards did Hayden buy?
We want 3/4 of 48. We [URL='https://www.mathcelebrity.com/fraction.php?frac1=48&frac2=3/4&pl=Multiply']type this statement into our calculator[/URL] and we get:
[B]36[/B]

Help Plz

Nick's age: x
John's age: x/2
Pip's age = 2/3 * x/2 = x/3
The sum is 26, so we have:
x + x/2 + x/3 = 26
Common denominator is (1 * 2 * 3) = 6
6x/6 + 3x/6 + 2x/6 = 26
Combine like terms:
11x/6 = 26
Cross multiply:
11x = 156
x = 14.1818
This doesn't make sense for age. Are you sure you wrote out the problem right?

Henrietta hired a tutor to help her improve her math scores. While working with the tutor, she took

Henrietta hired a tutor to help her improve her math scores. While working with the tutor, she took four tests. She scored 10 points better on the second test than she did on the first, 20 points better on the third test than on the first, and 30 points better on the fourth test than on the first. If the mean of these four tests was 70, what was her score on the third test?
Givens:
[LIST]
[*]Let the first test score be s:
[*]The second test score is: s + 10
[*]The third test score is: s + 20
[*]The fourth test score is: s + 30
[/LIST]
The mean of the four tests is 70, found below:
Sum of test scores / Number of Tests = Mean
Plugging in our number, we get:
(s + s + 10 + s + 20 + s + 30) / 4 = 70
Cross multiply and simplify:
4s + 60 = 70 * 4
4s + 60 = 280
To [URL='https://www.mathcelebrity.com/1unk.php?num=4s%2B60%3D280&pl=Solve']solve this equation for s, we type it in our search engine[/URL] and we get:
s = 55
So the third test score:
s + 20 = 55 + 20
[B]75[/B]

How many 1/4 sheets are there in 5 sheets

How many 1/4 sheets are there in 5 sheets
We divide 5 sheets by 1/4 sheets:
5/1/4
However, when we divide by a fraction, it's the same as multiplying by the reciprocal of the fraction:
The reciprocal of 1/4 is 4/1, so we have:
5 * 4/1 = 20/1 = [B]20[/B]

How many dimes must be added to a bag of 200 nickels so that the average value of the coins in the b

How many dimes must be added to a bag of 200 nickels so that the average value of the coins in the bag is 8 cents?
200 nickels has a value of 200 * 0.05 = $10.
Average value of coins is $10/200 = 0.05
Set up our average equation, where we have total value divided by total coins:
(200 * 0.05 + 0.1d)/(200 + d) = 0.08
Cross multiply:
16 + 0.08d = 10 + 0.1d
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=16%2B0.08d%3D10%2B0.1d&pl=Solve']equation solver[/URL], we get:
[B]d = 300[/B]

How many one-fifths are there in 200?

How many one-fifths are there in 200?
Using the rule of dividing by a fraction is the same as multiplying by the reciprocal, we have:
200 / 1/5 = 200 * 5 = [B]1000[/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]

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 (x - 1)/3 = k and k = 2, what is the value of x?

If (x - 1)/3 = k and k = 2, what is the value of x?
If k = 2, we have:
(x - 1)/3 = 2
Cross multiply:
x - 1 = 3 * 2
x - 1 = 6
[URL='https://www.mathcelebrity.com/1unk.php?num=x-1%3D6&pl=Solve']Type this equation into the search engine[/URL], we get:
[B]x = 7[/B]

If 1/2 cup of milk makes 8 donuts. How much cups it takes to make 28 donuts

If 1/2 cup of milk makes 8 donuts. How much cups it takes to make 28 donuts?
Set up a proportion of cups to donuts, where c is the number of cups required to make 28 donuts:
1/2/8 = c/28
Cross multiply:
28(1/2) = 8c
8c = 14
[URL='https://www.mathcelebrity.com/1unk.php?num=8c%3D14&pl=Solve']Plugging this equation into our search engine[/URL], we get:
[B]c = 1.75[/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 2 is added to the numerator and denominator it becomes 9/10 and if 3 is subtracted from the numer

If 2 is added to the numerator and denominator it becomes 9/10 and if 3 is subtracted from the numerator and denominator it become 4/5. Find the fractions.
Convert 2 to a fraction with a denominator of 10:
20/2 = 10, so we multiply 2 by 10/10:
2*10/10 = 20/10
Add 2 to the numerator and denominator:
(n + 2)/(d + 2) = 9/10
Cross multiply and simplify:
10(n + 2) = 9(d + 2)
10n + 20 = 9d + 18
Move constants to right side by subtracting 20 from each side and subtracting 9d:
10n - 9d = -2
Subtract 3 from the numerator and denominator:
(n - 3)/(d - 3) = 4/5
Cross multiply and simplify:
5(n - 3) = 4(d - 3)
5n - 15 = 4d - 12
Move constants to right side by adding 15 to each side and subtracting 4d:
5n - 4d = 3
Build our system of equations:
[LIST=1]
[*]10n - 9d = -2
[*]5n - 4d = 3
[/LIST]
Multiply equation (2) by -2:
[LIST=1]
[*]10n - 9d = -2
[*]-10n + 8d = -6
[/LIST]
Now add equation (1) to equation (2)
(10 -10)n (-9 + 8)d = -2 - 6
The n's cancel, so we have:
-d = -8
Multiply through by -1:
d = 8
Now bring back our first equation from before, and plug in d = 8 into it to solve for n:
10n - 9d = -2
10n - 9(8) = -2
10n - 72 = -2
To solve for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=10n-72%3D-2&pl=Solve']plug this equation into our search engine[/URL] and we get:
n = 7
So our fraction, n/d = [B]7/8[/B]

If 25% of 30% of x is 9, what is x?

If 25% of 30% of x is 9, what is x?
Convert percentages to decimals when multiplying:
25% = 0.25
30% = 0.3
0.25 * 0.3 * x = 9
0.075x = 9
Using our math engine, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.075x%3D9&pl=Solve']type this equation in[/URL] and we get:
x = [B]120
[MEDIA=youtube]5EwNxiBdLu0[/MEDIA][/B]

If 3 coins are flipped simultaneously, the probability of having three tails is

If 3 coins are flipped simultaneously, the probability of having three tails is...
The probability of flipping a head is 1/2. Since each coin flip is independent, we multiply the probabilities together of the three coin flips:
P(HHH) = 1/2 * 1/2 * 1/2
P(HHH) = [B]1/8[/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 4(x-9)=3x-8x, what is x?

[SIZE=5]If 4(x-9)=3x-8x, what is x?
[/SIZE]
[SIZE=4]Multiply through:
4x - 36 = 3x - 8x
Group like terms:
4x - 36 = -5x
[/SIZE]
[URL='https://www.mathcelebrity.com/1unk.php?num=4x-36%3D-5x&pl=Solve'][SIZE=4]Typing this equation into the search[/SIZE][/URL][SIZE=4][URL='https://www.mathcelebrity.com/1unk.php?num=4x-36%3D-5x&pl=Solve'] engine[/URL], we get:
[B]x = 4[/B][/SIZE]

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 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 a machine produces 100 bags per minute how long will it take to make 40,000

If a machine produces 100 bags per minute how long will it take to make 40,000
100 bags/ per minute = 40,000 bags / m
Cross multiply
100m = 40000
[URL='https://www.mathcelebrity.com/1unk.php?num=100m%3D40000&pl=Solve']Type this equation into the search engine[/URL] and we get:
m = [B]400[/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 increased by 16 and then divided by 3, the result is 8

If a number is increased by 16 and then divided by 3, the result is 8.
Let x be the number. We have:
(x + 16)/3 = 8
Cross multiply
x + 16 = 24
Using our equation calculator, we get:
[B]x = 8[/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 Frank’s age is double of Willis’ age and the sum of their ages is 42. What are their ages?

If Frank’s age is double of Willis’ age and the sum of their ages is 42. What are their ages?
Let Frank's age be f. Let Willis's age be w. We're given two equations:
[LIST=1]
[*]f = 2w <-- Double means multiply by 2
[*]f + w = 42
[/LIST]
Substitute equation (1) into equation (2):
2w + w = 42
To solve for w, [URL='https://www.mathcelebrity.com/1unk.php?num=2w%2Bw%3D42&pl=Solve']type this equation into our search engine[/URL]. We get:
w = [B]14
[/B]
Now, take w = 14, and substitute it back into equation (1) to solve for f:
f = 2(14)
f = [B]28[/B]

If half the number is added to twice the number, the answer is 50

If half the number is added to twice the number, the answer is 50.
Let the number be n. Half is also written as 0.5, and twice is written by multiplying by 2. We have:
0.5n + 2n = 50
[URL='https://www.mathcelebrity.com/1unk.php?num=0.5n%2B2n%3D50&pl=Solve']Plugging this equation into our search engine[/URL], we get:
[B]n = 20[/B]

If I add 8 to the number and then multiply the result by 6 I get the same answer as when I add 58 to

If I add 8 to the number and then multiply the result by 6 I get the same answer as when I add 58 to a number. Form an equation
Let the number be n. We're given:
6(n + 8) = n + 58
Multiply through:
6n + 48 = n + 58
To solve this equation for n, [URL='https://www.mathcelebrity.com/1unk.php?num=6n%2B48%3Dn%2B58&pl=Solve']we type it into our search engine[/URL] and we get:
n = [B]2[/B]

If m% of m is 36, then m is?

If m% of m is 36, then m is?
m% = m/100, so we have:
m/100 * m = 36
m^2/100 = 36
Cross multiply and we get:
m^2 = 3600
We use our [URL='https://www.mathcelebrity.com/radex.php?num=sqrt(3600%2F1)&pl=Simplify+Radical+Expression']radical expressions simplifier[/URL] to get:
m = [B]60[/B]

If one half of a number is 24, what is twice the number?

If one half of a number is 24, what is twice the number?
Let the number be n. We have:
n/2 = 24
Cross multiply, we get n = 48
The problem asks for 2n.
2(48) = [B]96[/B]

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[/B]

If Quinn has 4 times as many quarters as nickels and they have a combined value of 525 cents, how ma

If Quinn has 4 times as many quarters as nickels and they have a combined value of 525 cents, how many of each coin does he have?
Using q for quarters and n for nickels, and using 525 cents as $5.25, we're given two equations:
[LIST=1]
[*]q = 4n
[*]0.25q + 0.05n = 5.25
[/LIST]
Substitute equation (1) into equation (2) for q:
0.25(4n) + 0.05n = 5.25
Multiply through and simplify:
n + 0.05n = 5.25
To solve this equation for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=n%2B0.05n%3D5.25&pl=Solve']type it in our search engine[/URL] and we get:
n = [B]5
[/B]
To get q, we plug in n = 5 into equation (1) above:
q = 4(5)
q = [B]20[/B]

If tanx = 3/4 ,what is cosx?

If tanx = 3/4 ,what is cosx?
tan(x) = sin(x)/cos(x), so we have:
sin(x)/cos(x) = 3/4
cross multiply:
4sin(x) = 3cos(x)
Divide each side by 3 to isolate cos(x):
cos(x) = [B]4sin(x)/3 [/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 the number of professors in a college is P and the number is students S, and there are 14 times a

If the number of professors in a college is P and the number is students S, and there are 14 times as many students as professors
14 times as many means we multiply:
[B]S = 14P[/B]

If the perimeter of a rectangular sign is 44cm and the width is 2cm shorter than half the length, th

If the perimeter of a rectangular sign is 44cm and the width is 2cm shorter than half the length, then what are the length and width?
The perimeter (P) of a rectangle is:
2l + 2w = P
We're given P = 44, so we substitute this into the rectangle perimeter equation:
2l + 2w = 44
We're also given w = 0.5l - 2. Substitute the into the Perimeter equation:
2l + 2(0.5l - 2) = 44
Multiply through and simplify:
2l + l - 4 = 44
Combine like terms:
3l - 4 = 44
[URL='https://www.mathcelebrity.com/1unk.php?num=3l-4%3D44&pl=Solve']Type this equation into the search engine[/URL], and we get:
[B]l = 16[/B]
Substitute this back into the equation w = 0.5l - 2
w = 0.5(16) - 2
w = 8 - 2
[B]w = 6[/B]

if the ratio of 2x to 5y is 3 to 4, what is the ratio of x to y?

if the ratio of 2x to 5y is 3 to 4, what is the ratio of x to y?
Set up our given ratio:
2x/5y = 3/4
Cross multiply:
2x * 4 = 5y * 3
8x = 15y
Divide each side by 8:
8x/8 = 15y/8
x = 15y/8
Now divide each side by y to find x/y:
x/y = 15y/8y
x/y =[B] 15/8[/B]

If the speed of an aeroplane is reduced by 40km/hr, it takes 20 minutes more to cover 1200m. Find th

If the speed of an aeroplane is reduced by 40km/hr, it takes 20 minutes more to cover 1200m. Find the time taken by aeroplane to cover 1200m initially.
We know from the distance formula (d) using rate (r) and time (t) that:
d = rt
Regular speed:
1200 = rt
Divide each side by t, we get:
r = 1200/t
Reduced speed. 20 minutes = 60/20 = 1/3 of an hour. So we multiply 1,200 by 3
3600 = (r - 40)(t + 1/3)
If we multiply 3 by (t + 1/3), we get:
3t + 1
So we have:
3600 = (r - 40)(3t + 1)
Substitute r = 1200/t into the reduced speed equation:
3600 = (1200/t - 40)(3t + 1)
Multiply through and we get:
3600 = 3600 - 120t + 1200/t - 40
Subtract 3,600 from each side
3600 - 3600 = 3600 - 3600 - 120t + 1200/t - 40
The 3600's cancel, so we get:
- 120t + 1200/t - 40 = 0
Multiply each side by t:
-120t^2 - 40t + 1200 = 0
We've got a quadratic equation. To solve for t, [URL='https://www.mathcelebrity.com/quadratic.php?num=-120t%5E2-40t%2B1200%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']we type this in our search engine[/URL] and we get:
t = -10/3 or t = 3. Since time [I]cannot[/I] be negative, our final answer is:
[B]t = 3[/B]

If there are 52 cards in a pack, what is the probability of picking 2 kings in a row when the first

If there are 52 cards in a pack, what is the probability of picking 2 kings in a row when the first card picked is not put back?
4 kings in a deck, and 52 cards in a pack.
First draw, the probability of drawing a king is 4/52.
Second draw, we have 51 cards left since we do not put the first card back, and only 3 Kings left. So the second draw probability for a King is 3/51.
Since each draw is independent, we multiply the first and second draws:
4/52 * 3/51 = [B]12/2652 = 0.0045[/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 x = b/a, then ax = b

If x = b/a, then ax = b
Yes, because we [I]cross multiply[/I] to get:
x = b/a
ax = b

If x is divided by 9, the remainder is 5. What is the remainder if 3x is divided by 9?

If x is divided by 9, the remainder is 5. What is the remainder if 3x is divided by 9?
pick an integer x where when dividing by 9, we get a remainder of 5.
14/9 gives us a remainder of 5.
Now multiply 14 by 3:
14 * 3 = 42
[URL='https://www.mathcelebrity.com/modulus.php?num=42mod9&pl=Calculate+Modulus']42/9 gives a remainder of[/URL] [B]6[/B]

If x/2y = 3/4, what is the value of y/x?

If x/2y = 3/4, what is the value of y/x?
Cross multiply this proportion:
4x = 3(2y)
4x = 6y
Divide each side by x:
4x/x = 6y/x
The x's cancel, and we have:
6y/x = 4
Divide each side by 6:
6y/6x = 4/6
The 6's on the left cancel, we have:
y/x = 4/6
We can simplify this. [URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F6&frac2=3%2F8&pl=Simplify']Type in Simplify 4/6 into the search engine[/URL], and we get 2/3.
y/x = [B]2/3[/B]

If y varies directly as x and inversely as z, then which equation describes the relation?

