divide - To split into equal parts or groups
Formula: ÷
%60 of the freshman ate pizza at lunch today. If 180 freshman ate pizza, how many freshman are enro60% of the freshman ate pizza [URL='http://www.mathcelebrity.com/community/x-apple-data-detectors://3']at lunch today[/URL]. If 180 freshman ate pizza, how many freshman are enrolled at our school?
60% of x = 180
We write this as 0.6x = 180
Divide each side by 0.6 to isolate x.
We get x = 300 freshman
(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]
-1 3/5 divided by -2/3-1 3/5 divided by -2/3
We write this as:
-1&3/5 / 2/3
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=1%263%2F5&frac2=2%2F3&pl=Divide']fraction division calculator[/URL], we get:
[B]12/5[/B]
-3x<= -9 or 5+x<6-3x<= -9 or 5+x<6
Take each piece:
-3x<= -9
Divide each side by -3:
x>=3
Now take 5 + x < 6
5 + x < 6
Subtract 5 from each side:
x < 1
Joining together the two inequalities, we have:
x<1 or x>=3
Use our [URL='http://www.mathcelebrity.com/interval-notation-calculator.php?num=x%3C1orx%3E%3D3&pl=Show+Interval+Notation']interval notion calculator[/URL] to find the interval notation of this compound inequality
-5n - 5n - 5 = 5-5n - 5n - 5 = 5
Solve for [I]n[/I] in the equation -5n - 5n - 5 = 5
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(-5 - 5)n = -10n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
-10n - 5 = + 5
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -5 and 5. To do that, we add 5 to both sides
-10n - 5 + 5 = 5 + 5
[SIZE=5][B]Step 4: Cancel 5 on the left side:[/B][/SIZE]
-10n = 10
[SIZE=5][B]Step 5: Divide each side of the equation by -10[/B][/SIZE]
-10n/-10 = 10/-10
n = [B]- 1
[URL='https://www.mathcelebrity.com/1unk.php?num=-5n-5n-5%3D5&pl=Solve']Source[/URL][/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]
1 - n = n - 11 - n = n - 1
Solve for [I]n[/I] in the equation 1 - n = n - 1
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables -n and n. To do that, we subtract n from both sides
-n + 1 - n = n - 1 - n
[SIZE=5][B]Step 2: Cancel n on the right side:[/B][/SIZE]
-2n + 1 = -1
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 1 and -1. To do that, we subtract 1 from both sides
-2n + 1 - 1 = -1 - 1
[SIZE=5][B]Step 4: Cancel 1 on the left side:[/B][/SIZE]
-2n = -2
[SIZE=5][B]Step 5: Divide each side of the equation by -2[/B][/SIZE]
-2n/-2 = -2/-2
n = [B]1
[URL='https://www.mathcelebrity.com/1unk.php?num=1-n%3Dn-1&pl=Solve']Source[/URL][/B]
1 integer is 7 times another. If the product of the 2 integers is 448, then find the integers.1 integer is 7 times another. If the product of the 2 integers is 448, then find the integers.
Let the first integer be x and the second integer be y. We have the following two equations:
[LIST=1]
[*]x = 7y
[*]xy = 448
[/LIST]
Substitute (1) into (2), we have:
(7y)y = 448
7y^2 = 448
Divide each side by 7
y^2 = 64
y = -8, 8
We use 8, since 8*7 = 56, and 56*8 =448. So the answer is [B](x, y) = (8, 56)[/B]
1/2 the difference of x and 41/2 the difference of x and 4
The difference of x and 4:
x - 4
1/2 of the difference means we divide x -4 by 2:
[B](x - 4)/2[/B]
1/2n + 1&1/2n = -101/2n + 1&1/2n = -10
[URL='https://www.mathcelebrity.com/fraction.php?frac1=1%261%2F2&frac2=3%2F8&pl=Simplify']1&1/2 = 3/2[/URL] so we have:
n/2 + 3n/2 = -10
4n/2 = -10
2n = -10
Divide each side by 2:
2n/2 = -10/2
n = [B]-5[/B]
1/3ab^2=6 for a1/3ab^2=6 for a
Multiply each side by 3:
ab^2 = 18
Divide each side by b^2
a = 18/b^2
1/4 of the difference of 6 and a number is 2001/4 of the difference of 6 and a number is 200
Take this [B]algebraic expression[/B] in 4 parts:
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x
[*]The difference of 6 and a number means we subtract x from 6: 6 - x
[*]1/4 of the difference means we divide 6 - x by 4: (6 - x)/4
[*]Finally, the phrase [I]is[/I] means an equation, so we set (6 - x)/4 equal to 200
[/LIST]
[B](6 - x)/4 = 200[/B]
1/a + 1/b = 1/2 for a1/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 + 3/5 = 11/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 = 3/1921/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 divided by the sum of 4 and u10 divided by the sum of 4 and u
Take this algebraic expression in parts:
The sum of 4 and u means we add 4 to u:
4 + u
Next, we divide 10 by this sum:
[B]10/(4 + u)[/B]
10 more than a number z, divided by k10 more than a number z, divided by k
The phrase [I]a number[/I] means an arbitrary variable, lets call it x.
10 more than a number means we add 10 to x:
x + 10
We divide this quantity by k:
[B](x + 10)/k[/B]
10 times the square of a number w divided by 1210 times the square of a number w divided by 12
The square of a number w
w^2
10 times this
10w^2
Divided by 12
[B]10w^2/12[/B]
10 x 12 divided by 910 x 12 divided by 9
12/9
1.3333333
Then multiply by 10:
[B]13.33333333[/B]
100n = 100100n = 100
Solve for [I]n[/I] in the equation 100n = 100
[SIZE=5][B]Step 1: Divide each side of the equation by 100[/B][/SIZE]
100n/100 = 100/100
n = [B]1[/B]
10ac-x/11=3 for a10ac-x/11=3 for a
Add x/11 to each side of the equation to isolate a:
10ac - x/11 + x/11 = 3 + x/11
Cancelling the x/11 on the left side, we get:
10ac = 3 + x/11
Divide each side by 10c to isolate a:
10ac/10c = 3 + x/11
Cancelling the 10c on the left side, we get:
a = [B]3/10c + x/110c[/B]
10n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 610n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 6
Solve for [I]n[/I] in the equation 10n - 9n + 8n - 7n + 6n = 10 - 9 + 8 - 7 + 6
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(10 - 9 + 8 - 7 + 6)n = 8n
[SIZE=5][B]Step 2: Group the constant terms on the right hand side:[/B][/SIZE]
10 - 9 + 8 - 7 + 6 = 8
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
8n = + 8
[SIZE=5][B]Step 4: Divide each side of the equation by 8[/B][/SIZE]
8n/8 = 8/8
n = [B]1[/B]
10n = 0.510n = 0.5
Solve for [I]n[/I] in the equation 10n = .5
[SIZE=5][B]Step 1: Divide each side of the equation by 10[/B][/SIZE]
10n/10 = .5/10
n = [B]0.05
[URL='https://www.mathcelebrity.com/1unk.php?num=10n%3D.5&pl=Solve']Source[/URL][/B]
12 divided into groups of s12 divided into groups of s
We build our algebraic expression as follows:
[B]12/s[/B]
15y + 13/c = m for y15y + 13/c = m for y
Subtract 13/c from each side to isolate the y term:
15y + 13/c - 13/c = m - 13/c
Cancel the 13/c on the left side and we get
15y = m - 13/c
Now, divide each side by 15 to isolate y:
15y/15 = (m - 13/c)/15
Cancel the 15 on the left side, and we get:
y = [B](m - 13/c)/15[/B]
175 students separated into n classes is 25175/n = 25
25n = 175
Divide each side by 25
[B]n = 7 classes[/B]
175 students separated into n classes is 25175 students separated into n classes is 25
[U]Divide 175 by n[/U]
175/n
[U]The word [I]is[/I] means equal to, so set this expression equal to 25[/U]
175/n = 25
[U]Cross multiply[/U]
25n = 175
[U]Divide each side by 25[/U]
[B]n = 7[/B]
2 consecutive even integers that equal 118Let x be the first even integer. That means the next consecutive even integer must be x + 2.