If y varies directly as x and inversely as z, then which equation describes the relation?
Directly means we multiply, inversely means we divide, so we have a constant k such that:
[B]y = kx/z[/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 can buy 1?3 of a box of chocolates for 6 dollars, how much can you purchase for 4 dollars? Wr

If you can buy 1?3 of a box of chocolates for 6 dollars, how much can you purchase for 4 dollars? Write your answer as a fraction of a box.
Set up a proportion of dollars to boxes where b is the number of boxes for $4:
6/1/3 = 4/b
Cross multiply:
6b = 4/3
Multiply each side by 1/6 to isolate b:
b = 4/18
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=4&num2=18&num3=&pl=GCF+and+LCM']Type in GCF(4,18) into the search engine[/URL]. We get a greatest common factor of 2.
Divide 4 and 18 in the fraction by 2. We get the reduced fraction of:
[B]b = 2/9[/B]

If you divide my brother's age by 3 and then add 20, you will get my age, which is 31. What is my br

If you divide my brother's age by 3 and then add 20, you will get my age, which is 31. What is my brothers age?
Let b be the brother's age.
We're given the following relationship for the brother's age and my age:
b/3 + 20 = 31
Subtract 20 from each side:
b/3 + 20 - 20 = 31 - 20
Cancel the 20's on the left side and we get:
b/3 = 11
Cross multiply, and we get:
b = 3 * 11
b = [B]33
[/B]
Check our work using b = 33 for b/3 + 20 = 31:
33/3 + 20 ? 31
11 + 20 ? 31
31 = 31

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 2000 a company increased its workforce by 50%. In 2001 it decreased its workforce by 50%. How doe

In 2000 a company increased its workforce by 50%. In 2001 it decreased its workforce by 50%. How does the size of its workforce at the end of 2001 compare with the size of the workforce at the beginning of 2000?
Let w be the size of the workforce before any changes. We have:
[LIST]
[*]w(2000) = w(1999) * 1.5 [I](50% increase is the same as multiplying by 1.5)[/I]
[*]w(2001) = w(2000)/1.5 [I](50% decrease is the same as dividing by 1.5)[/I]
[/LIST]
Substitute the first equation back into the second equation
w(2001) = w(1999) * 1.5/1.5
Cancel the 1.5 on top and bottom
w(2001) = w(1999)
This means the workforce had [B]zero net change[/B] from the beginning of 2000 to the end of 2001.

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 a paper bag, 7 of the 15 marbles are yellow. In a cloth bag, 2 of the 15 marbles are yellow. If

In a paper bag, 7 of the 15 marbles are yellow. In a cloth bag, 2 of the 15 marbles are yellow. If Tim randomly draws one marble from each bag, what is the probability that they are both yellow?
Bag 1 probability of drawing yellow is 7/15
Bag 2 probability of drawing yellow is 2/15
Since each event is independent, we multiply each draw to get our final probability:
P(yellow Bag 1)(yellow Bag 2) = P(Yellow Bag 1) * P(Yellow Bag 2)
P(yellow Bag 1)(yellow Bag 2) = 7/15 * 2/15
P(yellow Bag 1)(yellow Bag 2) = [B]14/225[/B]
[URL='https://www.mathcelebrity.com/fraction.php?frac1=14%2F225&frac2=3%2F8&pl=Simplify']Since we cannot simplify this fraction anymore[/URL], our answer is [B]14/225[/B]

In how many ways can a committee consisting of 4 faculty members and 5 students be formed if there a

In how many ways can a committee consisting of 4 faculty members and 5 students be formed if there are 8 faculty members and 9 students eligible to serve on the committee?
We have 8 choose 4 * 9 choose 5 written as : 8C4 * 9C5
[LIST]
[*][URL='http://www.mathcelebrity.com/permutation.php?num=8&den=4&pl=Combinations']8C4[/URL] = 70
[*][URL='http://www.mathcelebrity.com/permutation.php?num=9&den=5&pl=Combinations']9C5[/URL] = 126
[/LIST]
Multiply these together to get [B]8,820[/B]

In the year 1999, Hicham El Guerrouj of Morocco set a new world record when he ran a mile in 3 minut

In the year 1999, Hicham El Guerrouj of Morocco set a new world record when he ran a mile in 3 minutes 43.13 seconds. What was his speed in miles per hour? (Round your answer to the nearest hundredth.)
3 minutes = 60 seconds per minute = 180 seconds
180 seconds + 43.13 seconds = 223.13 seconds
223.13 seconds/3600 seconds per hour = 1 mile/n miles
Cross multiply:
223.13n = 3600
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=223.13n%3D3600&pl=Solve']equation solver[/URL], we get:
n = [B]16.13 miles per hour[/B]

In this class of 4/5 students are right handed. if there are 20 right handed students, what is the t

In this class of 4/5 students are right handed. if there are 20 right handed students, what is the total number of students in this class?
Let x be the total number of students in the class. We have:
4/5x = 20
Cross multiplying or using our [URL='http://www.mathcelebrity.com/1unk.php?num=4x%3D100&pl=Solve']equation calculator[/URL], we get:
4x = 100
Divide each side by 4
[B]x = 25[/B]

In Trina's desk drawer, there are 15 paper clips and 18 rubber bands. In Kirk's office supply tray,

In Trina's desk drawer, there are 15 paper clips and 18 rubber bands. In Kirk's office supply tray, there are 13 paper clips and 16 rubber bands. Who has a higher ratio of paper clips to rubber bands?
Trina: 15/18
Kirk: 13/16
We want common denominators to compare, so we get a greatest common factor (GCF) for 16 and 18.
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=16&num2=18&num3=&pl=GCF+and+LCM']Running this through our search engine[/URL], we get GCF(16, 18) = 144
For Trina, 144/18 = 8
For Kirk, 144/16 = 9
We multiply Trina's fraction, top and bottom by 8:
15 * 8 / 18 * 8
120/144
We multiply Trina's fraction, top and bottom by 8:
13 * 8 / 16 * 8
104/144
[B]Trina[/B] has more in her numerator, so her ratio of paper clips to rubber bands is greater.

index form of (5^3)^6

Index form of (5^3)^6
Index form is written as a number raised to a power.
Let's simplify by multiply the exponents. Since 6*3 = 18, We have:
[B]5^18[/B]

It takes 60 minutes for 7 people to paint 5 walls. How many minutes does it take 10 people to paint

It takes 60 minutes for 7 people to paint 5 walls. How many minutes does it take 10 people to paint 10 walls?
Rate * Time = Output
Let "Rate" (r) be the rate at which [B]one person[/B] works.
So we have:
7r * 60 = 5
Multiply through and simplify:
420r = 5
Divide each side by 5 to isolate r:
r = 1/84
So now we want to find out how many minutes it takes 10 people to paint 10 walls using this rate:
10rt = 10
With r = 1/84, we have:
10t/84 = 10
Cross multiply:
10t = 840
To solve for t, we t[URL='https://www.mathcelebrity.com/1unk.php?num=10t%3D840&pl=Solve']ype this equation into our search engine[/URL] and we get:
t = [B]84 minutes[/B]

j - m/4 = 4k for m

j - m/4 = 4k for m
Multiply each side by 4:
4j - 4m/4 = 4(4k)
Simplify:
4j - m = 16k
Add m to each side:
4j - m + m = 16k + m
The m's cancel on the left side, so we have:
4j = 16k + m
Subtract 16k from each side:
4j - 16k = 16k - 16k + m
The 16k cancels on the right side, so we're left with:
[B]m = 4j - 16k or 4(j - 4k)[/B]

jamie needs 3 cups of flour and 4 cups of sugar how many cups of sugar will she need if she uses 9 c

jamie needs 3 cups of flour and 4 cups of sugar how many cups of sugar will she need if she uses 9 cups of flour?
Set up a proportion of flour/sugar:
3/4 = 9/x
[URL='http://www.mathcelebrity.com/prop.php?num1=3&num2=9&den1=4&den2=x&propsign=%3D&pl=Calculate+missing+proportion+value']Cross multiply or enter that into the search engine[/URL]
3x = 36
[B]x = 12[/B]

Jerry, an electrician, worked 7 months out the year. What percent of the year did he work?

Jerry, an electrician, worked 7 months out the year. What percent of the year did he work?
We know that there are 12 months in a year.
Percentage worked = Months worked in a year / months in a year * 100%
Percentage worked = 7/12 * 100%
Percentage worked = 0.5833333 * 100%
Multiplying by 100 means we shift the decimal place 2 spaces to the right:
Percentage worked = [B]58.33%[/B]

Joe had saved $264. He spent 3/8 of that to buy a camera. How much did the camera cost?

Joe had saved $264. He spent 3/8 of that to buy a camera. How much did the camera cost?
[URL='https://www.mathcelebrity.com/fraction.php?frac1=264&frac2=3%2F8&pl=Multiply']264 *3/8[/URL] = [B]99[/B]

joe plans to watch 3 movies each month. white an equation to represent the total number of movies n

joe plans to watch 3 movies each month. white an equation to represent the total number of movies n that he will watch in m months
Build movie equation. 3 movies per month at m months means we multiply:
[B]n = 3m[/B]

John read the first 114 pages of a novel, which was 3 pages less than 1/3

John read the first 114 pages of a novel, which was 3 pages less than 1/3
Set up the equation for the number of pages (p) in the novel
1/3p - 3 = 114
Add 3 to each side
1/3p = 117
Multiply each side by 3
[B]p = 351[/B]

John read the first 114 pages of a novel, which was 3 pages less than 1/3 of the novel.

John read the first 114 pages of a novel, which was 3 pages less than 1/3 of the novel.
Let n be the number of pages in the novel. We have:
1/3n - 3 = 114
Multiply each side by 3:
n - 9 = 342
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=n-9%3D342&pl=Solve']equation solver[/URL], we get [B]n = 351[/B].

k add 2 multiply by 6 then subtract 8

k add 2 multiply by 6 then subtract 8
k add 2:
k + 2
Multiply by 6:
6(k + 2)
Then subtract 8:
[B]6(k + 2) - 8[/B]

k add d , multiply by e , then subtract f .

k add d , multiply by e , then subtract f .
[LIST]
[*]k add d: k + d
[*]Multiply by e: e(k + d)
[*]Then subtract f: [B]e(k + d) - f[/B]
[/LIST]

K varies inversely with square root of m and directly with the cube of n.

K varies inversely with square root of m and directly with the cube of n.
[LIST]
[*]We take a constant c as our constant of proportionality.
[*]The word inversely means we divide
[*]The word directly means we multiply
[/LIST]
[B]k = cn^3/sqrt(m)[/B]

kate is twice as old as her sister mars. the sum of their ages is 24. find their ages.

kate is twice as old as her sister mars. the sum of their ages is 24. find their ages.
Let k be Kate's age
Let m be Mars's age
We're given two equations:
[LIST=1]
[*]k = 2m. (Because twice means multiply by 2)
[*]k + m = 24
[/LIST]
Substitute equation (1) for k into equation (2):
2m + m = 24
T o solve for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=2m%2Bm%3D24&pl=Solve']type this equation into our math engine[/URL]:
m = [B]8
[/B]
We want to solve for k using m= 8. Substitute this into equation 1
k = 2(8)
k = [B]16
[/B]
Check our work for equation 1
16 = 2 * 8
16 = 16
Check our work for equation 2
16 + 8 ? 24
24 = 24
[MEDIA=youtube]TJMTRYP-Ct8[/MEDIA]

Kelly took clothes to the cleaners 3 times last month. First, she brought 4 shirts and 1 pair of sla

Kelly took clothes to the cleaners 3 times last month. First, she brought 4 shirts and 1 pair of slacks and paid11.45. Then she brought 5 shirts, 3 pairs of slacks, and 1 sports coat and paid 27.41. Finally, she brought 5 shirts and 1 sports coat and paid 16.94. How much was she charged for each shirt, each pair of slacks, and each sports coat?
Let s be the cost of shirts, p be the cost of slacks, and c be the cost of sports coats. We're given:
[LIST=1]
[*]4s + p = 11.45
[*]5s + 3p + c = 27.41
[*]5s + c = 16.94
[/LIST]
Rearrange (1) by subtracting 4s from each side:
p = 11.45 - 4s
Rearrange (3)by subtracting 5s from each side:
c = 16.94 - 5s
Take those rearranged equations, and plug them into (2):
5s + 3(11.45 - 4s) + (16.94 - 5s) = 27.41
Multiply through:
5s + 34.35 - 12s + 16.94 - 5s = 27.41
[URL='https://www.mathcelebrity.com/1unk.php?num=5s%2B34.35-12s%2B16.94-5s%3D27.41&pl=Solve']Group like terms using our equation calculator [/URL]and we get:
[B]s = 1.99 [/B] <-- Shirt Cost
Plug s = 1.99 into modified equation (1):
p = 11.45 - 4(1.99)
p = 11.45 - 7.96
[B]p = 3.49[/B] <-- Slacks Cost
Plug s = 1.99 into modified equation (3):
c = 16.94 - 5(1.99)
c = 16.94 - 9.95
[B]c = 6.99[/B] <-- Sports Coat cost

Kendra is half as old as Morgan and 3 years younger than Lizzie. The total of their ages is 39. How

Kendra is half as old as Morgan and 3 years younger than Lizzie. The total of their ages is 39. How old are they?
Let k be Kendra's age, m be Morgan's age, and l be Lizzie's age. We're given:
[LIST=1]
[*]k = 0.5m
[*]k = l - 3
[*]k + l + m = 39
[/LIST]
Rearranging (1) by multiplying each side by 2, we have:
m = 2k
Rearranging (2) by adding 3 to each side, we have:
l = k + 3
Substituting these new values into (3), we have:
k + (k + 3) + (2k) = 39
Group like terms:
(k + k + 2k) + 3 = 39
4k + 3 = 39
[URL='https://www.mathcelebrity.com/1unk.php?num=4k%2B3%3D39&pl=Solve']Type this equation into the search engine[/URL], and we get:
[B]k = 9
[/B]
Substitute this back into (1), we have:
m = 2(9)
[B]m = 18
[/B]
Substitute this back into (2), we have:
l = (9) + 3
[B][B]l = 12[/B][/B]

Kimberly wants to become a member of the desert squad at a big catering company very badly, but she

Kimberly wants to become a member of the desert squad at a big catering company very badly, but she must pass three difficult tests to do so. On the first Terrifying Tiramisu test she scored a 68. On the second the challenging Chocalate-Sprinkled Creme Brulee she scored a 72. If kimberly needs an average of 60 on all three tests to become a member on the squad what is the lowest score she can make on her third and final test
This is a missing average problem.
Given 2 scores of 68, 72, what should be score number 3 in order to attain an average score of 60?
[SIZE=5][B]Setup Average Equation:[/B][/SIZE]
Average = (Sum of our 2 numbers + unknown score of [I]x)/[/I]Total Numbers
60 = (68 + 72 + x)/3
[SIZE=5][B]Cross Multiply[/B][/SIZE]
68 + 72 + x = 60 x 3
x + 140 = 180
[SIZE=5][B]Subtract 140 from both sides of the equation to isolate x:[/B][/SIZE]
x + 140 - 140 = 180 - 140
x = [B]40[/B]

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].

Kyle can walk ½ mile in ¼ of an hour. What is Kyle’s speed in miles per hour?

Kyle can walk ½ mile in ¼ of an hour. What is Kyle’s speed in miles per hour?
We write this in terms of miles per hour as:
1/2 / 1/4
We want 1 for the denominator to represent an hour, so we multiply top and bottom of the fraction by 4:
4/2 / 4/4
2 / 1
[B]2 miles per hour[/B]

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]

Layla buys 2 1/2pounds of chocolate for 3.50 how much is she paying for a pound of chocolate

Layla buys 2 1/2pounds of chocolate for 3.50 how much is she paying for a pound of chocolate?
[URL='https://www.mathcelebrity.com/fraction.php?frac1=2%261%2F2&frac2=3%2F8&pl=Simplify']Using our mixed fraction converter[/URL], 2&1/2 = 5/2
Cost per pound = 3.50 / 5/2 pounds
Dividing by 5/2 is the same as multiplying by the reciprocal 2/5:
3.50 * 2/5
7/5
[B]$1.40 per pound[/B]

Leilani can read 20 pages in 2 minutes. if she can maintain this page, how many pages can she read i

Leilani can read 20 pages in 2 minutes. if she can maintain this page, how many pages can she read in an hour?
We know that 1 hour is 60 minutes.
Let p be the number of pages Leilani can read in 1 hour (60 minutes)
The read rate is constant, so we can build a proportion.
20 pages /2 minutes = p/60
We can cross multiply:
Numerator 1 * Denominator 2 = Denominator 1 * Numerator 2
[SIZE=5][B]Solving for Numerator 2 we get:[/B][/SIZE]
Numerator 2 = Numerator 1 * Denominator 2/Denominator 1
[SIZE=5][B]Evaluating and simplifying using your input values we get:[/B][/SIZE]
p = 20 * 60/ 2
p = 1200/2
p = [B]600[/B]

Let x be the dog’s age in years. What is the dog’s age when he is thrice as old?