Set up our equation:
x + (x + 2) = 118
Group x terms
2x + 2 = 118
Subtract 2 from each side
2x = 116
Divide each side by 2
x = 58
Which means the next consecutive even integer is 58 + 2 = 60
So our two consecutive even integers are [B]58, 60[/B]
Check our work:
58 + 60 = 118
2 numbers that are equal have a sum of 602 numbers that are equal have a sum of 60
Let's choose 2 arbitrary variables for the 2 numbers
x, y
Were given 2 equations:
[LIST=1]
[*]x = y <-- Because we have the phrase [I]that are equal[/I]
[*]x + y = 60
[/LIST]
Because x = y in equation (1), we can substitute equation (1) into equation (2) for x:
y + y = 60
Add like terms to get:
2y = 60
Divide each side by 2:
2y/2 = 60/2
Cancel the 2's and we get:
y = [B]30
[/B]
Since x = y, x = y = 30
x = [B]30[/B]
2 times half of a numberA 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 the sum of 3 and 5 divided by 102 times the sum of 3 and 5 divided by 10
The sum of 3 and 5 is written as:
3 + 5
2 times this sum:
2(3 + 5)
Then, we divide this by 10:
[B]2(3 + 5)/10[/B]
[B][/B]
If the problem asks you to simplify this, you [URL='https://www.mathcelebrity.com/order-of-operations-calculator.php?num=2%283%2B5%29%2F10&pl=Perform+Order+of+Operations']type this expression into our search engine[/URL] and get:
[B]1.6[/B]
2 times x divided by 4 times y2 times x divided by 4 times y
2 times x:
2x
4 times y:
4y
2 times x divided by 4 times y
[B]2x/4y[/B]
20 teachers made a bulk purchase of some textbooks. The teachers received a 24% discount for the bul20 teachers made a bulk purchase of some textbooks. The teachers received a 24% discount for the bulk purchase, which originally cost $5230. Assuming the cost was divided equally among the teachers, how much did each teacher pay?
[U]Calculate Discount Percent:[/U]
If the teachers got a 24% discount, that means they paid:
100% - 24% = 76%
[URL='https://www.mathcelebrity.com/perc.php?num=+5&den=+8&num1=+16&pct1=+80&pct2=+90&den1=+80&pct=76&pcheck=4&decimal=+65.236&astart=+12&aend=+20&wp1=20&wp2=30&pl=Calculate']76% as a decimal[/URL] = 0.76 (Discount Percent)
[U]Calculate discount price:[/U]
Discount Price = Full Price * (Discount Percent)
Discount Price = 5230 * 0.76
Discount Price = 3974.80
Price per teacher = Discount Price / Number of Teachers
Price per teacher = 3974.80 / 20
Price per teacher = [B]$198.74[/B]
200 apples at $69.99 how much is each apple$69.99 per apple / 200 apples
We want the price per apple. Divide top and bottom by 200
$0.35 per apple.
21 apples and 49 pears. What is the largest number of baskets to make and still use up all the fruit21 apples and 49 pears. What is the largest number of baskets to make and still use up all the fruit
We use our [URL='https://www.mathcelebrity.com/gcflcm.php?num1=21&num2=49&num3=&pl=GCF+and+LCM']greatest common factor calculator for GCF(21, 49)[/URL] to get:
GCF(21, 49) = 7
This means with [B]7 baskets[/B]:
[LIST]
[*]We divide 21 apples by 7 to get 3 apples per basket
[*]We divide 49 pears by 7 to get 7 pears per basket
[/LIST]
22 & 1/2 / 1/8 =22 & 1/2 / 1/8 =
22 & 1/2 = 45/2
[URL='https://www.mathcelebrity.com/fraction.php?frac1=45%2F2&frac2=1%2F8&pl=Divide']45/2 / 1/8[/URL] = [B]180[/B]
2d = (a - b)/(b - c) for d2d = (a - b)/(b - c) for d
Divide each side by 2 to isolate d:
2d/2 = (a - b)/2(b - c)
Cancel the 2's on the left side, we get:
d = [B](a - b)/2(b - c)[/B]
2n + 10 = 3n + 52n + 10 = 3n + 5
Solve for [I]n[/I] in the equation 2n + 10 = 3n + 5
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 2n and 3n. To do that, we subtract 3n from both sides
2n + 10 - 3n = 3n + 5 - 3n
[SIZE=5][B]Step 2: Cancel 3n on the right side:[/B][/SIZE]
-n + 10 = 5
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 10 and 5. To do that, we subtract 10 from both sides
-n + 10 - 10 = 5 - 10
[SIZE=5][B]Step 4: Cancel 10 on the left side:[/B][/SIZE]
-n = -5
[SIZE=5][B]Step 5: Divide each side of the equation by -1[/B][/SIZE]
-1n/-1 = -5/-1
n = [B]5[/B]
2n + 8 = 242n + 8 = 24
Solve for [I]n[/I] in the equation 2n + 8 = 24
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 8 and 24. To do that, we subtract 8 from both sides
2n + 8 - 8 = 24 - 8
[SIZE=5][B]Step 2: Cancel 8 on the left side:[/B][/SIZE]
2n = 16
[SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE]
2n/2 = 16/2
n = [B]8[/B]
2n - 1&1/2n = 592n - 1&1/2n = 59
1&1/2n = 3/2n or 1.5n
So we have:
2n - 1.5n = 59
Solve for [I]n[/I] in the equation 2n - 1.5n = 59
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(2 - 1.5)n = 0.5n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
0.5n = + 59
[SIZE=5][B]Step 3: Divide each side of the equation by 0.5[/B][/SIZE]
0.5n/0.5 = 59/0.5
n = [B]118[/B]
2n - 7 = 02n - 7 = 0
Solve for [I]n[/I] in the equation 2n - 7 = 0
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -7 and 0. To do that, we add 7 to both sides
2n - 7 + 7 = 0 + 7
[SIZE=5][B]Step 2: Cancel 7 on the left side:[/B][/SIZE]
2n = 7
[SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE]
2n/2 = 7/2
n = [B]3.5[/B]
2n = 4n2n = 4n
Solve for [I]n[/I] in the equation 2n = 4n
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 2n and 4n. To do that, we subtract 4n from both sides
2n - 4n = 4n - 4n
[SIZE=5][B]Step 2: Cancel 4n on the right side:[/B][/SIZE]
-2n = 0
[SIZE=5][B]Step 3: Divide each side of the equation by -2[/B][/SIZE]
-2n/-2 = 0/-2
n = [B]0[/B]
2x - b/y = 4c for y2x - 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/5 - 9y = 6 for x2x/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 x2x/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]
2y divided by the sum of 3x and 52y divided by the sum of 3x and 5
The sum of 3x and 5 means we add 5 to 3x:
3x + 5
2y divided by the sum of 3x and 5:
[B]2y/(3x + 5)[/B]
3 children painted a local park track of 5000m, Alex painted 70m, Dell painted 15m, and Tony painted3 children painted a local park track of 5000m, Alex painted 70m, Dell painted 15m, and Tony painted 35m. This pattern continues to the end of the track. What percentage of the park did each child paint?