Let x be the dog’s age in years. What is the dog’s age when he is thrice as old?
Thrice means triple, or multiply by 3. So we have the future age as:
[B]3x[/B]

Lisa has 32 nickels this is one third of her coins how many coins does she have

Lisa has 32 nickels this is one third of her coins how many coins does she have
Let x be the total amount of coins. we have:
32 = x/3
Cross multiply, we get:
[B]x = 96[/B]

Local salesman receives a base salary of $650 monthly. He also receives a commission of 11% on all s

Local salesman receives a base salary of $650 monthly. He also receives a commission of 11% on all sales over $1500. How much would he have to sell in one month if he needed to have $3000
Let the Sales amount be s. We have:
Sales over 1,500 is written as s - 1500
11% is also 0.11 as a decimal, so we have:
0.11(s - 1500) + 650 = 3000
Multiply through:
0.11s - 165 + 650 = 3500
0.11s + 485 = 3500
To solve this equation for s, [URL='https://www.mathcelebrity.com/1unk.php?num=0.11s%2B485%3D3500&pl=Solve']we type it in our search engine[/URL] and we get:
s = [B]27,409.10[/B]

Lucas has nickels,dimes,and quarters in the ratio 1:3:2. If 10 of Lucas coins are quarters, how many

Lucas has nickels,dimes,and quarters in the ratio 1:3:2. If 10 of Lucas coins are quarters, how many nickels and dimes does Lucas have?
1 + 3 + 2 = 6.
Quarters account for 2/6 which is 1/3 of the total coin count. Let x be the total number of coins. We have:
1/3x = 10
Multiply each side by 3
x = 30
We have the following ratios and totals:
[LIST]
[*]Nickels: 1/6 * 30 = [B]5 nickels[/B]
[*]Dimes: 3/6 * 30 = [B]15 dimes[/B]
[*]Quarters: 2/6 * 30 = [B]10 quarters[/B]
[/LIST]

m is inversely proportional to the square of p-1 when p=4 m=5 find m when p=6

m is inversely proportional to the square of p-1 when p=4 and m=5. find m when p=6
Inversely proportional means there is a constant k such that:
m = k/(p - 1)^2
When p = 4 and m = 5, we have:
5 = k/(4 - 1)^2
5 = k/3^2
5 = k/9
[U]Cross multiply:[/U]
k = 45
[U]The problems asks for m when p = 6. And we also now know that k = 45. So plug in the numbers:[/U]
m = k/(p - 1)^2
m = 45/(6 - 1)^2
m = 45/5^2
m = 45/25
m = [B]1.8[/B]

M/n = p-6 for m

M/n = p-6 for m
Solve this literal equation by multiplying each side by n to isolate M:
Mn/n = n(p - 6)
Cancelling the n terms on the left side, we get:
[B]M = n(p - 6)[/B]

m/x = k-6 for m

m/x = k-6 for m
To solve this literal equation, multiply each side by x:
x(m/x) = x(k - 6)
The x's cancel on the left side, so we get:
m = [B]x(k - 6)[/B]

m=u/k-r/k for k

m=u/k-r/k for k
Multiply both sides by k to eliminate the k denominator:
km = uk/k - rk/k
Cancel the k's on the right side and we get
km = u - r
Divide each side by m:
km/m = (u - r)/m
Cancel the m on the left side:
[B]k = (u - r)/m[/B]

Marcus drives a machine that paints lines along the highway. He needs to paint a line that is 9/10 o

Marcus drives a machine that paints lines along the highway. He needs to paint a line that is 9/10 of a mile long. He is 2/3 of the way done when he runs out of paint. What fraction of a mile has he painted?
Marcus has painted 2/3 of 9/10.
If we [URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F3&frac2=9%2F10&pl=Multiply']type 2/3 of 91/20 in our search engine[/URL], we get:
[B]3/5[/B]

Maria bought 7 boxes. A week later half of all her boxes were destroyed in a fire. There are now onl

Maria bought 7 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?
[U]Let x be the starting box number. We have:[/U]
(x + 7)/2 = 22
[U]Cross multiply[/U]
x + 7 = 44
[U]Subtract 7 from each side[/U]
[B]x = 37[/B]

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. How many did she start with?
Take this in parts
[LIST=1]
[*]Maria starts with b boxes.
[*]She buys seven more. So she has b + 7 boxes
[*]A week later, half of all her boxes are destroyed in a fire. Which means she's left with 1/2. (b + 7)/2
[*]Now she has 22 boxes. So we set (b + 7)/2 = 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']Typing this equation into our search engine and solving for b[/URL], we get:
[B]b = 37[/B]

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]

Marissa has 24 coins in quarters and nickels. She has 3 dollars. How many of the coins are quarters?

Let n be the number of nickels and q be the number of quarters.
We have two equations:
(1) n + q = 24
(2) 0.05n + 0.25q = 3
Rearrange (1) to solve for n in terms of q for another equation (3)
(3) n = 24 - q
Plug (3) into (2)
0.05(24 - q) + 0.25q = 3
Multiply through:
1.2 - 0.05q + 0.25q = 3
Combine q terms
0.2q + 1.2 = 3
Subtract 1.2 from each side:
0.2q = 1.8
Divide each side by 0.2
[B]q = 9[/B]

Max's age is 2 more than his fathers age divided by 4. Max is 13 years old. How old is his dad?

Max's age is 2 more than his fathers age divided by 4. Max is 13 years old. How old is his dad?
Let Max's father be age f. We're given:
(f + 2)/4 = 13
Cross Multiply:
f + 2 = 52
[URL='https://www.mathcelebrity.com/1unk.php?num=f%2B2%3D52&pl=Solve']Typing this equation into the search engine[/URL], we get:
f = [B]50[/B]

mcubemultipliedbyntothefourthpower

mcubemultipliedbyntothefourthpower
m cubed means we raise m to the 3rd power:
m^3
n to the fourth power:
n^4
Multiply both expressions together:
[B]m^3n^4[/B]

Michael invited 30 of his friends to his part and a third of guest arrived late how many arrived on

Michael invited 30 of his friends to his part and a third of guest arrived late how many arrived on time
If 1/3 arrived late, then
[URL='https://www.mathcelebrity.com/fraction.php?frac1=1&frac2=1%2F3&pl=Subtract']1 - 1/3[/URL] = 2/3 arrived on time
Guests who arrived on time = 2/3 of 30
[URL='https://www.mathcelebrity.com/fraction.php?frac1=30&frac2=2/3&pl=Multiply']Guests who arrived on time[/URL] = [B]20[/B]

Mindy and troy combined ate 9 pieces of the wedding cake. Mindy ate 3 pieces of cake troy had 1/4 of

Mindy and troy combined ate 9 pieces of the wedding cake. Mindy ate 3 pieces of cake troy had 1/4 of the total cake. Write an equation to determine how many pieces of cake (c) that were in total
Let c be the total number of pieces of cake. Let m be the number of pieces Mindy ate. Let t be the number of pieces Troy ate. We have the following given equations:
[LIST]
[*]m + t = 9
[*]m = 3
[*]t = 1/4c
[/LIST]
Combining (2) and (3) into (1), we have:
3 + 1/4c = 9
Subtract 3 from each side:
1/4c = 6
Cross multiply:
[B]c = 24[/B]

Monomials

Free Monomials Calculator - This calculator will raise a monomial to a power,multiply monomials, or divide monomials.

Multiply 0 by 3 and add 4

Multiply 0 by 3 and add 4
multiply 0 by 3:
0 * 3
Then add 4:
[B]0 * 3 + 4 <--- [/B][I]This is our algebraic expression.[/I]
If we want to evaluate this expression, we [URL='https://www.mathcelebrity.com/order-of-operations-calculator.php?num=0%2A3%2B4&pl=Perform+Order+of+Operations']type it in our search engine[/URL] and we get:
[B]4[/B]

multiply 3 by the difference of u and t

multiply 3 by the difference of u and t
Take this algebraic expression in parts:
The difference of u and t means we subtract t from u
u - t
Multiply this difference by 3:
[B]3(u - t)[/B]

Multiply 3w by the sum of v and 2u

Multiply 3w by the sum of v and 2u
the sum of v and 2u:
v + 2u
Multiply 3w by the sum of v and 2u
[B]3w(v + 2u)[/B]

multiply 5 and sum of twice of d and 10

multiply 5 and sum of twice of d and 10
Twice d means we multiply d by 2:
2d
The sum of twice d and 10 means we add 2d to 10
2d + 10
We multiply this quantity by 5:
[B]5(2d + 10)[/B]

multiply 9 by 3, subtract y from the result

multiply 9 by 3, subtract y from the result
Multiply 9 by 3
9 * 3
Subtract y from the result
[B]9 * 3 - y[/B]

multiply 9 by the quotient of 4 and z

multiply 9 by the quotient of 4 and z
Quotient of 4 and z is written as:
4/z
Multiply this quotient by 9:
9(4)/z
Multiplying the top, we get:
[B]36/z[/B]

multiply a number by 4 and then subtract the answer from 30

multiply a number by 4 and then subtract the answer from 30
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
Multiply this number by 4:
4x
Subtract the answer from 30:
[B]30 - 4x[/B]

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]

Multiply c by five and square the answer

Multiply c by five and square the answer
Multiply c by five:
5c
Square the answer means we raise 5c to the power of 2:
[B](5c)^2 [/B]
This can also be written as [B]25c^2[/B]

multiply k by 5.8, and then subtract 3.09 from the product

multiply k by 5.8, and then subtract 3.09 from the product
Take this algebraic expression in pieces:
[U]Multiply k by 5.8:[/U]
5.8k
[U]Then subtract 3.09 from the product[/U]
[B]5.8k - 3.09[/B]

multiply k by 6q

multiply k by 6q
[B]6qk[/B]

multiply m by 2, then add n to the result

[U]Multiply m by 2[/U]
2m
[U]Add n to the result[/U]
[B]2m + n[/B]

multiply m by 5, double the result, then multiply 10 by what you have

multiply m by 5, double the result, then multiply 10 by what you have
Take this algebraic expression in parts:
[LIST]
[*]Multiply m by 5: 5m
[*]double the result means multiply 5m by 2: 2(5m) = 10m
[*]Multiply 10 by what you have means multiply 10 by the result of 10m above:
[/LIST]
10(10m) = [B]100m[/B]

multiply p by itself

multiply p by itself
Two ways to express this
[B]p * p
p^2[/B]

multiply r by t, add the result to u, then multiply what you have by s

multiply r by t, add the result to u, then multiply what you have by s
Take this algebraic expression in parts:
[LIST=1]
[*]Multiply r by t: rt
[*]Add the result to u means we add rt to u: u + r
[*]Multiply what you have by s. This means we take the result in #2, u + r, and multiply it by s:
[/LIST]
[B]s(u + r)[/B]

Multiply t by 4 then multiply s by the result

Multiply t by 4 then multiply s by the result
Multiply t by 4
4t
Then multiply s by the result
[B]4st[/B]

multiply t by u, add the to v, then triple what you have

multiply t by u, add the to v, then triple what you have
Multiply t by u:
tu
Add this to v:
v + tu
Then triple what you have - This means we multiply the expression above by 3:
[B]3(v + tu)[/B]

Multiply the difference of 3 and q by p

Multiply the difference of 3 and q by p.
Take this algebraic expression in pieces:
[B][U]Step 1: The difference of 3 and q[/U][/B]
The word [I]difference[/I] means we subtract the variable q from 3
3 - q
[B][U]Step 2: Multiply the expression 3 - q by p:[/U]
p(3 - q)[/B]

multiply the sum of 2p and q by3

multiply the sum of 2p and q by3
The sum of 2p and q:
2p + q
Multiply the sum by 3:
[B]3(2p + q)[/B]

multiply u by s, multiply the result by v, then multiply t

multiply u by s, multiply the result by v, then multiply t
Take this algebraic expression in parts:
[LIST]
[*]Multiply u by s: us
[*]Multiply the result by v: usv
[*]Then multiply by t: [B]usvt[/B]
[/LIST]

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.

mx=ac/np for n

mx=ac/np for n
Cross multiply:
mnpx = ac
Divide each side by mpx:
mnpx/mpx = ac/mpx
Cancel the mpx on the right side:
n = [B]ac/mpx[/B]

n + n/2 + n/4 + n/8 + n/16 = 19,375

n + n/2 + n/4 + n/8 + n/16 = 19,375
Convert to like fractions with a denominator of 16:
16n/16 + 8n/16 + 4n/16 + +2n/16 + n/16 = 19,375
31n/16 = 19,375
Cross multiply:
31n = 19,375 * 16
31n = 310000
Divide each side by 1:
31n/31 = 310000/31
n = [B]10,000[/B]

n is tripled then decreased by 5

n is tripled then decreased by 5
n is tripled means we multiply n by 3:
3n
Decreased by 5 means we subtract 5 from 3n:
[B]3n - 5[/B]

N squared multiplied by the difference of n and 3

N squared multiplied by the difference of n and 3
n squared means we raise n to the power of 2:
n^2
The difference of n and 3 means we subtract 3 from n:
n - 3
Now we multiply both terms together:
[B]n^2(n - 3)[/B]

n subtract m, multiply by c, then add w

n subtract m, multiply by c, then add w
Take this algebraic expression in pieces:
[LIST]
[*]n subtract m: n - m
[*]multiply by c: c(n - m)
[*]Then add w: [B]c(n - m) + w[/B]
[/LIST]

n/11 = 11

n/11 = 11
Cross multiply:
n = 11 * 11
n = [B]121[/B]

Nate jars 2 liters of jam everyday. How many days did Nate spend making jam if he jarred 8 liters of

Nate jars 2 liters of jam everyday. How many days did Nate spend making jam if he jarred 8 liters of jam?
2 liters per 1 day and 8 liters per x days.
Set up a proportion:
2/1 = 8/x
Cross multiply:
2x = 8
Divide each side by 2
x = [B]4 days[/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]

numerator of a fraction is 5 less than its denominator. if 1 is added to the numerator and to the de

numerator of a fraction is 5 less than its denominator. if 1 is added to the numerator and to the denominator the new fraction is 2/3. find the fraction.
Let n be the numerator.
Let d be the denominator.
We're given 2 equations:
[LIST=1]
[*]n = d - 5
[*](n + 1)/(d + 1) = 2/3
[/LIST]
Substitute equation (1) into equation (2) for n:
(d - 5 + 1) / (d + 1) = 2/3
(d - 4) / (d + 1) = 2/3
Cross multiply:
3(d - 4) = 2(d + 1)
To solve this equation for d, we type it in our search engine and we get:
d = 14
Substitute d = 14 into equation (1) to solve for n:
n = 14 - 5
n = 9
Therefore, our fraction n/d is:
[B]9/14[/B]

ohn read the first 114 pages of a novel, which was 3 pages less than1/3 of the novel

Let p be the novel pages.
We have 1/3p - 3 = 114
Add 3 to each side
1/3p = 117
Multiply each side by 3
p = 351

One third of the bagels in a bakery are sesame bagels. There are 72 sesame bagels.