70 + 15 + 35 = 120
When we take[URL='https://www.mathcelebrity.com/modulus.php?num=5000mod120&pl=Calculate+Modulus'] 5000 divided by 120[/URL], we get:
41 remainder 80
So we have:
[LIST]
[*]Alex: 70 * 41 = 2870
[*]Dell: 15 * 41 = 615
[*]Tony: 35 * 41 = 1435
[/LIST]
Now Alex goes next, and paints the full 70. So he has:
2870 + 70 = 2940
Dell goes next, and paints the last 10
615 + 10 = 625
Now for percentages:
[LIST]
[*]Alex: 2940/5000 = [B]58.8%[/B]
[*]Dell: 625/5000 = [B]12.5%[/B]
[*]Tony: 1435/5000 = [B]28.7%[/B]
[/LIST]
3 quarts of oil is $6.99 how much is one quart of oil?3 quarts of oil is $6.99 how much is one quart of oil?
$6.99 / 3 quarts
Divide top and bottom by 3:
[B]$2.33 / 1 quart[/B]
3/4 a number b divided by 53/4 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]
32 girls and 52 boys were on an overseas learning trip . They were divided into as many groups as po32 girls and 52 boys were on an overseas learning trip . They were divided into as many groups as possible where the number of groups of girls and the number of groups of boys is the same .how many boys and how many girls were in each group
We want a number such that our total members divided by this number equals our group size.
We take the greatest common factor (32,52) = 4
Therefore, we have:
[LIST]
[*][B]32/4 = 8 girls in each group[/B]
[*][B]52/4 = 13 boys in each group[/B]
[/LIST]
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]
3f,subtract g from the result, then divide what you have by h3f,subtract g from the result, then divide what you have by h
Take this algebraic expression in pieces:
3f subtract g means we subtract the variable g from the expression 3f:
3f - g
Divide what we have by h, means we take the result above, 3f - g, and divide it by h:
[B](3f - g)/h[/B]
3k^3 = rt for t3k^3 = rt for t
This is a literal equation. Let's divide each side of the equation by r, to isolate t:
3k^3/r = rt/r
Cancel the r's on the right side, and we get:
t = [B]3k^3/r[/B]
3n/5 = 13n/5 = 1
Cross multiply:
3n = 5 * 1
3n = 5
Divide each side by 3:
3n/3 = 5/3
n = 5/3
4 divided by sin60 degrees4 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]
4d/a - 9 = g for a4d/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]
4n - 8 = n + 14n - 8 = n + 1
Solve for [I]n[/I] in the equation 4n - 8 = n + 1
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 4n and n. To do that, we subtract n from both sides
4n - 8 - n = n + 1 - n
[SIZE=5][B]Step 2: Cancel n on the right side:[/B][/SIZE]
3n - 8 = 1
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -8 and 1. To do that, we add 8 to both sides
3n - 8 + 8 = 1 + 8
[SIZE=5][B]Step 4: Cancel 8 on the left side:[/B][/SIZE]
3n = 9
[SIZE=5][B]Step 5: Divide each side of the equation by 3[/B][/SIZE]
3n/3 = 9/3
n = [B]3[/B]
5 -8| -2n|=-75Subtract 5 from each side:
-8|-2n| = -80
Divide each side by -8
|-2n| = 10
Since this is an absolute value equation, we need to setup two equations:
-2n = 10
-2n = -10
Solving for the first one by dividing each side by -2, we get:
n = -5
Solving for the second one by dividing each side by -2, we get:
n = 5
5 books and 5 bags cost $175. What is the cost of 2 books and 2 bags5 books and 5 bags cost $175. What is the cost of 2 books and 2 bags
Let the cost of each book be b and the cost of each bag be c. We're given
5b + 5c = 175
We can factor this as:
5(b + c) = 175
Divide each side of the equation by 5, we get:
(b + c) = 35
The problem asks for 2b + 2c
Factor out 2:
2(b + c)
we know from above that (b + c) = 35, so we substitute:
2(35)
[B]70[/B]
5 girls share 4 sandwiches. What fraction of the sandwich does each girl get5 girls share 4 sandwiches. What fraction of the sandwich does each girl get?
We want to know sandwiches per girls. So we divide:
4 sandwiches per 5 girls
[B]4/5[/B]
5 more than the reciprocal of a number5 more than the reciprocal 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 reciprocal of this number means we divide 1 over x:
1/x
5 more means we add 5 to 1/x
[B]1/x + 5[/B]
5 times a number increased by 4 is divided by 6 times the same number5 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 x, divided by 75 times x
5x
Divided by 7
5x/7
5 times y, divided by 85 times y
5y
Divided by 8
5y/8
5n - 5 = 855n - 5 = 85
Solve for [I]n[/I] in the equation 5n - 5 = 85
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -5 and 85. To do that, we add 5 to both sides
5n - 5 + 5 = 85 + 5
[SIZE=5][B]Step 2: Cancel 5 on the left side:[/B][/SIZE]
5n = 90
[SIZE=5][B]Step 3: Divide each side of the equation by 5[/B][/SIZE]
5n/5 = 90/5
n = [B]18[/B]
6 is divided by square of a number6 is divided by square of a number
The phrase [I]a number [/I]means an arbitrary variable, let's call it x.
x
the square of this means we raise x to the power of 2:
x^2
Next, we divide 6 by x^2:
[B]6/x^2[/B]
6 times a number multiplied by 3 all divided by 46 times a number multiplied by 3 all divided by 4
Take this algebraic expression in parts:
[LIST]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x
[*]6 times a number: 6x
[*]Multiplied by 3: 3(6x) = 18x
[*]All divided by 4: 18x/4
[/LIST]
We can simplify this:
We type 18/4 into our search engine, simplify, and we get 9/2. So our answer is:
[B]9x/2[/B]
6 times the reciprocal of a number equals 3 times the reciprocal of 7 .6 times the reciprocal of a number equals 3 times the reciprocal of 7 .
This is an algebraic expression. Let's take it in parts:
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The reciprocal of a number x means we divide 1 over x:
1/x
6 times the reciprocal means we multiply 6 by 1/x:
6/x
The reciprocal of 7 means we divide 1/7
1/7
3 times the reciprocal means we multiply 1/7 by 3:
3/7
Now, the phrase [I]equals[/I] mean an equation, so we set 6/x = 3/7
[B]6/x = 3/7[/B] <-- This is our algebraic expression
If the problem asks you to solve for x, then [URL='https://www.mathcelebrity.com/prop.php?num1=6&num2=3&den1=x&den2=7&propsign=%3D&pl=Calculate+missing+proportion+value']we type this proportion in our search engine[/URL] and get:
x = 14
6 times the sum of a number and 5 is 166 times the sum of a number and 5 is 16
A number represents an arbitrary variable, let's call it x
x
The sum of x and 5
x + 5
6 times the sum of x and 5
6(x + 5)
Is means equal to, so set 6(x + 5) equal to 16
[B]6(x + 5) = 16 <-- This is our algebraic expression
Solve for x[/B]
Multiply through:
6x + 30 = 16
Subtract 30 from each side:
6x - 30 + 30 = 16 - 30
6x = -14
Divide each side by 6
6x/6 = -14/6
Simplify this fraction by dividing top and bottom by 2:
x = [B]-7/3
[MEDIA=youtube]oEx5dsYK7DY[/MEDIA][/B]
6 times y divided by x squared6 times y divided by x squared
6 times y:
6y
x squared means we raise x to the power of 2:
x^2
The phrase [I]divided by[/I] means we have a fraction:
[B]6y/x^2[/B]
6 times y divided by x squared6 times y divided by x squared
6 times y:
6y
x squared means we raise x to the power of 2:
x^2
The phrase [I]divided by[/I] means we divide 6y by x^2:
[B]6y/x^2[/B]
64 divided by the cube of y64 divided by the cube of y
The cube of y means y raised to the 3rd power:
y^3
64 divided by this:
[B]64/y^3[/B]
69 divided by the sum of 10 and y69 divided by the sum of 10 and y
The sum of 10 and y
10 + y
69 divided by this
[B]69/(10 + y)[/B]
7 minus a number all divided by 47 minus a number all divided by 4
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
7 minus a number
7 - x
All divided by 4:
[B](7 - x)/4[/B]
7n + 4 + n - 5 = 637n + 4 + n - 5 = 63
Solve for [I]n[/I] in the equation 7n + 4 + n - 5 = 63
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(7 + 1)n = 8n
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
4 - 5 = -1
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
8n - 1 = + 63
[SIZE=5][B]Step 4: Group constants:[/B][/SIZE]
We need to group our constants -1 and 63. To do that, we add 1 to both sides
8n - 1 + 1 = 63 + 1
[SIZE=5][B]Step 5: Cancel 1 on the left side:[/B][/SIZE]
8n = 64
[SIZE=5][B]Step 6: Divide each side of the equation by 8[/B][/SIZE]
8n/8 = 64/8
n = [B]8[/B]
8 sweets are shared among 4 pupils. how many does each pupil get8 sweets divided by 4 pupils = 2 sweets per pupil.
We can also write this as a proportion:
8 sweets x sweets
---------- = ------------
4 pupils 1 pupil
Express this as 8/4 = x/1.
[URL='http://www.mathcelebrity.com/prop.php?num1=8&num2=x&den1=4&den2=1&propsign=%3D&pl=Calculate+missing+proportion+value']Enter that into the search engine[/URL]
x = 2
8 to the power of 8 divided by 88 to the power of 8 divided by 8
8 to the power of 8
8^8
Divided by 8
[B]8^8/8[/B]
8 to the power of x over 2 to the power of y8 to the power of x over 2 to the power of y
Step 1: 8 to the power of x means we take 8 and raise it to an exponent of x:
8^x
Step 2: 2 to the power of y means we take 2 and raise it to an exponent of y:
2^y
Step 3: The word [I]over[/I] means a quotient, also known as divided by, so we have:
[B]8^x/2^y
[MEDIA=youtube]SPQKOt5EoqA[/MEDIA][/B]
85 bags of nuts are to be divided among 18 friends. Each bag contains 15 nuts. How many nuts will ea85 bags of nuts are to be divided among 18 friends. Each bag contains 15 nuts. How many nuts will each friend get?
[B][U]Calculate the total nuts:[/U][/B]
Total Nuts = Total Bags * Nuts Per Bag
Total Nuts = 85 * 15
Total Nuts = 1,275
[B][U]Figure out how many nuts each person gets:[/U][/B]
Nuts per person = Total Nuts / Friends
Nuts per person = 1,275 / 18
Nuts per person = [B]70.83[/B]
9 divided by the quantity x plus y9 divided by the quantity x plus y
The quantity x plus y:
x + y
9 divided by this quantity:
[B]9/(x + y)[/B]
9 divided by the sum of x and 4 is equal to 6 divided by x minus 49 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 rulers cost the same as 11 erasers. One eraser cost 0.09 cents. What is the cost of 1 ruler9 rulers cost the same as 11 erasers. One eraser cost 0.09 cents. What is the cost of 1 ruler
Let the cost of a ruler be r. We're given:
9r = 11(0.09)
9r = 0.99
Divide each side by 9 and we get:
r = [B]0.11[/B]
993 cold drinks bottles are to be placed in crates. Each crate can hold 9 bottles. How many crates w993 cold drinks bottles are to be placed in crates. Each crate can hold 9 bottles. How many crates would be needed and how many bottles will remain?
Let c equal the number of crates
9 bottles per crate * c = 993
9c = 993
Solve for [I]c[/I] in the equation 9c = 993
[SIZE=5][B]Step 1: Divide each side of the equation by 9[/B][/SIZE]
9c /9 = 993/9
c = 110.33333333333
Since we can't have fractional crates, we round up 1 to the next full crate
c = [B]111[/B]
A 128 ounce carton of milk states that there are 20 servings. How many ounces are in a serving?A 128 ounce carton of milk states that there are 20 servings. How many ounces are in a serving?
We divide 128 ounces by 20 servings to get ounces per serving:
128 ounces / 20 servings
[B]6.4 ounces / serving[/B]
a 24 sheet of aluminum is to be cut into 2.5 strips. how wide is the remaining wasted piece of alumia 24 sheet of aluminum is to be cut into 2.5 strips. how wide is the remaining wasted piece of aluminum
Divide 24 by 2.5 to get number of sheets:
24/2.5 = 9.6
So we have 9 full sheets. Which means each strip is [B]0.6 wide[/B]
A 3 gallon bottle of bleach cost $16.32. What is the price per cup?A 3 gallon bottle of bleach cost $16.32. What is the price per cup?
We're given 16.32 / 3 gallons
Divide the top and bottom of the fraction by 3 to get the cost per gallon:
16.32/3 = 5.44 gallon
Using our [URL='https://www.mathcelebrity.com/liqm.php?quant=1&pl=Calculate&type=gallon']measurement converter[/URL], we see that:
1 gallon = 16 cups
So 5.44 /16 cups=[B]$0.34 per cup[/B]
A 3-gallon bucket of paint costs $87.12. What is the price per quart?A 3-gallon bucket of paint costs $87.12. What is the price per quart?