One third of the bagels in a bakery are sesame bagels. There are 72 sesame bagels.
Set up our equation where b is the number of total bagels
72 = b/3
Multiply each side by 3
[B]b = 216[/B]

one third of the sum of 4 and P

The sum of 4 and p is written as:
4 + p
We then take 1/3 of that, or multiply:
1/3(4 + p)

one-fifth of forty-five

one-fifth of forty-five
one-fifth is 1/4
forty-five is 45
When you see a fraction then the word of and then a number, it means you multiply:
1/5 * 45
45/5
[B]9[/B]

One-half the sum of 5 and t

One-half the sum of 5 and t
The sum of 5 and t:
5 + t
One-half of this means we multiply 5 + t by 1/2
[B](5 + t)/2[/B]

opposite of twice the quotient of a and a

opposite of twice the quotient of a and a
the quotient of a and a:
a/a
1
Twice the quotient of a and a
2(1)
2
Opposite means multiply 2 by -1:
-1 * 2
[B]-2[/B]

Out of the 485 Cookies for the bake sale, 2/5 were chocolate chip. Estimate the number of chocolate

Out of the 485 Cookies for the bake sale, 2/5 were chocolate chip. Estimate the number of chocolate chips
We want 2/5 of 485. We [URL='https://www.mathcelebrity.com/fraction.php?frac1=485&frac2=2/5&pl=Multiply']type this in our search engine[/URL] and we get;
[B]194[/B]

P is twice the length plus twice the width

P is twice the length plus twice the width
Let the length be l. Let the width be w. The phrase [I]twice[/I] means we multiply by 2. We have:
[B]2l + 2w = P[/B]

P varies directly as q and the square of r and inversely as s

P varies directly as q and the square of r and inversely as s
There exists a constant k such that:
p = kqr^2/s
[I]Note: Direct variations multiply and inverse variations divide[/I]

p varies directly as the square of r and inversely as q and s and p = 40 when q = 5, r = 4 and s = 6

p varies directly as the square of r and inversely as q and s and p = 40 when q = 5, r = 4 and s = 6, what is the equation of variation?
Two rules of variation:
[LIST=1]
[*]Varies directly means we multiply
[*]Varies inversely means we divide
[/LIST]
There exists a constant k such that our initial equation of variation is:
p = kr^2/qs
[B][/B]
With p = 40 when q = 5, r = 4 and s = 6, we have:
4^2k / 5 * 6 = 40
16k/30 = 40
Cross multiply:
16k = 40 * 30
16k = 1200
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=16k%3D1200&pl=Solve']equation calculator[/URL], we get:
k = [B]75[/B]
So our final equation of variation is:
[B]p = 75r^2/qs[/B]

p/q = f/q- f for f

p/q = f/q- f for f
Isolate f in this literal equation.
Factor out f on the right side:
p/q = f(1/q - 1)
Rewriting the term in parentheses, we get:
p/q = f(1 - q)/q
Cross multiply:
f = pq/q(1 - q)
Cancelling the q/q on the right side, we get:
f = [B]p/(1 - q)[/B]

P/v=nr/t for r

P/v=nr/t for r
Cross multiply to solve this literal equation:
Pt = nrv
Divide each side of the equation by nv:
Pt/nv = nrv/nv
Cancel the nv's on the right side, we get:
r = [B]Pt/nv[/B]

P=15+5d/11 for d

Subtract 15 from each side:
5d/11 = P - 15
Multiply each side by 11
5d = 11p - 165
Divide each side of the equation by d:
d = (11p - 165)
------------
5

P=ab/c, for c

P=ab/c, for c
Cross multiply:
cP = ab
Divide each side by P
[B]c = (ab)/P[/B]

Pablo is saving money to buy a game. So far he has saved $22, which is one-half of the total cost

Pablo is saving money to buy a game. So far he has saved $22, which is one-half of the total cost of the game. How much does the game cost?
22 is 1/2 of the cost, so multiply 22 * 2 to get the [B]full cost of $44[/B].

Penny bought a new car for $25,000. The value of the car has decreased in value at rate of 3% each

Penny bought a new car for $25,000. The value of the car has decreased in value at rate of 3% each year since. Let x = the number of years since 2010 and y = the value of the car. What will the value of the car be in 2020? Write the equation, using the variables above, that represents this situation and solve the problem, showing the calculation you did to get your solution. Round your answer to the nearest whole number.
We have the equation y(x):
y(x) = 25,000(0.97)^x <-- Since a 3 % decrease is the same as multiplying the starting value by 0.97
The problem asks for y(2020). So x = 2020 - 2010 = 10.
y(10) = 25,000(0.97)^10
y(10) = 25,000(0.73742412689)
y(10) = [B]18,435.60[/B]

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

product of r plus 7 and 4

product of r plus 7 and 4
r plus 7 means we add 7 to r:
r + 7
The product means we multiply the expression r + a 7 by 4:
[B]4(r + 7)[/B]

pv/t = ab/c for c

pv/t = ab/c for c
Cross multiply:
cpv = abt
Divide each side of the equation by pv to isolate c:
cpv/pv = abt/pv
Cancel the pv terms on the left side and we get:
c = [B]abt/pv[/B]

q=c+d/5 for d

q=c+d/5 for d
Subtract c from each side to solve this literal equation:
q - c = c - c + d/5
Cancel the c's on the right side, we get
d/5 = q - c
Multiply each side by 5:
5d/5 = 5(q - c)
Cancel the 5's on the left side, we get:
[B]d = 5(q - c)[/B]

r squared plus the product of 3 and s plus 5

r squared plus the product of 3 and s plus 5
r squared means we raise r to the power of 2
r^2
The product of 3 and s means we multiply s by 3:
3s
plus 5 means we add
3s + 5
R squared plus means we add r^2:
[B]r^2 + 3s + 5[/B]

r varies directly with s and inversely with the square root of t

r varies directly with s and inversely with the square root of t
Varies directly means we multiply
Varies inversely means we divide
There exists a constant k such that:
[B]r = ks/sqrt(t)[/B]

r=l^2w/2 for w

r=l^2w/2 for w
Solve this literal equation by isolating w.
Cross multiply:
2r = l^2w
Divide each side by l^2
w = [B]2r/l^2[/B]

Raise 3 to 9th power then multiply b by the result

Raise 3 to 9th power then multiply b by the result
3^9th power:
3^9
Multiply b by the result?
[B]3^9 * b[/B]

raise 6 to the 4th power, add h to the result, then multiply what you have by 8

raise 6 to the 4th power, add h to the result, then multiply what you have by 8
Raise 6 to the 4th power:
6^4
add h to the result:
6^4 + h
Then multiply what we have by 8:
[B]8(6^4 + h)[/B]

Raise c to the 7th power, divide the result by 4, then triple what you have

Raise c to the 7th power, divide the result by 4, then triple what you have.
Take this algebraic expression in pieces.
Raise c to the 7th power:
c^7
Divide the result by 4, means we divide c^7 by 4
c^7 / 4
Triple what you have means multiply c^7 / 4 by 3
[B]3(c^7 / 4)[/B]

raise f to the 8th power then multiply the result by g

f to the 8th power:
f^8
Multiply the result by g
(f^8) * g

Raise f to the 8th power, divide the result by 5, then multiply 10

Raise f to the 8th power, divide the result by 5, then multiply 10
f to the 8th power means we raise f to the power of 8 using an exponent:
f^8
Divide f^8 by 5
(f^8)/5
Now multiply this by 10:
10(f^8)/5
We can simplify this algebraic expression by dividing 10/5 to get 2 on top:
2[B](f^8)[/B]

Raise p to the 9th power, multiply the result by q, then divide what you have by r

Raise p to the 9th power, multiply the result by q, then divide what you have by r.
Take this in steps:
[LIST]
[*]Raise p to the 9th power: p^9
[*]Multiply the result by q: qp^9
[*]Divide what you have (the result) by r: qp^9/r
[/LIST]
[B](qp^9)/r[/B]

raise z to the 2nd power, multiply 8 by the result then subtract what you have from 4

raise z to the 2nd power, multiply 8 by the result then subtract what you have from 4
Take this algebraic expression in pieces:
[LIST]
[*]Raise z to the 2nd power: z^2
[*]Multiply by 8: 8z^2
[*]Subtract what you have from 4:
[/LIST]
[B]4 - 8z^2[/B]

Rearrange the following equation to make x the subject, and select the correct rearrangement from th

Rearrange the following equation to make x the subject, and select the correct rearrangement from the list below
3x + 2y 1
-------- = ---
4x + y 3
[LIST]
[*]x = 7y/13
[*]x = 7y/5
[*]x = -7y
[*]x = -3y
[*]x = 3y/5
[*]x = -5y/13
[*]x = -y
[/LIST]
Cross multiply:
3(3x - 2y) = 4x + y
Multiply the left side through
9x - 6y = 4x + y
Subtract 4x from each side and add 6y to each side
5x = 7y
Divide each side by 5 to isolate x, the subject of an equation is the variable to the left
[B]x = 7y/5[/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]

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

Sam bought 8 new basketball trading cards to add to his collection. The next day his dog ate half of his collection. There are now only 40 cards left. How many cards did Sam start with?
Let the starting about of cards be s.
Sam adds 8 new cards, so he has s + 8.
Then the dog ate half, so he's left with half. Sam is left with 40 cards:
(s + 8)/2 = 40
Cross multiply:
s + 8 = 80
[URL='https://www.mathcelebrity.com/1unk.php?num=s%2B8%3D80&pl=Solve']Type s + 8 = 80 into the search engine[/URL], and we get [B]s = 72[/B]

Sara has a box of candies. In the box there are 8 pink candies, 7 purple candies and 5 blue candies.

Sara has a box of candies. In the box there are 8 pink candies, 7 purple candies and 5 blue candies. She takes one candy and records its color. She then puts it back in the box and draws another candy. What is the probability of taking out a pink candy followed by a blue candy?
[B][U]Calculate the total number of candies:[/U][/B]
Total candies = Pink + Purple + Blue
Total candies = 8 + 7 + 5
Total candies = 20
[B][U]Calculate the probability of drawing one pink candy:[/U][/B]
P(Pink) = 8/20
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=8%2F20&frac2=3%2F8&pl=Simplify']fraction reduction calculator[/URL], we get:
P(Pink) = 2/5
[B][U]Calculate the probability of drawing one blue candy:[/U][/B]
P(Blue) = 5/20 <-- [I]20 options since Sara replaced her first draw[/I]
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=5%2F20&frac2=3%2F8&pl=Simplify']fraction reduction calculator[/URL], we get:
P(Blue) = 1/4
The problem asks for the probability of a Pink followed by a Blue. Since each event is independent, we multiply:
P(Pink, Blue) = P(Pink) * P(Blue)
P(Pink, Blue) = 2/5 * 1/4
P(Pink, Blue) = 2/20
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F20&frac2=3%2F8&pl=Simplify']fraction reduction calculator[/URL], we get:
P(Pink, Blue) = [B]1/10 or 10%[/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]

Serial numbers for a product are to be using 3 letters followed by 4 digits. The letters are to be t

Serial numbers for a product are to be using 3 letters followed by 4 digits. The letters are to be taken from the first 8 letters of the alphabet with no repeats. The digits are taken from numbers 0-9 with no repeats. How many serial numbers can be generated
The serial number is organized with letters (L) and digits (D) like this:
LLLDDDD
Here's how we get the serial number:
[LIST=1]
[*]The first letter can be any of 8 letters A-H
[*]The second letter can be any 7 of 8 letters A-H
[*]The third letter can be any 6 of 8 letters A-H
[*]The fourth digit can be any of 10 digits 0-9
[*]The fifth digit can be any 9 of 10 digits 0-9
[*]The sixth digit can be any 8 of 10 digits 0-9
[*]The seventh digit can be any 7 of 10 digits 0-9
[/LIST]
We multiply all possibilities:
8 * 7 * 6 * 10 * 9 * 8 * 7
[B]1,693,440[/B]

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]

She earns $20 per hour as a carpenter and $25 per hour as a blacksmith, last week Giselle worked bot

She earns $20 per hour as a carpenter and $25 per hour as a blacksmith, last week Giselle worked both jobs for a total of 30 hours, and a total of $690. How long did Giselle work as a carpenter and how long did she work as a blacksmith?
Assumptions:
[LIST]
[*]Let b be the number of hours Giselle worked as a blacksmith
[*]Let c be the number of hours Giselle worked as a carpenter
[/LIST]
Givens:
[LIST=1]
[*]b + c = 30
[*]25b + 20c = 690
[/LIST]
Rearrange equation (1) to solve for b by subtracting c from each side:
[LIST=1]
[*]b = 30 - c
[*]25b + 20c = 690
[/LIST]
Substitute equation (1) into equation (2) for b
25(30 - c) + 20c = 690
Multiply through:
750 - 25c + 20c = 690
To solve for c, we [URL='https://www.mathcelebrity.com/1unk.php?num=750-25c%2B20c%3D690&pl=Solve']type this equation into our search engine[/URL] and we get:
c = [B]12
[/B]
Now, we plug in c = 12 into modified equation (1) to solve for b:
b = 30 - 12
b = [B]18[/B]

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

Solve 100 / 1/2

Solve 100 / 1/2
Dividing by a fraction is the same as multiplying by the reciprocal of the fraction:
[URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F2&pl=Reciprocal']Reciprocal of 1/2[/URL] = 2
100 * 2 = [B]200[/B]

Solve 11 - 1/2y = 3 + 6x for y

Solve 11 - 1/2y = 3 + 6x for y
Subtract 11 from each side so we can isolate the y term:
11 -11 - 1/2y = 3 + 6x - 11
Cancelling the 11's on the left side, we get:
-1/2y = 6x - 8 <-- Since 3 - 11 = -8
Multiply both sides of the equation by -2 to remove the -1/2 on the left side:
-2(-1/2)y = -2(6x - 8)
Simplifying, we get:
y = [B]-12x + 16[/B]

Solve a= (a + b + c + d)/4 for c

Solve a= (a + b + c + d)/4 for c
Cross multiply:
4a = a + b + c + d
Subtract a + b+ d from each side to isolate c:
4a - a - b - d = a + b + c + d - a - b - d
Canceling the a, b, and d from the right side, we get:
c = [B]3a - b - d [/B]

Solve mgh=1/2mv^2+1/2(2/5)mr^2(v^2/r^2) for v

Solve mgh=1/2mv^2+1/2(2/5)mr^2(v^2/r^2) for v
1/2(2/5) = 1/5 since the 2's cancel
r^2/r^2 = 1
So we simplify, and get:
mgh=1/2mv^2+1/5(mv^2) for v
Divide each side by m, so m's cancel in each term on the left and right side:
gh = 1/2v^2 + 1/5(v^2)
Combine like terms for v^2 on the right side:
1/2 + 1/5 = 7/10 from our [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%2F2&frac2=1%2F5&pl=Add']fraction calculator[/URL]
So we have:
gh = 7v^2/10
Multiply each side by 10:
10gh = 7v^2
Now divide each side by 7
10gh/7 = v^2
Take the square root of each side:
[B]v = sqrt(10gh/7)[/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]

standard deviation of 545 dollars. Find the sample size needed to have a confidence level of 95% and

Standard Error (margin of Error) = Standard Deviation / sqrt(n)
128 = 545/sqrt(n)
Cross multiply:
128sqrt(n) = 545
Divide by 128
sqrt(n) = 4.2578125
Square both sides:
[B]n = 18.1289672852 But we need an integer, so the answer is 19[/B]

Start with q. Multiply by p. Add 3. Divide A

Start with q. Multiply by p. Add 3. Divide A
Start with q:
q
Multiply by p:
pq
Add 3:
pq + 3
Divide A means divide by A. We wrap pq + 3 in parentheses to divide by the sum
(pq + 3)/A

Start with x , subtract 6, then times by 3.

Start with x , subtract 6, then times by 3.
We start with x:
x
Subtract 6:
x - 6
The phrase [I]times by[/I] means we multiply (x - 6) by 3
[B]3(x - 6) [/B] <-- This is our algebraic expression
If the problem asks you to multiply through, then you'd have:
3x - 18

Steven has some money. If he spends $9, then he will have 3/5 of the amount he started with.

Steven has some money. If he spends $9, then he will have 3/5 of the amount he started with.
Let the amount Steven started with be s. We're given:
s - 9 = 3s/5
Multiply each side through by 5 to eliminate the fraction:
5(s - 9) = 5(3s/5)
Cancel the 5's on the right side and we get:
5s - 45 = 3s
To solve for s, we [URL='https://www.mathcelebrity.com/1unk.php?num=5s-45%3D3s&pl=Solve']type this equation into our search engine[/URL] and we get:
s = [B]22.5[/B]

Subtract 7 from p, then multiply 5 by the result

Subtract 7 from p, then multiply 5 by the result.
Subtract 7 from p
p - 7
Multiply 5 by the result:
[B]5(p - 7)[/B]

subtract the product of 5 and x from 7

subtract the product of 5 and x from 7
The product of 5 and x means we multiply 5 by x:
5x
We subtract this product, 5x, from 7
[B]7 - 5x[/B]

subtract w from u, triple the result, then multiply v by what you have

subtract w from u, triple the result, then multiply v by what you have
Take this algebraic expression in 3 parts:
[U]1) subtract w from u:[/U]
u - w
[U]2) Triple the result means we multiply u - w by 3:[/U]
3(u - w)
[U]3) Multiply v by what you have. [I]What you have[/I] means the result from step 2:[/U]
[B]3v(u - w)[/B]

subtract w from v, add the result to u, then triple what you have

subtract w from v, add the result to u, then triple what you have
Take this algebraic expression in parts:
[LIST=1]
[*]Subtract w from v: v - w
[*]Add the result to u (the result is #1): u + v - w
[*]Triple what you have. This means multiply the result in #2 by 3:
[/LIST]
[B]3(u + v - w)[/B]

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 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 twice f plus h

sum of twice f plus h
Twice f means we multiply f by 2:
2f
sum of twice f plus h
[B]2f + h[/B]

sum of twice w and 3 times l

sum of twice w and 3 times l
Twice w means we multiply w by 2:
2w
3 times l:
3l
When we see the phrase [I]sum of[/I], we add:
[B]2w + 3l[/B]

Suppose Briley has 10 coins in quarters and dimes and has a total of 1.45. How many of each coin doe

Suppose Briley has 10 coins in quarters and dimes and has a total of 1.45. How many of each coin does she have?
Set up two equations where d is the number of dimes and q is the number of quarters:
(1) d + q = 10
(2) 0.1d + 0.25q = 1.45
Rearrange (1) into (3) to solve for d
(3) d = 10 - q
Now plug (3) into (2)
0.1(10 - q) + 0.25q = 1.45
Multiply through:
1 - 0.1q + 0.25q = 1.45
Combine q terms
0.15q + 1 = 1.45
Subtract 1 from each side
0.15q = 0.45
Divide each side by 0.15
[B]q = 3[/B]
Plug our q = 3 value into (3)
d = 10 - 3
[B]d = 7[/B]

T = mg - mf for f

T = mg - mf for f
Subtract mg from each side:
T - mg = mg - mg - mf
Cancel the mg on the right side and we get:
T - mg = -mf
Multiply each side by -1:
-(T - mg) = -(-mf)
mg - T = mf
Now Divide each side by m to isolate f:
(mg - T)/m = mf/m
Cancel the m on the right side and we get:
f = [B](mg - T)/m[/B]

t varies directly with the square of r and inversely with w

t varies directly with the square of r and inversely with w
There exists a constant k such that:
[B]t = kr^2/w[/B]
[I]Directly means multiply and inversely means divide[/I]

take away the product of 12 and p from 25

take away the product of 12 and p from 25
The product of 12 and p means we multiply 12 by p:
12p
Take away this product means we subtract 12p from 25:
[B]25 - 12p[/B]

The 4/7 part of a number is 84 . What is the number?

The 4/7 part of a number is 84 . What is the number?
We multiply 4/7 * 84.
7 goes into 84 12 times, so we have:
4 * 12 = [B]48[/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 average age of 15 men is 25 years. What is their total age in years?

The average age of 15 men is 25 years. What is their total age in years?
Average Age = Total Ages/Total Men
25 = Total Ages / 15
Cross multiply and we get:
Total Ages = 15 * 25
Total Ages = [B]375[/B]

The average height of a family of 6 is 6 feet. After the demise of the mother, the average height re

The average height of a family of 6 is 6 feet. After the demise of the mother, the average height remained the same. What is the height of the mother?
[LIST]
[*]Let the height of the family without the mom be f. Let the height of the mother be m.
[*]Averages mean we add the heights and divide by the number of people who were measured.
[/LIST]
We're given two equations:
[LIST=1]
[*](f + m)/6 = 6
[*]f/5 = 6
[/LIST]
Cross multiplying equation (2), we get:
f = 5 * 6
f = 30
Plug f = 30 into equation (1), we get:
(30 + m)/6 = 6
Cross multiplying, we get:
m + 30 = 6 * 6
m + 30 = 36
To solve this equation for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=m%2B30%3D36&pl=Solve']type it in our search engine[/URL] and we get:
m = [B]6[/B]
[SIZE=3][FONT=Arial][COLOR=rgb(34, 34, 34)][/COLOR][/FONT][/SIZE]

The average of 16 and x is 21. Find x.

The average of 16 and x is 21. Find x.
The average of 2 numbers is the sum of the 2 numbers divided by 2. So we have:
(16 + x)/2 = 21
Cross multiply:
16 + x = 21*2
16 + x = 42
To solve this equation, [URL='https://www.mathcelebrity.com/1unk.php?num=16%2Bx%3D42&pl=Solve']we type this expression into the search engine[/URL] and get [B]x = 26[/B].
Check our work by restating our answer:
The average of 16 and 26 is 21. TRUE.

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 of eighty-five and a number m is ninety

the average of eighty-five and a number m is ninety
Average of 2 numbers means we add both numbers and divide by 2:
(85 + m)/2 = 90
Cross multiply:
m + 85 = 90 * 2
m + 85 = 180
To solve this equation for m, we [URL='https://www.mathcelebrity.com/1unk.php?num=m%2B85%3D180&pl=Solve']type it in our math engine [/URL]and we get:
m = [B]95[/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 cost of purchasing a hockey stick and puck if the stick costs 6 less than twice the cost of the

The cost of purchasing a hockey stick and puck if the stick costs 6 less than twice the cost of the puck.
Let the hockey stick cost h, and puck cost p.
Twice the cost of the puck means we multiply p by 2:
2p
6 less than this means we subtract 6:
h = 2p - 6
[B][/B]
The total cost of the hockey stick and puck is:
p + 2p - 6
[B]3p - 6[/B]

The denominator of a fraction is 4 more than the numerator. If 4 is added to the numerator and 7 is

The denominator of a fraction is 4 more than the numerator. If 4 is added to the numerator and 7 is added to the denominator, the value of the fraction is 1/2. Find the original fraction.
Let the original fraction be n/d.
We're given:
[LIST=1]
[*]d = n + 4
[*](n + 4) / (d + 7) = 1/2
[/LIST]
Cross multiply Equation 2:
2(n + 4) = d + 7
2n + 8 = d + 7
Now substitute equation (1) into tihs:
2n + 8 = (n + 4) + 7
2n + 8 = n + 11
[URL='https://www.mathcelebrity.com/1unk.php?num=2n%2B8%3Dn%2B11&pl=Solve']Type this equation into our search engine[/URL], and we get:
n = 3
This means from equation (1), that:
d = 3 + 4
d = 7
So our original fraction n/d = [B]3/7[/B]

the difference between 7 times a number and 9 less than a number

the difference between 7 times a number and 9 less than a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
7 times a number means we multiply x by 7
7x
9 less than a number means we subtract 9 from x
x - 9
The difference between the two expressions means we subtract (x - 9) from 7x
7x - (x - 9)
Simplifying this, we have:
7x - x + 9
Grouping like terms, we get:
[B]6x + 9[/B]

The difference between the opposite of a number and 6.

The difference between the opposite of a number and 6.
The phrase [I]a number means[/I] an arbitrary variable, let's call it x.
x
The opposite of a number means we multiply by x by -1
-x
The phrase [I]the difference between[/I] means we subtract 6 from -x:
[B]-x - 6[/B]

The difference between the quotient of x and y, and twice z

The difference between the quotient of x and y, and twice z
The quotient of x and y means we divide x by y:
x/y
Twice z means we multiply z by 2:
2z
The difference between the quotient of x and y, and twice z means we subtract 2z from x/y
[B]x/y - 2z[/B]

the difference between triple a number and double a number

the difference between triple a number and double a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
Triple a number means we multiply x by 3:
3x
Double a number means we multiply x by 2:
2x
The difference means we subtract 2x from 3x:
3x - 2x
Simplifying like terms, we have:
(3 - 2)x = [B]x[/B]

The difference between two numbers is 25. The smaller number is 1/6th of the larger number. What is

The difference between two numbers is 25. The smaller number is 1/6th of the larger number. What is the value of the smaller number
Let the smaller number be s. Let the larger number be l. We're given two equations:
[LIST=1]
[*]l - s = 25
[*]s = l/6
[/LIST]
Plug in equation (2) into equation (1):
l - l/6 = 25
Multiply each side of the equation by 6 to remove the fraction:
6l - l = 150
To solve for l, we [URL='https://www.mathcelebrity.com/1unk.php?num=6l-l%3D150&pl=Solve']type this equation into our search engine[/URL] and we get:
l = 30
To solve for s, we plug in l = 30 into equation (2) above:
s = 30/6
[B]s = 5[/B]

The difference of 2 positive numbers is 54. The quotient obtained on dividing the 1 by the other is

The difference of 2 positive numbers is 54. The quotient obtained on dividing the 1 by the other is 4. Find the numbers.
Let the numbers be x and y. We have:
[LIST]
[*]x - y = 54
[*]x/y = 4
[*]Cross multiply x/y = 4 to get x = 4y
[*]Now substitute x = 4y into the first equation
[*](4y) - y = 54
[*]3y = 54
[*]Divide each side by 3
[*][B]y = 18[/B]
[*]If x = 4y, then x = 4(18)
[*][B]x = 72[/B]
[/LIST]

The difference of 25 and a number added to triple another number

The difference of 25 and a number added to triple another number
The phrase [I]a number [/I]means an arbitrary variable, let's call it x:
x
The difference of 25 and a number means we subtract x from 25:
25 - x
The phrase [I]another number[/I] means a different arbitrary variable, let's call it y:
y
Triple another number means we multiply y by 3:
3y
The phrase [I]added to[/I] means we add 25 - x to 3y
[B]25 - x + 3y[/B]

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 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 two numbers is 12 and their mean is 15. Find the two numbers

The difference of two numbers is 12 and their mean is 15. Find the two numbers.
Let the two numbers be x and y. We're given:
[LIST=1]
[*]x - y = 12
[*](x + y)/2 = 15. <-- Mean is an average
[/LIST]
Rearrange equation 1 by adding y to each side:
x - y + y = y + 12
Cancelling the y's on the left side, we get:
x = y + 12
Now substitute this into equation 2:
(y + 12 + y)/2 = 15
Cross multiply:
y + 12 + y = 30
Group like terms for y:
2y + 12 = 30
[URL='https://www.mathcelebrity.com/1unk.php?num=2y%2B12%3D30&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]y = 9[/B]
Now substitute this into modified equation 1:
x = y + 12
x = 9 + 12
[B]x = 21[/B]

the difference of x and y added to twice the sum of a and b

the difference of x and y added to twice the sum of a and b
Take this algebraic expression in parts:
[LIST]
[*]The difference of x and y: x - y
[*]The sum of a and b: a + b
[*]Twice the sum of a and b means we multiply a + b by 2: 2(a + b)
[*]The phrase [I]added to[/I] means we add:
[/LIST]
[B]x - y + 2(a + b)[/B]

The dimensions of a rectangle are 30 cm and 18 cm. When its length decreased by x cm and its width i

The dimensions of a rectangle are 30 cm and 18 cm. When its length decreased
by x cm and its width is increased by x cm, its area is increased by 35 sq. cm.
a. Express the new length and the new width in terms of x.
b. Express the new area of the rectangle in terms of x.
c. Find the value of x.
Calculate the current area. Using our [URL='https://www.mathcelebrity.com/rectangle.php?l=30&w=18&a=&p=&pl=Calculate+Rectangle']rectangle calculator with length = 30 and width = 18[/URL], we get:
A = 540
a) Decrease length by x and increase width by x, and we get:
[LIST]
[*]length = [B]30 - x[/B]
[*]width = [B]18 + x[/B]
[/LIST]
b) Our new area using the lw = A formula is:
(30 - x)(18 + x) = 540 + 35
Multiplying through and simplifying, we get:
540 - 18x + 30x - x^2 = 575
[B]-x^2 + 12x + 540 = 575[/B]
c) We have a quadratic equation. To solve this, [URL='https://www.mathcelebrity.com/quadratic.php?num=-x%5E2%2B12x%2B540%3D575&pl=Solve+Quadratic+Equation&hintnum=+0']we type it in our search engine, choose solve[/URL], and we get:
[B]x = 5 or x = 7[/B]
Trying x = 5, we get:
A = (30 - 5)(18 + 5)
A = 25 * 23
A = 575
Now let's try x = 7:
A = (30 - 7)(18 + 7)
A = 23 * 25
A = 575
They both check out.
So we can have

The doubling time of a population of flies is 8 hours by what factor does a population increase in 2

The doubling time of a population of flies is 8 hours.
a) By what factor does a population increase in 24 hours?
b) By what factor does the population increase in 2 weeks?
a) If the population doubles every 8 hours, then it doubles 3 times in 24 hours, since 24/8 = 3.
So 2 * 3 = 6. The increase factor is [B]6[/B]
b) Since a) is one full day, we now need to know how much it doubles in 14 days, which is 2 weeks. We take our factor of 6 for each day, and multiply it by 14: 14 * 6 = [B]84[/B]

The fraction has a value of 3/5. The sum of the numerator and the denominator was 40. What was the f

The fraction has a value of 3/5. The sum of the numerator and the denominator was 40. What was the fraction?
We're given two equations with a fraction with numerator (n) and denominator (d):
[LIST=1]
[*]n + d = 40
[*]n/d = 3/5
[/LIST]
Cross multiply equation 2, we get:
5n = 3d
Divide each side by 5:
5n/5 = 3d/5
n = 3d/5
Substitute this into equation 1:
3d/5 + d = 40
Multiply through both sides of the equation by 5:
5(3d/5) = 5d = 40 * 5
3d + 5d =200
To solve this equation for d, we [URL='https://www.mathcelebrity.com/1unk.php?num=3d%2B5d%3D200&pl=Solve']type it in our search engine and we get[/URL]:
d = [B]25
[/B]
Now substitute that back into equation 1:
n + 25 = 40
Using [URL='https://www.mathcelebrity.com/1unk.php?num=n%2B25%3D40&pl=Solve']our equation solver again[/URL], we get:
n = [B]15[/B]

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 length of a rectangular building is 6 feet less than 3 times the width. The perimeter is 120 fee

The length of a rectangular building is 6 feet less than 3 times the width. The perimeter is 120 feet. Find the width and length of the building.
P = 2l + 2w
Since P = 120, we have:
(1) 2l + 2w = 120
We are also given:
(2) l = 3w - 6
Substitute equation (2) into equation (1)
2(3w - 6) + 2w = 120
Multiply through:
6w - 12 + 2w = 120
Combine like terms:
8w - 12 = 120
Add 12 to each side:
8w = 132
Divide each side by 8 to isolate w:
w =16.5
Now substitute w into equation (2)
l = 3(16.5) - 6
l = 49.5 - 6
l = 43.5
So (l, w) = (43.5, 16.5)

The length of a wooden frame is 1 foot longer than its width and its area is equal to 12ft²

The length of a wooden frame is 1 foot longer than its width and its area is equal to 12ft²
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 length of Sally’s garden is 4 meters greater than 3 times the width. The perimeter of her garden

The length of Sally’s garden is 4 meters greater than 3 times the width. The perimeter of her garden is 72 meters. Find the dimensions of Sally’s garden.
Gardens have a rectangle shape. Perimeter of a rectangle is 2l + 2w. We're given:
[LIST=1]
[*]l = 3w + 4 [I](3 times the width Plus 4 since greater means add)[/I]
[*]2l + 2w = 72
[/LIST]
We substitute equation (1) into equation (2) for l:
2(3w + 4) + 2w = 72
Multiply through and simplify:
6w + 8 + 2w = 72
(6 +2)w + 8 = 72
8w + 8 = 72
To solve this equation for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=8w%2B8%3D72&pl=Solve']type it in our search engine[/URL] and we get:
w = [B]8
[/B]
To solve for l, we substitute w = 8 above into Equation (1):
l = 3(8) + 4
l = 24 + 4
l = [B]28[/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 mean age of 5 people in a room is 28 years. A person enters the room. The mean age is now 32. W

The mean age of 5 people in a room is 28 years. A person enters the room. The mean age is now 32. What is the age of the person who entered the room?
The sum of the 5 people's scores is S. We know:
S/5 = 28
Cross multiply:
S = 140
We're told that:
(140 + a)/6 = 32
Cross multiply:
140 + a = 192
[URL='https://www.mathcelebrity.com/1unk.php?num=140%2Ba%3D192&pl=Solve']Type this equation into our search engine[/URL], we get:
a = [B]52[/B]

The mean age of 5 people in a room is 32 years. A person enters the room. The mean age is now 40. Wh