3 gallons equals 12 quarts with our [URL='https://www.mathcelebrity.com/liqm.php?quant=3&pl=Calculate&type=gallon#quart']conversion calculator[/URL]. We divide 87.12 for 12 quarts by 12:
[URL='https://www.mathcelebrity.com/perc.php?num=87.12&den=12&pcheck=1&num1=16&pct1=80&pct2=70&den1=80&idpct1=10&hltype=1&idpct2=90&pct=82&decimal=+65.236&astart=12&aend=20&wp1=20&wp2=30&pl=Calculate']87.12 / 12[/URL] = [B]$7.26 per quart[/B]
A 6000 seat theater has tickets for sale at $24 and $40. How many tickets should be sold at each priA 6000 seat theater has tickets for sale at $24 and $40. How many tickets should be sold at each price for a sellout performance to generate a total revenue of $188,800?
Let x be the number of $24 tickets, and y be the number of $40 tickets. We have:
[LIST=1]
[*]24x + 40y = 188,800
[*]x + y = 6,000
[*]Rearrange (2) to solve for x: x = 6000 - y
[*]Plug in (3) to (1):
[/LIST]
24(6000 - y) + 40y = 188800
144,000 - 24y + 40y = 188,800
16y + 144,000 = 188,800
Subtract 144,000 from each side:
16y = 44,800
Divide each side by 16
y = 2,800 ($40 tickets)
Plug this into (2)
x + 2,800 = 6000
Subtract 2,800 from each side:
x = 3,200 ($24 tickets)
A 7-foot piece of cotton cloth costs $3.36. What is the price per inch?A 7-foot piece of cotton cloth costs $3.36. What is the price per inch?
Using [URL='https://www.mathcelebrity.com/linearcon.php?quant=7&pl=Calculate&type=foot']our length converter[/URL], we see that:
7 feet = 84 inches
So $3.36 for 84 inches. We [URL='https://www.mathcelebrity.com/perc.php?num=3.36&den=84&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']divide $3.36 by 84[/URL] to get the cost per inch:
$3.36/84 = [B]0.04 per inch[/B]
A 98-inch piece of wire must be cut into two pieces. One piece must be 10 inches shorter than the otA 98-inch piece of wire must be cut into two pieces. One piece must be 10 inches shorter than the other. How long should the pieces be?
The key phrase in this problem is [B]two pieces[/B].
Declare Variables:
[LIST]
[*]Let the short piece length be s
[*]Let the long piece length be l
[/LIST]
We're given the following
[LIST=1]
[*]s = l - 10
[*]s + l = 98 (Because the two pieces add up to 98)
[/LIST]
Substitute equation (1) into equation (2) for s:
l - 10+ l = 98
Group like terms:
2l - 10 = 98
Solve for [I]l[/I] in the equation 2l - 10 = 98
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -10 and 98. To do that, we add 10 to both sides
2l - 10 + 10 = 98 + 10
[SIZE=5][B]Step 2: Cancel 10 on the left side:[/B][/SIZE]
2l = 108
[SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE]
2l/2 = 108/2
l = [B]54[/B]
To solve for s, we substitute l = 54 into equation (1):
s = 54 - 10
s = [B]44[/B]
Check our work:
The shorter piece is 10 inches shorter than the longer piece since 54 - 44 = 10
Second check: Do both pieces add up to 98
54 + 44 ? 98
98 = 98
a = v^2/r for ra = 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 barn contains cows, ducks, and a 3-legged dog named Tripod. There are twice as many cows as ducksA barn contains cows, ducks, and a 3-legged dog named Tripod. There are twice as many cows as ducks in the barn and a total of 313 legs. How many ducks are there in the barn?
[LIST]
[*]Let the number of ducks be d. Duck legs = 2 * d = 2d
[*]Number of cows = 2d. Cow legs = 4 * 2d = 8d
[*]1 dog Tripod has 3 legs
[/LIST]
Total legs:
2d + 8d + 3 = 313
Solve for [I]d[/I] in the equation 2d + 8d + 3 = 313
[SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE]
(2 + 8)d = 10d
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
10d + 3 = + 313
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 3 and 313. To do that, we subtract 3 from both sides
10d + 3 - 3 = 313 - 3
[SIZE=5][B]Step 4: Cancel 3 on the left side:[/B][/SIZE]
10d = 310
[SIZE=5][B]Step 5: Divide each side of the equation by 10[/B][/SIZE]
10d/10 = 310/10
d = [B]31[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=2d%2B8d%2B3%3D313&pl=Solve']Source[/URL]
a baseball park charges $4.50 per admission ticket. the park has already sold 42 tickets. how many ma baseball park charges $4.50 per admission ticket. the park has already sold 42 tickets. how many more tickets would they need to sell to earn at least $441?
Let the number of tickets above 42 be t.
A few things to note on this question:
[LIST]
[*]The phrase [I]at least[/I] means greater than or equal to, so we have an inequality.
[*]Earnings = Price * Quantity
[/LIST]
We're given:
Earnings = 4.50 * 42 + 4.5t >= 441
Earnings = 189 + 4.5t >= 441
We want to solve this inequality for t:
Solve for [I]t[/I] in the inequality 189 + 4.5t ? 441
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 189 and 441. To do that, we subtract 189 from both sides
4.5t + 189 - 189 ? 441 - 189
[SIZE=5][B]Step 2: Cancel 189 on the left side:[/B][/SIZE]
4.5t ? 252
[SIZE=5][B]Step 3: Divide each side of the inequality by 4.5[/B][/SIZE]
4.5t/4.5 ? 252.4.5
[B]t ? 56[/B]
a boat traveled 336 km downstream with the current. The trip downstream took 12 hours. write an equaa boat traveled 336 km downstream with the current. The trip downstream took 12 hours. write an equation to describe this relationship
We know the distance (d) equation in terms of rate (r) and time (t) as:
d = rt
We're given d = 336km and t = 12 hours, so we have:
[B]336 km = 12t [/B] <-- this is our equation
Divide each side by 12 to solve for t:
12t/12 = 336/12
t = [B]28 km / hour[/B]
A book publishing company has fixed costs of $180,000 and a variable cost of $25 per book. The booksA book publishing company has fixed costs of $180,000 and a variable cost of $25 per book. The books they make sell for $40 each.
[B][U]Set up Cost Function C(b) where b is the number of books:[/U][/B]
C(b) = Fixed Cost + Variable Cost x Number of Units
C(b) = 180,000 + 25(b)
[B]Set up Revenue Function R(b):[/B]
R(b) = 40b
Set them equal to each other
180,000 + 25b = 40b
Subtract 25b from each side:
15b = 180,000
Divide each side by 15
[B]b = 12,000 for break even[/B]
A cake is to be divided into 8 equal parts. After division, each equal portion is again divided intoA cake is to be divided into 8 equal parts. After division, each equal portion is again divided into 2 equal individual parts. How big is each of the new equal parts?