The mean age of 5 people in a room is 32 years. A person enters the room. The mean age is now 40. What is the age of the person who entered the room?
Mean = Sum of Ages in Years / Number of People
32 = Sum of Ages in Years / 5
Cross multiply:
Sum of Ages in Years = 32 * 5
Sum of Ages in Years = 160
Calculate new mean after the next person enters the room.
New Mean = (Sum of Ages in Years + New person's age) / (5 + 1)
Given a new Mean of 40, we have:
40 = (160 + New person's age) / 6
Cross multiply:
New Person's Age + 160 = 40 * 6
New Person's Age + 160 = 240
Let the new person's age be n. We have:
n + 160 = 240
To solve for n, [URL='https://www.mathcelebrity.com/1unk.php?num=n%2B160%3D240&pl=Solve']we type this equation into our search engine[/URL] and we get:
n = [B]80[/B]

The mean age of 5 people in a room is 38 years. A person enters the room. The mean age is now 39. Wh

The mean age of 5 people in a room is 38 years. A person enters the room. The mean age is now 39. What is the age of the person who entered the room?
The mean formulas is denoted as:
Mean = Sum of Ages / Total People
We're given Mean = 38 and Total People = 5, so we plug in our numbers:
28 = Sum of Ages / 5
Cross multiply, and we get:
Sum of Ages = 28 * 5
Sum of Ages = 140
One more person enters the room. The mean age is now 39. Set up our Mean formula:
Mean = Sum of Ages / Total People
With a new Mean of 39 and (5 + 1) = 6 people, we have:
39 = Sum of Ages / 6
But the new sum of Ages is the old sum of ages for 5 people plus the new age (a):
Sum of Ages = 140 + a
So we have:
29 = (140 + a)/6
Cross multiply:
140 + a = 29 * 6
140 + a = 174
To solve for a, [URL='https://www.mathcelebrity.com/1unk.php?num=140%2Ba%3D174&pl=Solve']we type this equation into our search engine[/URL] and we get:
a = [B]34[/B]

The mean height of a class of 20 children is 1.27 the mean height of 12 boys in the class is 1.29 wh

The mean height of a class of 20 children is 1.27 the mean height of 12 boys in the class is 1.29 what is the mean height of the girls in the class?
The mean of sums is the sum of means. So we have:
Total Height / 20 = 1.27
Cross multiplying, we get:
Total Height = 20 * 1.27
Total Height = 25.4
Boys Height / 12 = 1.29
Cross multiplying, we get:
Boys Height = 12 * 1.29
Boys Height = 15.48
The Problem asks for mean height for girls. The formula is:
Girls Height / # of Girls = Mean of Girls Height
# of Girls = Total children - # of boys
# of Girls = 20 - 12
# of Girls = 8
Girls Height = Total Height - Boys Height
Girls Height = 25.4 - 15.48
Girls Height = 9.92
Plugging this into the Mean of girls height, we get:
9.92 /8 = [B]1.24[/B]

the mean of 12 scores is 8.8 . what is the sum of the scores ?

the mean of 12 scores is 8.8 . what is the sum of the scores ?
The Mean is denoted as:
Mean = Sum / count
We're given:
8.8 = Sum / 12
Cross multiply and we get:
Sum = 8.8*12
Sum = [B]105.6[/B]

The mean of 3 numbers is 20. Two of the numbers are 21, and 35. What is the 3rd number?

The mean of 3 numbers is 20. Two of the numbers are 21, and 35. What is the 3rd number?
The mean of 3 numbers is the sum of 3 numbers divided by 3. Let the 3rd number be n. We have:
Mean = (21 + 35 + n) / 3
The Mean is given as 20, so we have:
20 = (n + 56) / 3
Cross multiply:
n + 56 = 20 * 3
n + 56 = 60
To solve for n, we [URL='https://www.mathcelebrity.com/1unk.php?num=n%2B56%3D60&pl=Solve']type this number in our search engine [/URL]and we get:
n = [B]4[/B]

The mean of two numbers is 49.1. The first number is 18.3. What is the second number

The mean of two numbers is 49.1. The first number is 18.3. What is the second number
We call the second number n. Since the mean is an average, in this case 2 numbers, we have:
(18.3 + n)/2 = 49.1
Cross multiply:
18.3 + n = 98.2
[URL='https://www.mathcelebrity.com/1unk.php?num=18.3%2Bn%3D98.2&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]n = 79.9[/B]

The Oakdale High School Speech and Debate Club hosted its annual car wash fundraiser. Each club memb

The Oakdale High School Speech and Debate Club hosted its annual car wash fundraiser. Each club member brought a bottle of car wash soap, so there were 8 total bottles. 6 of the bottles contained orange soap. If a club member randomly selects 5 bottles to pour into the first soap bucket, what is the probability that all of them contain orange soap?
This is assumed to be draw without replacement, so we have:
[LIST=1]
[*]Draw 1: 6/8
[*]Draw 2: 5/7
[*]Draw 3: 4/6
[*]Draw 4: 3/5
[*]Draw 5: 2/4
[/LIST]
Since they are independent events, we multiply:
6/8 * 5/7 * 4/6 * 3/5 * 2/4
(6 * 5 * 4 * 3 * 2)/(8 * 7 * 6 * 5 * 4)
720/6720
[B]0.1071[/B]

The opposite of the difference of h and 5

The opposite of the difference of h and 5
The difference of h and 5
h - 5
The opposite of the difference of h and 5 means we multiply the difference of h and 5 by -1:
-(h - 5)
Distribute the negative sign:
[B]5 - h[/B]

the output is double the input

the output is double the input
Double means multiply by 2. So this means a function with input of x and output of y such that:
[B]y = 2x[/B]

The perimeter of a poster is 20 feet. The poster is 6 feet tall. How wide is it?

The perimeter of a poster is 20 feet. The poster is 6 feet tall. How wide is it?
[U]Assumptions and givens:[/U]
[LIST]
[*]The poster has a rectangle shape
[*]l = 6
[*]P = 20
[*]The perimeter of a rectangle (P) is: 2l + 2w = P
[/LIST]
Plugging in our l and P values, we get:
2(6) + 2w = 20
Multiplying through and simplifying, we get:
12 + 2w = 20
To solve for w, we [URL='https://www.mathcelebrity.com/1unk.php?num=12%2B2w%3D20&pl=Solve']type this equation into our search engine [/URL]and we get:
w = [B]4[/B]

The perimeter of a rectangle parking lot is 340 m. If the length of the parking lot is 97 m, what is

The perimeter of a rectangle parking lot is 340 m. If the length of the parking lot is 97 m, what is it’s width?
The formula for a rectangles perimeter P, is:
P = 2l + 2w where l is the length and w is the width.
Plugging in our P = 340 and l = 97, we have:
2(97) + 2w = 340
Multiply through, we get:
2w + 194 = 340
[URL='https://www.mathcelebrity.com/1unk.php?num=2w%2B194%3D340&pl=Solve']Type this equation into our search engine[/URL], we get:
[B]w = 73[/B]

The perpendicular height of a right-angled triangle is 70 mm longer than the base. Find the perimete

The perpendicular height of a right-angled triangle is 70 mm longer than the base. Find the perimeter of the triangle if its area is 3000.
[LIST]
[*]h = b + 70
[*]A = 1/2bh = 3000
[/LIST]
Substitute the height equation into the area equation
1/2b(b + 70) = 3000
Multiply each side by 2
b^2 + 70b = 6000
Subtract 6000 from each side:
b^2 + 70b - 6000 = 0
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=b%5E2%2B70b-6000%3D0&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic calculator[/URL], we get:
b = 50 and b = -120
Since the base cannot be negative, we use b = 50.
If b = 50, then h = 50 + 70 = 120
The perimeter is b + h + hypotenuse
Using the [URL='http://www.mathcelebrity.com/righttriangle.php?angle_a=&a=70&angle_b=&b=50&c=&pl=Calculate+Right+Triangle']right-triangle calculator[/URL], we get hypotenuse = 86.02
Adding up all 3 for the perimeter: 50 + 70 + 86.02 = [B]206.02[/B]

The points -5, -24 and 5,r lie on a line with slope 4. Find the missing coordinate r. Slope = (y2 -

The points -5, -24 and 5,r lie on a line with slope 4. Find the missing coordinate r.
Slope = (y2 - y1)/(x2 - x1)
Plugging in our numbers, we get:
4 = (r - -24)/(5 - -5)
4 = (r +24)/10
Cross multiply:
r + 24 = 40
Subtract 24 from each side:
[B]r = 16[/B]

The points 6,4 and 9,r lie on a line with slope 3. Find the missing coordinate r.

The points 6,4 and 9,r lie on a line with slope 3. Find the missing coordinate r.
Slope = (y2 - y1)/(x2 - x1)
Plugging in our numbers, we get:
3 = (r - 4)/(9 - 6)
3 = (r - 4)/3
Cross multiply:
r - 4 = 9
Add 4 to each side:
[B]r = 13[/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 3 and the sum of m and 2n

the product of 3 and the sum of m and 2n
The sum of m and 2n means we add 2n to m:
m + 2n
The product of 3 means we multiply the sum m + 2n by 3:
[B]3(m + 2n)[/B]

the product of 8 and 15 more than a number

the product of 8 and 15 more than a number.
Take this algebraic expression in pieces.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
15 more than x means we add 15 to x:
x + 15
The product of 8 and 15 more than a number means we multiply 8 by x + 15
[B]8(x + 15)[/B]

the product of k and 70, minus 15

the product of k and 70, minus 15
Take this algebraic expression in pieces:
The product of k and 70 means we multiply 70 times k
70k
The word [I]minus[/I] means we subtract 15 from 70k
[B]70k - 15[/B]

The product of the 2 numbers x and y

The product of the 2 numbers x and y
The phrase [I]product [/I]means we multiply the two variables, x and y.
[B]xy[/B]

The quotient of 9-x and twice x

The quotient of 9-x and twice x
Twice x means we multiply x by 2:
2x
The quotient of 9 - x and twice x is formed by the fraction:
[B](9 - x)/2x[/B]

the quotient of a number and twice another number

the quotient of a number and 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 quotient means we divide x by 2y:
[B]x/2y[/B]

the quotient of triple m and n squared

the quotient of triple m and n squared
Triple m means we multiply m by 3:
3m
n squared means we raise n to the 2nd power:
n^2
The quotient is formed as follows:
[B]3m/n^2[/B]

The ratio of the number of carabaos, goats, and cows in a farm is 5:1:2. If there are 48 animals of

The ratio of the number of carabaos, goats, and cows in a farm is 5:1:2. If there are 48 animals of these kinds in his backyard how many of them are goats
Calculate total ratio:
5 + 1 + 2 = 8
Multiply fractional portion of goats by total animals in the backyard.
1/8 * 48 = [B]6 goats[/B]

the ratio of twice c to d

the ratio of twice c to d
Twice c means we multiply c by 2:
2c
The ratio is formed by the quotient:
[B]2c/d[/B]

the reciprocal of the product a and b

the reciprocal of the product a and b
Take this algebraic expression in pieces:
The product a and b means we multiply a times b
ab
The [I]reciprocal[/I] means we take 1 over ab
[B]1/ab[/B]

the result of quadrupling a number is 80

the result of quadrupling a number is 80
Let our number be x. Quadrupling any number means multiplying it by 4. We have:
4x = 80
[URL='https://www.mathcelebrity.com/1unk.php?num=4x%3D80&pl=Solve']Typing this problem into our search engine[/URL], we get:
[B]x = 20[/B]

The sales tax for an item was $21.50 and it cost $430 before tax. Find the sales tax rate. Write you

The sales tax for an item was $21.50 and it cost $430 before tax. Find the sales tax rate. Write your answer as a percentage.
Sales tax percentage is:
21.50/430 = 0.05
To get a percentage, multiply the decimal by 100
0.05 * 100 = [B]5%[/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 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 difference of n and 2, increased by twice n

The square of the difference of n and 2, increased by twice n
The difference of n and 2:
n - 2
The square of the difference of n and 2 means we raise (n - 2) to the 2nd power:
(n - 2)^2
Twice n means we multiply n by 2:
2n
The square of the difference of n and 2, increased by twice n
[B](n - 2)^2 + 2n[/B]

The square of the sum of twice a number x and y

The square of the sum of twice a number x and y
Take this in algebraic expression in 3 parts:
[LIST=1]
[*]Twice a number x means we multiply x by 2: 2x
[*]The sum of twice a number x and y means we add y to 2x above: 2x + y
[*]The square of the sum means we raise the sum (2x + y) to the second power below:
[/LIST]
[B](2x + y)^2[/B]

The sum of 24 and twice Chau’s score . Use c to represent Chau’s score.

The sum of 24 and twice Chau’s score . Use c to represent Chau’s score.
Twice Chau's score means we multiply his score of c by 2:
2c
The sum of 24 and 2c means we add:
[B]24 + 2c[/B]

the sum of 3 numbers divided by its product

the sum of 3 numbers divided by its product
The phrase [I]3 numbers[/I] means we choose [I]3[/I] arbitrary variables. Let's call them x, y, z.
The sum of of these 3 numbers is:
x + y + z
The phrase [I]its product[/I] means we multiply all 3 arbitrary variables together:
xyz
Now, the phrase [I]divided by[/I] means we divide x + y + z by xyz:
[B](x + y + z)/xyz[/B]

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 a and b divided by their product

The sum of a and b divided by their product
The sum of a and b means we add b to a:
a + b
The product of a and b means we multiply a by b:
ab
To get our final algebraic expression, we divide the sum (a + b) by the product ab:
[B](a + b)/ab[/B]

the sum of a and b, divided by the product of c and d

the sum of a and b, divided by the product of c and d
The sum of a and b, means we add b to a
a + b
The product of c and d means we multiply c by d
cd
Divided by means we divide a + b by cd
[B](a + b)/cd[/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 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 d and twice g

Twice g means we multiply g by 2.
2g
The sum of d and twice g means we add:
d + 2g

The sum of d and v, all multiplied by 8

The sum of d and v, all multiplied by 8
This is an algebraic expression.
The sum of d and v:
d + v
Multiply this sum by 8:
[B]8(d + v)[/B]

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 the ages of levi and renee is 89 years. 7 years ago levi's age was 4 times renees age. Ho

The sum of the ages of levi and renee is 89 years. 7 years ago levi's age was 4 times renees age. How old is Levi now?
Let Levi's current age be l. Let Renee's current age be r. Were given two equations:
[LIST=1]
[*]l + r = 89
[*]l - 7 = 4(r - 7)
[/LIST]
Simplify equation 2 by multiplying through:
[LIST=1]
[*]l + r = 89
[*]l - 7 = 4r - 28
[/LIST]
Rearrange equation 1 to solve for r and isolate l by subtracting l from each side:
[LIST=1]
[*]r = 89 - l
[*]l - 7 = 4r - 28
[/LIST]
Now substitute equation (1) into equation (2):
l - 7 = 4(89 - l) - 28
l - 7 = 356 - 4l - 28
l - 7 = 328 - 4l
To solve for l, we [URL='https://www.mathcelebrity.com/1unk.php?num=l-7%3D328-4l&pl=Solve']type the equation into our search engine[/URL] and we get:
l = [B]67[/B]

The sum of the square of a number and 7 times a number

The sum of the square of a number and 7 times a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
Square the number:
x^2
7 times the number means we multiply x by 7:
7x
The sum means we add x^2 and 7x
[B]x^2 + 7x[/B]

the sum of twice a and b

the sum of twice a and b
Twice a means multiply a by 2
2a
The sum of means add Twice a to b
[B]2a + b[/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 numbers multiplied by 9

Choose two variables as arbitrary numbers, let's say x and y
[U]The sum of x and y is:[/U]
x + y
[U]Multiply that by 9[/U]
[B]9(x + y)[/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 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 multiplied by 6.