1 cake * 8 parts * 2 parts = 16 parts.
So each slice is 1/16 of a cake.
A car is purchased for $24,000 . Each year it loses 30% of its value. After how many years will tA 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 - 0.3)^t
D(t) = 24000 * (0.7)^t
The problem asks for D(t)<=7300
24000 * (0.7)^t = 7300
Divide each side by 24000
(0.7)^t = 7300/24000
(0.7)^t= 0.30416666666
Take the natural log of both sides:
LN(0.7^t) = -1.190179482215518
Using the natural log identities, we have:
t * LN(0.7) = -1.190179482215518
t * -0.35667494393873245= -1.190179482215518
Divide each side by -0.35667494393873245
t = 3.33687437943
[B]Rounding this up, we have t = 4[/B]
A car repair bill was $441. This included $153 for parts and four hours of labor . Find the hourly rA car repair bill was $441. This included $153 for parts and four hours of labor . Find the hourly rate I was charge for labor
Subtract the cost of parts from the total repair bill to get the labor cost:
Labor Cost = Total Bill - Parts Cost
Labor Cost = 441 - 153
Labor Cost = 288
Labor Cost can be broken down into Labor divided by hours
Hourly Labor Rate = Labor Cost / Labor Hours
Hourly Labor Rate = = 288 / 4
Hourly Labor Rate = [B]72[/B]
A cell phone provider is offering an unlimited data plan for $70 per month or a 5 GB plan for $55 peA cell phone provider is offering an unlimited data plan for $70 per month or a 5 GB plan for $55 per month. However, if you go over your 5 GB of data in a month, you have to pay an extra $10 for each GB. How many GB would be used to make both plans cost the same?
Let g be the number of GB.
The limited plan has a cost as follows:
C = 10(g - 5) + 55
C = 10g - 50 + 55
C = 10g + 5
We want to set the limited plan equal to the unlimited plan and solve for g:
10g + 5 = 70
Solve for [I]g[/I] in the equation 10g + 5 = 70
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 5 and 70. To do that, we subtract 5 from both sides
10g + 5 - 5 = 70 - 5
[SIZE=5][B]Step 2: Cancel 5 on the left side:[/B][/SIZE]
10g = 65
[SIZE=5][B]Step 3: Divide each side of the equation by 10[/B][/SIZE]
10g/10 = 65/10
g = [B]6.5[/B]
Check our work for g = 6.5:
10(6.5) + 5
65 + 5
70
A certain Illness is spreading at a rate of 10% per hour. How long will it take to spread to 1,200 pA certain Illness is spreading at a rate of 10% per hour. How long will it take to spread to 1,200 people if 3 people initially exposes? Round to the nearest hour.
Let h be the number of hours. We have the equation:
3 * (1.1)^h = 1,200
Divide each side by 3:
1.1^h = 400
[URL='https://www.mathcelebrity.com/natlog.php?num=1.1%5Eh%3D400&pl=Calculate']Type this equation into our search engine [/URL]to solve for h:
h = 62.86
To the nearest hour, we round up and get [B]h = 63[/B]
A company has 12,600 employees. Of these, 1/4 drive alone to work, 1/6 car pool, 1/8 use public tranA company has 12,600 employees. Of these, 1/4 drive alone to work, 1/6 car pool, 1/8 use public transportation, 1/9 cycle, and the remainder use other methods of transportation. How many employees use each method of transportation?
Find the remainder fraction:
Remainder = 1 - (1/4 + 1/6 + 1/8 + 1/9)
The least common multiple of 4, 6, 8, 9 is 72. So we divide 72 by each fraction denominator to get our multiplier:
1/4 = 18/72
1/6 = 12/72
1/8 = 9/72
1/9 = 8/72
Add those all up:
(18 + 12 + 9 + 8)/72
47/72
Now subtract the other methods out from 1 to get the remainder of who use other methods:
Remainder = 1 - 47/72
Since 1 = 72/72, we have:
(72 - 47)/72
[B]25/72[/B]
A company now has 4900 employees nationwide. It wishes to reduce the number of employees by 300 perA company now has 4900 employees nationwide. It wishes to reduce the number of employees by 300 per year through retirements, until its total employment is 2560. How long will this take?
Figure out how many reductions are needed
4900 - 2560 = 2340
We want 300 per year for retirements, so let x equal how many years we need to get 2340 reductions.
300x = 2340
Divide each side by 300
x = 7.8 years.
If we want full years, we would do 8 full years
A construction crew has just built a new road. They built 43.75 kilometers of road at a rate of 7 kiA construction crew has just built a new road. They built 43.75 kilometers of road at a rate of 7 kilometers per week. How many weeks did it take them?
Let w = weeks
7 kilometers per week * w = 43.75
To solve for w, we divide each side of the equation by 7:
7w/7 = 43.75/7
Cancel the 7's, we get:
w = [B]6.25 [/B]
A construction worker can lift 220 lb, while an architect can only lift 40 lb. The construction workA construction worker can lift 220 lb, while an architect can only lift 40 lb. The construction worker can lift how many times what the architect can lift?
[URL='https://www.mathcelebrity.com/perc.php?num=220&den=40&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']We divide 220 by 40 to get the multiplier:[/URL]
220/40 = [B]5.5 times[/B]
A cookie recipe uses 10 times as much flour as sugar. If the total amount of these ingredients is 8A 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 dad gave his 3 sons each the same amount of money in an envelope. He took $20 from one son for getA dad gave his 3 sons each the same amount of money in an envelope. He took $20 from one son for getting a D on a math test and he gave another son an extra $35 for doing extra chores. Combined, the sons had $81. Figure out how much each son had.
Let x, y, and z be the money each son received.
To begin, x = y = z.
But then, Dad took 20 from son X and gave it to son Y.
So now, x = y - 20
Next, he gave Son Z an extra $35 for chores
So z is now y + 35 since y and z used to be equal
Combined, they all have 81.
x + y + z = 181
But with the changes, it is:
(y - 20) + y + (y + 35)
Combine like terms:
3y - 20 + 35 = 81
3y + 15 = 81
Subtract 15 from each side:
3y = 66
Divide each side by 3 to isolate y
y = 22
Since x = y - 20, x = 2
Since z = y + 35, we have z = 57
Checking our work, 2 + 22 + 57 = 81.
A desk drawer contains 10 blue pencils, 7 red pencils, and 8 green pencils. Without looking, you draA desk drawer contains 10 blue pencils, 7 red pencils, and 8 green pencils. Without looking, you draw out a pencil and then draw out a second pencil without returning the first pencil. What is the probability that the first pencil and the second pencil are both green?
We are drawing without replacement. Take each draw probability:
[LIST=1]
[*]First draw, we have a total of 10 + 7 + 8 = 25 pencils to choose from. P(Green) = 8/25
[*]Next draw, we only have 24 total pencils, and 7 green pencils since we do not replace. Therefore, we have P(Green)= 7/24
[/LIST]
Since both events are independent, we have:
P(Green) * P(Green) = 8/25 * 7/24
P(Green) * P(Green) = 56/600
Using our [URL='http://www.mathcelebrity.com/gcflcm.php?num1=56&num2=600&num3=&pl=GCF']GCF Calculator[/URL], we see the greatest common factor of 56 and 600 is 8. So we divide top and bottom of the fraction by 8.
[B]P(Green) * P(Green) = 7/75[/B]
A farmer is taking her eggs to the market in a cart, but she hits a pothole, which knocks over allA farmer is taking her eggs to the market in a cart, but she hits a
pothole, which knocks over all the containers of eggs. Though she is
unhurt, every egg is broken. So she goes to her insurance agent, who
asks her how many eggs she had. She says she doesn't know, but she
remembers somethings from various ways she tried packing the eggs.
When she put the eggs in groups of two, three, four, five, and six
there was one egg left over, but when she put them in groups of seven
they ended up in complete groups with no eggs left over.
What can the farmer figure from this information about the number of
eggs she had? Is there more than one answer?
We need a number (n) that leaves a remainder of 1 when divided by 2, 3, 4, 5, 6 but no remainder when divided by 7.
217 + 84 = [B]301[/B].
Other solutions are multiples of 3 x 4 x 5 x 7, but we want the lowest one here.
A father is K years old and his son is M years younger. The sum of their ages is 53.A father is K years old and his son is M years younger. The sum of their ages is 53.
Father's Age = K
Son's Age = K - M
and we know K + (K - M) = 53
Combine like terms:
2K - M = 53
Add M to each side:
2K - M + M = 53 + M
Cancel the M's on the left side, we get:
2K = 53+ M
Divide each side by 2:
2K/2 = (53 + M)/2
Cancel the 2's on the left side:
K = [B](53 + M)/2[/B]
A financial analyst computed the ROI for all companies listed on the NYSE. She found that the mean oA financial analyst computed the ROI for all companies listed on the NYSE. She found that the mean of this distribution was 10% with standard deviation of 5%. She is interested in examining further those companies whose ROI is between 14% and 16% of the approximately 1,500 companies listed on the exchange, how many are of interest of her?
First, use our [URL='http://www.mathcelebrity.com/zscore.php?z=p%280.14%3Cz%3C0.16%29&pl=Calculate+Probability']z-score calculator[/URL] to get P(0.14 < z < 0.16) = 0.007889
Divide that by 2 for two-tail test to get0.003944729
Use the NORMSINV(0.003944729) in Excel to get the Z value of 2.66
Therefore, the companies of interest are 2.66 * 1500 * 0.10 = [B]399[/B]
A first number plus twice a second number is 6. Twice the first number plus the second totals 15. FiA 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. Twice the first number plus the second totals 23. FiA 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 football gained 52 yards during the possession. In the next 3 possessions they gained the same amoA football gained 52 yards during the possession. In the next 3 possessions they gained the same amount of yards each time. If they gained a total of 256 yards, write and solve an equation for how many yards they gained in each of the last 3 possessions.
Subtract 52 initial yards
256 - 52 = 204
Now, divide 204 by 3 possessions
204/3 = [B]68 yards[/B]
A fraction has a value of 3/4. If 7 is added to the numerator, the resulting fraction is equal to thA 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, whA 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 30 students from your school is part of the audience for a TV game show. The total numberA group of 30 students from your school is part of the audience for a TV game show. The total number of people in the audience is 120. What theoretical probability of 5 students from your school being selected as contestants out of 9 possible contestant spots?
We want the probability a student from your school is chosen out of total students times total ways to choose students from your school:
[U]a) P(5 students being selected):[/U]
5/30 * 4/(120 - 30)
5/30 * 4/90
20/2700
[URL='https://www.mathcelebrity.com/fraction.php?frac1=20%2F2700&frac2=3%2F8&pl=Simplify']Simplifying this fraction[/URL], we get:
1/135
[U]b) Total Ways 9 students can be picked from your school:[/U]
9/120
[URL='https://www.mathcelebrity.com/fraction.php?frac1=9%2F120&frac2=3%2F8&pl=Simplify']Simplifying this fraction[/URL], we get:
3/40
Divide a by b:
1/135 / 3/40
40/405
[URL='https://www.mathcelebrity.com/fraction.php?frac1=40%2F405&frac2=3%2F8&pl=Simplify']Simplifying[/URL], we get:
[B]8/81[/B]
A gym has 18 exercise stations, including 2 rowing machines. What is the probability that a randomlA gym has 18 exercise stations, including 2 rowing machines. What is the probability that a randomly selected exercise station will be a rowing machine?
The probability is 2/18.
We can simplify this fraction. Divide top and bottom by 2:
[B]1/9[/B]
A helicopter blade does 3206 full turns in 7 minutes , work out the number of full turns per minuteA helicopter blade does 3206 full turns in 7 minutes , work out the number of full turns per minute
3206 full turns / 7 minutes
[URL='https://www.mathcelebrity.com/fraction.php?frac1=3206%2F7&frac2=3%2F8&pl=Simplify']Divide the fraction by 7 to get turns per minute[/URL]
[B]458 turns per minute[/B]
A helicopter rose vertically 300 m and then flew west 400 m how far was the helicopter from it’s staA helicopter rose vertically 300 m and then flew west 400 m how far was the helicopter from it’s starting point?
The distance forms a right triangle. We want the distance of the hypotenuse.
Using our [URL='http://www.mathcelebrity.com/pythag.php?side1input=300&side2input=400&hypinput=&pl=Solve+Missing+Side']right triangle calculator[/URL], we get a distance of [B]500[/B].
We also could use a shortcut on this problem. If you divide 300 and 400 by 100, you get 3 and 4. Since we want the hypotenuse, you get the famous 3-4-5 triangle ratio. So the answer is 5 * 100 = 500.
A home is to be built on a rectangular plot of land with a perimeter of 800 feet. If the length is 2A home is to be built on a rectangular plot of land with a perimeter of 800 feet. If the length is 20 feet less than 3 times the width, what are the dimensions of the rectangular plot?
[U]Set up equations:[/U]
(1) 2l + 2w = 800
(2) l = 3w - 20
[U]Substitute (2) into (1)[/U]
2(3w - 20) + 2w = 800
6w - 40 + 2w = 800
[U]Group the w terms[/U]
8w - 40 = 800
[U]Add 40 to each side[/U]
8w = 840
[U]Divide each side by 8[/U]
[B]w = 105
[/B]
[U]Substitute w = 105 into (2)[/U]
l = 3(105) - 20
l = 315 - 20
[B]l = 295[/B]
A house costs 3.5 times as much as the lot. Together they sold for $135,000. Find the cost of eachA house costs 3.5 times as much as the lot. Together they sold for $135,000. Find the cost of each.
Let the house cost be h, and the lot cost be l. We have the following equations:
[LIST=1]
[*]h = 3.5l
[*]h + l = 135,000
[/LIST]
Substitute (1) into (2)
3.5l + l = 135,000
Combine like terms:
4.5l = 135,000
Divide each side by 4.5 to isolate l
[B]l = 30,000[/B]
Substitute this back into equation (1)
h = 3.5(30,000)
[B]h = 105,000[/B]
A jet plane traveling at 550 mph over takes a propeller plane traveling at 150 mph that had a 3 hourA jet plane traveling at 550 mph over takes a propeller plane traveling at 150 mph that had a 3 hours head start. How far from the starting point are the planes?