The sum of x and y is multiplied by 6.
[LIST]
[*]The sum of x and y: x + y
[*]Multiply the sum by 6:
[/LIST]
[B]6(x + y)[/B]

The total age of three cousins is 48. Suresh is half as old as Hakima and 4 years older than Saad. h

The total age of three cousins is 48. Suresh is half as old as Hakima and 4 years older than Saad. How old are the cousins?
Let a be Suresh's age, h be Hakima's age, and c be Saad's age. We're given:
[LIST=1]
[*]a + h + c = 48
[*]a = 0.5h
[*]a = c + 4
[/LIST]
To isolate equations in terms of Suresh's age (a), let's do the following:
[LIST]
[*]Rewriting (3) by subtracting 4 from each side, we get c = a - 4
[*]Rewriting (2) by multiply each side by 2, we have h = 2a
[/LIST]
We have a new system of equations:
[LIST=1]
[*]a + h + c = 48
[*]h = 2a
[*]c = a - 4
[/LIST]
Plug (2) and (3) into (1)
a + (2a) + (a - 4) = 48
Group like terms:
(1 + 2 + 1)a - 4 = 48
4a - 4 = 48
[URL='https://www.mathcelebrity.com/1unk.php?num=4a-4%3D48&pl=Solve']Typing this equation into our search engine[/URL], we get:
[B]a = 13 [/B]<-- Suresh's age
This means that Hakima's age, from modified equation (2) above is:
h = 2(13)
[B]h = 26[/B] <-- Hakima's age
This means that Saad's age, from modified equation (3) above is:
c = 13 - 4
[B]c = 9[/B] <-- Saad's age
[B]
[/B]

The total of z and 12 multiplied by the difference of 9 and y

The total of z and 12 multiplied by the difference of 9 and y
The total of z and 12:
z + 12
The difference of 9 and y:
9 - y
Now we multiply z + 12 by 9 - y:
[B](z + 12)(9 - y)[/B]

there are $4.20 in nickel and quarters. There are 6 more nickels than quarters there. How many coins

there are $4.20 in nickel and quarters. There are 6 more nickels than quarters there. How many coins of each are there
We're given two equations:
[LIST=1]
[*]n = q + 6
[*]0.05n + 0.25q = 4.2
[/LIST]
Substitute equation (1) into equation (2):
0.05(q + 6) + 0.25q = 4.2
Multiply through and simplify:
0.05q + 0.3 + 0.25q
0.3q + 0.3 = 4.2
To solve for q, we [URL='https://www.mathcelebrity.com/1unk.php?num=0.3q%2B0.3%3D4.2&pl=Solve']type this equation into the search engine[/URL] and we get:
q = [B]13
[/B]
To solve for n, we plug in q = 13 into equation (1):
n = 13 + 6
n = [B]19[/B]

There are 10 true or false questions on a test. You do not know the answer to 4 of the questions, so

There are 10 true or false questions on a test. You do not know the answer to 4 of the questions, so you guess. What is the probability that you will get all 4 answers right?
Probability you guess right is 1/2 or 0.5.
Since each event is independent of the other events, we multiply 1/2 4 times:
1/2 * 1/2 * 1/2 * 1/2 = [B]1/16[/B]

there are 120 calories in 3/4 cup serving of cereal. How many Calories are there in 6 cups of cereal

120/3/4 = x/6
Cross multiply:
0.75x = 720
Divide each side of the equation by 0.75
[B]x = 960[/B]

There are 24 hours in a day and scientists tell us that we should sleep for 3/8 of the day. How much

There are 24 hours in a day and scientists tell us that we should sleep for 3/8 of the day. How much time should we spend sleeping?
Multiply 24 hours per day * 3/8 day
Since 24/8 = 3, we have:
3 * 3 = [B]9 hours of sleep[/B].

There are 24 hours in a day and scientists tell us that we should sleep for 3/8 of the day. How much

There are 24 hours in a day and scientists tell us that we should sleep for 3/8 of the day. How much time should we spend sleeping?
3/8 of the day means we take 3/8 of 24 also written as:
3/8 * 24
We [URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F8&frac2=24&pl=Multiply']type this expression into our search engine [/URL]and get:
[B]9 hours[/B]

There are 32 female performers in a dance recital. The ratio of men to women is 3:8. How many men ar

There are 32 female performers in a dance recital. The ratio of men to women is 3:8. How many men are in the dance recital?
3:8 = x:32
3/8 = x/32
Cross multiply:
8x = 96
Divide each side by 8
x = 12
Check our work:
12:32
Divide each part by 4
12/4 = 3 and 32/4 = 8 so we have 3:8 :)

There are 377 baseball teams at the tournament and each team has 228 players. How many players are a

There are 377 baseball teams at the tournament and each team has 228 players. How many players are at the tournament?
Key words are "There are", "each team", and "how many".
We multiply teams by players per team to get number of players.
377 * 228 = [B]85,956[/B]

There are 5 orange books, 12 black books, and 8 tan books on Mr. Johnsons bookshelf. Calculate the p

There are 5 orange books, 12 black books, and 8 tan books on Mr. Johnsons bookshelf. Calculate the probability of randomly selecting a black book and then a tan book without replacement. Write your answer as a percent.
P(black book first draw)
P(black book first draw) = 12 black / (5 orange + 12 black + 8 tan)
P(black book first draw) = 12 / 25
P(tan book second draw)
P(tan book second draw) = 8 tan / (5 orange + 11 black + 8 tan) <-- 11 black because we already drew one black
P(tan book second draw) = 8 / 24
Using our fraction reduction calculator, this simplifies to 1/3
Since each draw is independent, we multiply both probabilities:
P(black book first draw, tan book second draw) = 12/25 * 1/3
P(black book first draw, tan book second draw) = 12/75
P(black book first draw, tan book second draw) = [B]16%[/B]

There are thrice as many girls (g) as there are boys (b)

There are thrice as many girls (g) as there are boys (b)
Thrice means we multiply by 3, so we have the following algebraic expression:
[B]g = 3b[/B]

There is a bag filled with 3 blue, 4 red and 5 green marbles. A marble is taken at random from the

There is a bag filled with 3 blue, 4 red and 5 green marbles. A marble is taken at random from the bag, the colour is noted and then it is not replaced. Another marble is taken at random. What is the probability of getting exactly 1 green?
Calculate Total marbles
Total marbles = Blue + Red + Green
Total marbles = 3 + 4 + 5
Total marbles = 12
Probability of a green = 5/12
Probability of not green = 1 - 5/12 = 7/12
To get exactly one green in two draws, we either get a green, not green, or a not green, green
[U]First Draw Green, Second Draw Not Green[/U]
[LIST]
[*]1st draw: Probability of a green = 5/12
[*]2nd draw: Probability of not green = 7/11 <-- 11 since we did not replace the first marble
[*]To get the probability of the event, since each draw is independent, we multiply both probabilities
[*]Probability of the event is (5/12) * (7/11) = 35/132
[/LIST]
[U]First Draw Not Green, Second Draw Not Green[/U]
[LIST]
[*]1st draw: Probability of not a green = 7/12
[*]2nd draw: Probability of not green = 5/11 <-- 11 since we did not replace the first marble
[*]To get the probability of the event, since each draw is independent, we multiply both probabilities
[*]Probability of the event is (7/12) * (5/11) = 35/132
[/LIST]
To get the probability of exactly one green, we add both of the events:
First Draw Green, Second Draw Not Green + First Draw Not Green, Second Draw Not Green
35/132 + 35/132 = 70/132
[URL='https://www.mathcelebrity.com/fraction.php?frac1=70%2F132&frac2=3%2F8&pl=Simplify']Using our fraction simplify calculator[/URL], we get:
[B]35/66[/B]

There is a bag filled with 5 blue, 6 red and 2 green marbles. A marble is taken at random from the b

There is a bag filled with 5 blue, 6 red and 2 green marbles. A marble is taken at random from the bag, the colour is noted and then it is replaced. Another marble is taken at random. What is the probability of getting exactly 1 blue?
Find the total number of marbles in the bag:
Total marbles = 5 blue + 6 red + 2 green
Total marbles = 13
The problem asks for exactly one blue in 2 draws [I]with replacement[/I]. Which means you could draw as follows:
Blue, Not Blue
Not Blue, Blue
The probability of drawing a blue is 5/13, since we replace the marbles in the bag each time.
The probability of not drawing a blue is (6 + 2)/13 = 8/13
And since each of the 2 draws are independent of each other, we multiply the probability of each draw:
Blue, Not Blue = 5/13 * 8/13 =40/169
Not Blue, Blue = 8/13 * 5/13 = 40/169
We add both probabilities since they both count under our scenario:
40/169 + 40/169 = 80/169
Checking our [URL='https://www.mathcelebrity.com/fraction.php?frac1=80%2F169&frac2=3%2F8&pl=Simplify']fraction simplification calculator[/URL], we see you cannot simplify this fraction anymore.
So our probability stated in terms of a fraction is 80/169
[URL='https://www.mathcelebrity.com/perc.php?num=80&den=169&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']Stated in terms of a decimal[/URL], it's 0.4734

There were 286,200 graphic designer jobs in a country in 2010. It has been projected that there will

There were 286,200 graphic designer jobs in a country in 2010. It has been projected that there will be 312,500 graphic designer jobs in 2020. (a) Using the data, find the number of graphic designer jobs as a linear function of the year.
[B][U]Figure out the linear change from 2010 to 2020[/U][/B]
Number of years = 2020 - 2010
Number of years = 10
[B][U]Figure out the number of graphic designer job increases:[/U][/B]
Number of graphic designer job increases = 312,500 - 286,200
Number of graphic designer job increases = 26,300
[B][U]Figure out the number of graphic designer jobs added per year[/U][/B]
Graphic designer jobs added per year = Total Number of Graphic Designer jobs added / Number of Years
Graphic designer jobs added per year = 26,300 / 10
Graphic designer jobs added per year = 2,630
[U][B]Build the linear function for graphic designer jobs G(y) where y is the year:[/B][/U]
G(y) = 286,200 + 2,630(y - 2010)
[B][U]Multiply through and simplify:[/U][/B]
G(y) = 286,200 + 2,630(y - 2010)
G(y) = 286,200 + 2,630y - 5,286,300
[B]G(y) = 2,630y - 5,000,100[/B]

There were 500 balloons in 2 buckets. Rahul bursted 350 balloons of 1st bucket and Raghav bursted 4

There were 500 balloons in 2 buckets. Rahul bursted 350 balloons of 1st bucket and Raghav bursted 4 by 5 of the balloons of 2nd bucket. Who bursted more balloons and how many did Raghav burst?
If Rahul bursted 350 balloons, then we have:
Remaining Balloons = 500 - 350
Remaining Balloons = 150
Raghav bursted [URL='https://www.mathcelebrity.com/fraction.php?frac1=150&frac2=4/5&pl=Multiply']4/5 * 150[/URL] = [B]120[/B]
Since 350 for Rahul > 120 for Raghav, [B]Rahul bursted more balloons[/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

Three out of every 7 houses in a neighborhood are painted white. There are 224 houses in the neighbo

Three out of every 7 houses in a neighborhood are painted white. There are 224 houses in the neighborhood. How many houses are white?
[URL='https://www.mathcelebrity.com/fraction.php?frac1=224&frac2=3/7&pl=Multiply']3/7 of 224[/URL] = [B]96 houses are white[/B]

Three-fourths of 500

Three-fourths of 500
Three-fourths is written as 3/4
3/4 of 500 means we multiply 3/4 by 500
3 * 500/4
500/4 = 125, so we have:
3 * 125
[B]375[/B]

thrice the sum of x y and z

thrice the sum of x y and z
The sum of x, y, and z
x + y + z
Thrice the sum means multiply by 3
[B]3(x + y + z)[/B]

thrice the sum of x y and z

thrice the sum of x y and z
The sum of x, y, and z means we add all 3 variables together:
x + y + z
The word [I]thrice[/I] means we multiply the sum of x, y, and z by 3:
3(x + y +z)

thrice the sum of x y and z

thrice the sum of x y and z
The sum of x, y, and z:
x + y + z
Thrice means multiply the sum by 3:
[B]3(x + y + z)[/B]

thrice y plus x minus 2z

thrice y plus x minus 2z
Thrice y means we multiply y by 3:
3y
plus x
3y + x
minus 2z
[B]3y + x - 2z[/B]

To convert from Celsius to Fahrenheit, multiply by 1.8 and add 32. Write a formula to describe this

To convert from Celsius to Fahrenheit, multiply by 1.8 and add 32. Write a formula to describe this relationship.
Given C as Celsius and F as Fahrenheit, we have the following equation:
[B]F = 1.8C + 32[/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]

Translate this phrase into an algebraic expression. 57 decreased by twice Mais savings Use the varia

Translate this phrase into an algebraic expression. 57 decreased by twice Mais savings Use the variable m to represent Mais savings.
Twice means multiply by 2
2m
57 decreased by means subtract 2m from 57
[B]57 - 2m[/B]

triple 5, raise the result to the 10th power, then divide p by what you have

triple 5, raise the result to the 10th power, then divide p by what you have
Triple 5, means multiply 5 by 3
3 * 5 --> Simplified, this is 15
Raise the result to the 10th power, means we raise 15 to the 10 power:
15^10
Then divide it by p:
[B]15^10/p[/B]

triple a number and another number

triple a number and another number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
Triple a number means we multiply x by 3:
3x
The phrase [I]another number[/I] means another arbitrary variable, let's call it y:
y
The word [I]and[/I] means we add y to 3x:
[B]3x + y[/B]

triple c divide the result by a

triple c divide the result by a
Take this algebraic expression in pieces.
Triple c means we multiply the variable c by 3
3c
Divide the result by a, means we take 3c, and divide by a
[B]3c/a[/B]

triple c, multiply the result by a, then subtract b

triple c, multiply the result by a, then subtract b
Triple c means we multiply c by 3:
3c
Multiply the result by a means we multiply 3c by a
3ac
Then, we subtract b from 3ac:
[B]3ac - b[/B]

triple h then raise the result to the 8th power

triple h then raise the result to the 8th power
[U]Triple h means we multiply h by 3:[/U]
3h
[U]Raise the result to the 8th power:[/U]
[B](3h)^8[/B]

triple s add the result to q then divide what you have by r

triple s add the result to q then divide what you have by r.
Triple s means multiply s by 3:
3s
Add the result to q:
3s + q
Divide what you have by r:
[B](3s + q)/r[/B]

triple t multiply g

triple t multiply g
triple t means we multiply t by 3:
3t
Multiply g:
[B]3tg[/B]

triple the sum of 36 and 6 then add 4

triple the sum of 36 and 6 then add 4
Take this algebraic expression in parts:
The sum of 36 and 6:
36 + 6
Triple the sum means we multiply the sum by 3:
3(36 + 6)
Then add 4:
[B]3(36 + 6) + 4[/B]
If the problem asks you to simplify the algebraic expression, we have:
3(42) + 4
126 + 4
[B]130[/B]

Triple the sum of 4 and y

The sum of 4 and y is written as (4 + y)
Triple that means we multiply that entire sum by 3.
3(4 + y)

Triple the sum of 7 and m

The sum of 7 and m is written as 7 + m
Triple that means multiply by 3:
3(7 + m)

triple the sum of y and six

The sum of y and six is denoted as:
y + 6
We triple that sum by multiplying it by 3
3(y + 6)

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

tripled square of the difference of a and b

The difference of a and b is written as:
a - b
Square the difference means raise the difference to the power of 2
(a - b)^2
Triple this expression means multiply by 3:
[B]3(a - b)^2[/B]

twice a number subtracted from the square root of the same number

twice a number subtracted from the square root of the same number
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
Square root of the same number:
sqrt(x)
twice a number subtracted from the square root of the same number
[B]sqrt(x) - 2x[/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 product of p q and r

twice the product of p q and r
The product of p q and r means we multiply all 3 variables together:
pqr
The word [I]twice[/I] means we multiply pqr by 2:
[B]2pqr[/B]

twice the product of p q and r

twice the product of p q and r
The product of p q and r:
pqr
Twice means we multiply pqr by 2:
[B]2pqr[/B]

twice the square of the product of x and y

twice the square of the product of x and y
Take this algebraic expression in pieces:
[LIST]
[*]The product of x and y means we multiply x and y: xy
[*]The square of the product means we raise xy to the power of 2: (xy)^2 = x^2y^2
[*]Twice the square means we multiply the square by 2: [B]2x^2y^2[/B]
[/LIST]

twice the square of the product of x and y

twice the square of the product of x and y
[LIST]
[*]The product of x and y: xy
[*]The square of the product means we raise xy to the power of 2: (xy)^2
[*]Twice the square means we multiply by 2
[/LIST]
[B]2(xy)^2
or
2x^2y^2[/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[/B]

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 numbers have an average of 2100 and one number is $425 more than the other number. What are the

two numbers have an average of 2100 and one number is $425 more than the other number. What are the numbers
Let the first number be x and the second number be y. We're given two equations:
[LIST=1]
[*](x + y)/2 = 2100 (Average)
[*]y = x + 425
[/LIST]
Rearrange equation (1) by cross multiplying
x + y = 2 * 2100
x + y = 4200
So we have our revised set of equations:
[LIST=1]
[*]x + y = 4200
[*]y = x + 425
[/LIST]
Substituting equation (2) into equation (1) for y, we get:
x + (x + 425) = 4200
Combining like terms, we get:
2x + 425 = 4200
Using our [URL='https://www.mathcelebrity.com/1unk.php?num=2x%2B425%3D4200&pl=Solve']equation solver[/URL], we get:
x = [B]1887.5[/B]
Which means using equation (2), we get
y = 1887.5 + 425
y = [B]2312.5[/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]

two-thirds the difference of c and d

two-thirds the difference of c and d
The difference of c and d:
c - d
two-thirds the difference means we multiply c - d by 2/3:
[B]2(c - d)/3[/B]

u varies jointly as q and the square of m

u varies jointly as q and the square of m
Varies jointly means we multiply. There exists a constant k such that:
[B]u = kqm^2[/B]

u=ak/b for a

Cross multiply:
ub = ak
Divide each side of the equation by k to isolate a:
a = ub/k

U=ak/b, for a

U=ak/b, for a
[U]Cross multiply:[/U]
Ub = ak
[U]Divide each side by k[/U]
[B]a = Ub/k[/B]

Use c for unknown variable : Sam's age plus twice his age

Use c for unknown variable : Sam's age plus twice his age
Sam's age:
c
Twice his age means we multiply c by 2:
2c
Sam's age plus twice his age
[B]c + 2c[/B]

Use k as the constant of variation. L varies jointly as u and the square root of v.

Use k as the constant of variation. L varies jointly as u and the square root of v.
Since u and v vary jointly, we multiply by the constant of variation k:
[B]l = ku * sqrt(v)[/B]

What does $10.00 worth of 0.25 cents tickets how many would you get

What does $10.00 worth of 0.25 cents tickets how many would you get?
10/0.25 = [B]40 tickets
[/B]
Since 0.25 is 1/4, we could say:
10/1/4
Multiply top and bottom by 4
10 * 4 = 40

what integer is tripled when 9 is added to 3 fourths of it?

what integer is tripled when 9 is added to 3 fourths of it?
Let the integer be n. Tripling an integer means multiplying it by 3. We're given:
3n = 3n/4 + 9
Since 3 = 12/4, we have:
12n/4 = 3n/4 + 9
Subtract 3n/4 from each side:
9n/4 = 9
[URL='https://www.mathcelebrity.com/prop.php?num1=9n&num2=9&den1=4&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']Typing this equation into the search engine[/URL], we get:
[B]n = 4[/B]

What is the inverse of dividing by 3

What is the inverse of dividing by 3
[B]Multiplying by 3[/B]
Suppose we have 2 divided by 3:
2/3
To undo this operation to get to 2 again, we'd multiply by 3:
2/3 * 3 = 2

What is the probability of rolling 12, 5 times in a row?

The only way you can roll a 12 with two dice is 6 and 6. Since each die roll is independent, we have:
[LIST]
[*]P(12) = P(6) * P(6)
[*]P(12) = 1/6 * 1/6
[*]P(12) = 1/36.
[/LIST]
Now, what is the probability we roll a 12 five times in a row? The same rules apply, each new roll is independent of the last, so we multiply:
[LIST]
[*]P(12, 12, 12, 12, 12) = 1/36 * 1/36 * 1/36 * 1/36 * /36
[*]P(12, 12, 12, 12, 12) = [B]1/60,466,176[/B] or [B]1.65381717e-8[/B]
[/LIST]

what’s the probability of rolling a 5 and then rolling a number less then 2

what’s the probability of rolling a 5 and then rolling a number less then 2
[U]Roll a 5:[/U]
There's only one 5 on a six sided die
P(X = 5) = 1/6
A number less than 2 is only 1:
P(X < 2) = P(X = 1)
P(X = 1) = 1/6
Since each event is independent, we multiply:
P(X = 5) * P(X = 1) = 1/6 * 1/6
P(X = 5) * P(X = 1) = [B]1/36[/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 finding the power of a power, you _____________________ the exponents

When finding the power of a power, you _____________________ the exponents
[B]Multiply
[/B]
Example:
(a^b)^c = a^bc

When five people are playing a game called hearts, each person is dealt ten cards and the two remain

When five people are playing a game called hearts, each person is dealt ten cards and the two remaining cards are put face down on a table. Because of the rules of the game, it is very important to know the probability of either of the two cards being a heart. What is the probability that at least one card is a heart?
Probability that first card is not a heart is 3/4 since 4 suits in the deck, hearts are 1/4 of the deck.
Since we don't replace cards, the probability of the next card drawn without a heart is (13*3 - 1)/51 = 38/51
Probability of both cards not being hearts is found by multiplying both individual probabilities:
3/4 * 38/51 = 114/204
Having at least one heart is found by subtracting this from 1 which is 204/204:
204/204 - 114/204 = 90/204
[URL='https://www.mathcelebrity.com/search.php?q=90%2F204&x=0&y=0']This reduces to[/URL] [B]15/34[/B]

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 equivalent to 3(2x + 1)(4x + 1)?

Which of the following is equivalent to 3(2x + 1)(4x + 1)?
[LIST]
[*]A) 45x
[*]B) 24x^2 + 3
[*]C) 24x^2 + 18x + 3
[*]D) 18x^2 + 6
[/LIST]
First, [URL='https://www.mathcelebrity.com/binomult.php?term1=2x%2B1&term2=4x%2B1&pl=Expand+Product+of+2+Binomials+using+FOIL']multiply the binomials[/URL]:
We get 8x^2 + 6x + 1
Now multiply this polynomial by 3:
3(8x^2 + 6x + 1) = [B]24x^2 + 18x + 3, answer C[/B]

Wilbie had candy to give to his 3 children. He first took 5 pieces and evenly divided the rest among

Wilbie had candy to give to his 3 children. He first took 5 pieces and evenly divided the rest among each child. Each child received 3 pieces. With how many pieces did he start?
Let the starting candy amount be c. We're given:
(c - 5)/3 = 3
Cross multiply:
c - 5 = 3*3
c - 5 = 9
[URL='https://www.mathcelebrity.com/1unk.php?num=c-5%3D9&pl=Solve']Type this equation into the search engine[/URL], and we get:
c = 14

Write .02 as a percent

Write .02 as a percent
Multiply by 100 and add a percent:
.02 * 100% = [B]2%[/B]

Write a fraction with a denominator of 9. The fraction should be less than 1/2

Write a fraction with a denominator of 9. The fraction should be less than 1/2
Let n be the numerator. We have:
n/9 < 1/2
multiply each side by 9:
9n/9 < 9/2
n < 9/2
Examples are 8/2, 7/2, 6/2, 5/2, 4/2, 3/2,

Write an algebraic expression for 8 multiplied by the result of u reduced by 11.

Write an algebraic expression for 8 multiplied by the result of u reduced by 11.
u [I]reduced by[/I] 11
Reduced by means subtract 11 from u. So we have:
u - 11
We multiply this expression by 8 to get our algebraic expression of:
[B]8(u - 11)[/B]

Write an equation that relates the quantities. G varies jointly with t and q and inversely with the

Write an equation that relates the quantities. G varies jointly with t and q and inversely with the cube of w . The constant of proportionality is 8.25 .
[LIST]
[*]Varies jointly or directly means we multiply
[*]Varies inversely means divide
[*]The cube of w means we raise w to the 3rd power: w^3
[/LIST]
Using k = 8.25 as our constant of proportionality, we have:
[B]g = 8.25qt/w^3[/B]

x add y, multiply by z then subtract d

x add y, multiply by z then subtract d
Take this algebraic expression in pieces:
[LIST]
[*]x add y: x + y
[*]multiply by z: z(x + y)
[*]Subtract d: [B]z(x + y) - d[/B]
[/LIST]

x is tripled and then 2 is added

x is tripled and then 2 is added
Tripled means multiply x by 3:
3x
Then add 2 to this:
[B]3x + 2[/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].

X varies directly with the cube root of y when x=1 y=27. Calculate y when x=4

X varies directly with the cube root of y when x=1 y=27. Calculate y when x=4
Varies directly means there is a constant k such that:
x = ky^(1/3)
When x = 1 and y = 27, we have:
27^1/3(k) = 1
3k = 1
To solve for k, we[URL='https://www.mathcelebrity.com/1unk.php?num=3k%3D1&pl=Solve'] type in our equation into our search engine[/URL] and we get:
k = 1/3
Now, the problem asks for y when x = 4. We use our variation equation above with k = 1/3 and x = 4:
4 = y^(1/3)/3
Cross multiply:
y^(1/3) = 4 * 3
y^(1/3) =12
Cube each side:
y^(1/3)^3 = 12^3
y = [B]1728[/B]

X+y/3=5 for x

(X+y)/3=5 for x
Cross multiply:
x + y = 15
Subtract y from each side:
[B]x = 15 - y[/B]

x/3 - g = a for x

x/3 - g = a for x
Add g to each side so we can isolate the x term:
x/3 - g + g = a + g
Cancel the g terms on the left side and we get:
x/3 = a + g
Multiply each side by 3 to isolate x:
3(x/3) = 3(a + g)
Cancelling the 3's on the left side, we get:
x = [B]3(a + g)[/B]

x/5-7=2q for x

x/5-7=2q for x
Add 7 to each side:
x/5 -7 + 7 = 2q + 7
Cancel the 7's on the left side, we get:
x/5 = 2q + 7
Cross multiply the 5:
x = 5(2q + 7)
x = [B]10q + 35[/B]

x/r - h = 4 for x

x/r - h = 4 for x
Add h to each side:
x/r - h + h = h + 4
Cancel the h's on the left side, we get:
x/r = h + 4
Multiply each side by r to isolate x:
xr/r = r(h + 4)
Cancel the r's on the left side, we get:
x = [B]r(h + 4)[/B]

x/y + 9 = n for x

x/y + 9 = n for x
Subtract 9 from each side to isolate the x term:
x/y + 9 - 9 = n - 9
Cancel the 9's on the left side and we get:
x/y = n - 9
Because we have a fraction on the left side, we can cross multiply the denominator y by n - 9
[B]x =[/B] [B]y(n - 9)[/B]

x/y + 9 = n for y

x/y + 9 = n for y
First, subtract 9 from each side to isolate the y term:
x/y + 9 - 9 = n - 9
Cancel the 9's on the left side, and we get:
x/y = n - 9
Cross multiply:
x = y(n - 9)
Divide each side by (n - 9):
x/(n - 9) = y(n - 9)/(n - 9)
Cancel the (n - 9) on the right side, and we get:
y = [B]x/(n - 9)[/B]

x/y = z - 8 for x

x/y = z - 8 for x
Cross multiply:
[B]x = y(z - 8)[/B]

x/y = z - 8 for x

x/y = z - 8 for x
Multiply each side by y to isolate x:
y*(x/y) = y(z - 8)
The y's cancel out on the left side, so we have:
x = [B]y(z - 8)[/B]

x/y = z - 8 for x

x/y = z - 8 for x
We start by seeing that x is isolated.
To remove y from the left side, we multiply each side of the equation by y:
xy/y = y(z - 8)
Cancelling y on the left side, we get our answer of:
x = [B]y(z - 8)
[MEDIA=youtube]_HNyGlnnQdQ[/MEDIA][/B]

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]

y varies directly as x and inversely as i

y varies directly as x and inversely as I
Note:
Direct variation means we multiply. Inverse variation means we divide.
There exists a constant k such that:
[B]y = kx/i[/B]

y/2+c=d for y

Multiply each side by 2 to isolate y.
y +2c = 2d
Subtract 2c from each side of the equation:
y = 2d - 2c
This can also be written y = 2(d - c)

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 earned $141 last week babysitting and cleaning. You earned $5 per hour babysitting and $7 per ho

You earned $141 last week babysitting and cleaning. You earned $5 per hour babysitting and $7 per hour cleaning. You worked 9 more hours babysitting than cleaning. How many hours did you work last week?
Let b be the hours of babysitting and c be the hours of cleaning. We're given two equations:
[LIST=1]
[*]b = c + 9
[*]5b + 7c = 141
[/LIST]
Substitute equation (1) into (2):
5(c + 9) + 7c = 141
Multiply through:
5c + 45 + 7c = 141
Combine like terms:
12c + 45 = 141
[URL='https://www.mathcelebrity.com/1unk.php?num=12c%2B45%3D141&pl=Solve']Typing this equation into our search engine[/URL], we get:
c = 8
Now substitute this value of c back into Equation (1):
b = 8 + 9
b = 17
The total hours worked (t) is:
t = b + c
t = 17 + 8
t = [B]25[/B]

You put $5500 in a bond fund which has an annual yield of 4.8%. How much interest will be earned in

You put $5500 in a bond fund which has an annual yield of 4.8%. How much interest will be earned in 23 years?
Build the accumulation of principal. We multiply 5,500 times 1.048 raised to the 23rd power.
Future Value = 5,500 (1.048)^23
Future Value =5,500(2.93974392046)
Future Value = 16,168.59
The question asks for interest earned, so we find this below:
Interest Earned = Future Value - Principal
Interest Earned = 16,168.59 - 5,500
Interest Earned = [B]10,668.59[/B]

Your salary after a 9% salary increase if your salary before the increase was s

Your salary after a 9% salary increase if your salary before the increase was s
9% increase means we multiply s by 1.09
[B]1.09s[/B]

z , subtract 5 then times by 3

z , subtract 5 then times by 3
Take this algebraic expression two parts:
[LIST]
[*]z subtract 5: z - 5
[*][I]Then times by 3[/I] means we multiply the expression z - 5 by 3
[/LIST]
[B]3(z - 5)[/B]

z = (x + y)/mx; Solve for x

z = (x + y)/mx; Solve for x
Cross multiply:
zmx = x + y
Subtract x from each side
zmx - x = y
Factor out x
x(zm - 1) = y
Divide each sdie by zm - 1
x = y/(zm - 1)

z varies directly with x and inversely with y

z varies directly with x and inversely with y
[LIST]
[*]The phrase directly means we multiply.
[*]The phrase inversely means we divide
[*]Variation means there exists a constant k such that:
[/LIST]
[B]z = kx/y[/B]

z varies inversely as the square of t. if z=4 when t=2, find z when t is 10

z varies inversely as the square of t. if z=4 when t=2, find z when t is 10
Varies inversely means there exists a constant k such that:
z = k/t^2
If z = 4 when t = 2, we have:
4 = k/2^2
4 = k/4
Cross multiply and we get:
k = 4 * 4
k = 16
Now the problem asks to find z when t is 10:
z = k/t^2
z = 16/10^2
z = 16/100
z = [B]0.16[/B]

z/w=x+z/x+y for z

z/w=x+z/x+y for z
This is a literal equation. Let's isolate z on one side.
Subtract z/x from each side.
z/w - z/x = x + y
Factor our z on the left side:
z(1/w - 1/x) = x + y
Divide each side by (1/w - 1/x)
z = x + y/(1/w - 1/x)
To remove reciprocals in the denominator, we rewrite 1/w - 1/x with a common denominator xw
(x - w)/xw
Then multiply x + y by the reciprocal
z = [B](x + y)xw/(x - w)[/B]

z=m-x+y, for x

z=m-x+y, for x
This is a literal equation. Let's add subtract (m + y) from each side:
z - (m + y) = m - x + y - (m + y)
The m + y terms cancel on the right side, so we have:
z - m - y = -x
Multiply each side by -1 to isolate x:
-1(z - m - y) = -(-x)
x = [B]m + y - z[/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]

zy-dm=ky/t for y

zy-dm=ky/t for y
Isolate terms with y to solve this literal equation.
Subtract zy from each side:
zy - dm - zy = ky/t - zy
Cancel the zy terms on the left side, we get:
-dm = ky/t - zy
Factor out y:
y(k/t - z) = -dm
Divide each side by (k/t - z)
y = -dm/(k/t - z)
(k/t - z) can be rewritten as (k - tz)/t
We multiply -dm by the reciprocal of this quotient to get our simplified literal equation:
y = [B]-dmt/(k - tz)[/B]

|(x-7)/5|<=4

|(x-7)/5|<=4
Set up two equations:
[LIST=1]
[*](x-7)/5 <= 4
[*](x-7)/5 > -4
[/LIST]
Cross Multiply (1):
x - 7 <= 20
Add 7 to each side:
x <= 27
Cross Multiply (2):
x - 7 > -20
Add 7 to each side:
x > -13