Use the formula D = rt where
[LIST]
[*]D = distance
[*]r = rate
[*]t = time
[/LIST]
The plan traveling 150 mph for 3 hours:
Time 1 = 150
Time 2 = 300
Time 3 = 450
Now at Time 3, the other plane starts
Time 4 = 600
Time 5 = 750
Time 6 =
450 + 150t = 550t
Subtract 150t
400t = 450
Divide each side by 400
t = 1.125
Plug this into either distance equation, and we get:
550(1.125) = [B]618.75 miles[/B]
a landscaper buys 1 gallon of plant fertilizer. he uses 1/5 of the fertilizer, and then divides thea landscaper buys 1 gallon of plant fertilizer. he uses 1/5 of the fertilizer, and then divides the rest into 3 smaller bottles. how many gallons does he put into each bottle?
First, we find the remaining fraction of fertilizer after using 1/5. [URL='https://www.mathcelebrity.com/fraction.php?frac1=1&frac2=1%2F5&pl=Subtract']Using our fraction calculator[/URL], we see:
1 - 1/5 = 4/5
To find the amount of fertilizer per bottle, we then [URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F5&frac2=3&pl=Divide']divide 4/5 by 3 and we get[/URL]:
[B]4/15 gallon per bottle[/B]
A laundry basket contains 30 socks, of which 9 are black. What is the probability that a randomly sA laundry basket contains 30 socks, of which 9 are black. What is the probability that a randomly selected sock will be black?
P(Black) = 9/30
Simplifying, we can divide top and bottom by 3:
[B]3/10
3/10 as a percentage is 30%[/B]
A local Dunkin’ Donuts shop reported that its sales have increased exactly 16% per year for the lastA local Dunkin’ Donuts shop reported that its sales have increased exactly 16% per year for the last 2 years. This year’s sales were $80,642. What were Dunkin' Donuts' sales 2 years ago?
Declare variable and convert numbers:
[LIST]
[*]16% = 0.16
[*]let the sales 2 years ago be s.
[/LIST]
s(1 + 0.16)(1 + 0.16) = 80,642
s(1.16)(1.16) = 80,642
1.3456s = 80642
Solve for [I]s[/I] in the equation 1.3456s = 80642
[SIZE=5][B]Step 1: Divide each side of the equation by 1.3456[/B][/SIZE]
1.3456s/1.3456 = 80642/1.3456
s = 59930.142687277
s = [B]59,930.14[/B]
A local shop sold 499 hamburgers and cheese burgers. There were 51 fewer cheese burgers sold. How maA local shop sold 499 hamburgers and cheese burgers. There were 51 fewer cheese burgers sold. How many hamburgers were sold?
Let h = number of hamburgers sold and c be the number of cheeseburgers sold.
We have two equations:
(1) c = h - 51
(2) c + h = 499
Substitute (1) into (2)
h - 51 + h = 499
Combine like terms
2h - 51 = 499
Add 51 to both sides
2h = 550
Divide each side by 2 to isolate h
[B]h = 275[/B]
A man left a warehouse at 9:00 a.m. and travels 150 km to reach his home. What is the speed if he aA man left a warehouse at 9:00 a.m. and travels 150 km to reach his home. What is the speed if he arrives at 11:00 a.m.?
[LIST]
[*]His trip took 2 hours (11 - 9)
[*]He traveled 150 km in 2 hours
[*]His speed is measured in km per hour
[/LIST]
If we have 150km/2 hours, we want his speed in km per hour
Divide top and bottom by 2
[B]75km/hr[/B]
A man purchased 20 tickets for a total of $225. The tickets cost $15 for adults and $10 for childrenA man purchased 20 tickets for a total of $225. The tickets cost $15 for adults and $10 for children. What was the cost of each ticket?
Declare variables:
[LIST]
[*]Let a be the number of adult's tickets
[*]Let c be the number of children's tickets
[/LIST]
Cost = Price * Quantity
We're given two equations:
[LIST=1]
[*]a + c = 20
[*]15a + 10c = 225
[/LIST]
Rearrange equation (1) in terms of a:
[LIST=1]
[*]a = 20 - c
[*]15a + 10c = 225
[/LIST]
Now that I have equation (1) in terms of a, we can substitute into equation (2) for a:
15(20 - c) + 10c = 225
Solve for [I]c[/I] in the equation 15(20 - c) + 10c = 225
We first need to simplify the expression removing parentheses
Simplify 15(20 - c): Distribute the 15 to each term in (20-c)
15 * 20 = (15 * 20) = 300
15 * -c = (15 * -1)c = -15c
Our Total expanded term is 300-15c
Our updated term to work with is 300 - 15c + 10c = 225
We first need to simplify the expression removing parentheses
Our updated term to work with is 300 - 15c + 10c = 225
[SIZE=5][B]Step 1: Group the c terms on the left hand side:[/B][/SIZE]
(-15 + 10)c = -5c
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
-5c + 300 = + 225
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 300 and 225. To do that, we subtract 300 from both sides
-5c + 300 - 300 = 225 - 300
[SIZE=5][B]Step 4: Cancel 300 on the left side:[/B][/SIZE]
-5c = -75
[SIZE=5][B]Step 5: Divide each side of the equation by -5[/B][/SIZE]
-5c/-5 = -75/-5
c = [B]15[/B]
Recall from equation (1) that a = 20 - c. So we substitute c = 15 into this equation to solve for a:
a = 20 - 15
a = [B]5[/B]
A members-only speaker series allows people to join for $16 and then pay $1 for every event attendedA members-only speaker series allows people to join for $16 and then pay $1 for every event attended. What is the maximum number of events someone can attend for a total cost of $47?
Subtract the join fee from the total cost:
$47 - $16 = $31
Now divide this number by the cost per event:
$31 / $1 = [B]31 events[/B]
A motorboat travels 408 kilometers in 8 hours going upstream and 546 kilometers in 6 hours going dowA motorboat travels 408 kilometers in 8 hours going upstream and 546 kilometers in 6 hours going downstream. What is the rate of the boat in still water and what is the rate of the current?
[U]Assumptions:[/U]
[LIST]
[*]B = the speed of the boat in still water.
[*]S = the speed of the stream
[/LIST]
Relative to the bank, the speeds are:
[LIST]
[*]Upstream is B - S.
[*]Downstream is B + S.
[/LIST]
[U]Use the Distance equation: Rate * Time = Distance[/U]
[LIST]
[*]Upstream: (B-S)6 = 258
[*]Downstream: (B+S)6 = 330
[/LIST]
Simplify first by dividing each equation by 6:
[LIST]
[*]B - S = 43
[*]B + S = 55
[/LIST]
Solve this system of equations by elimination. Add the two equations together:
(B + B) + (S - S) = 43 + 55
Cancelling the S's, we get:
2B = 98
Divide each side by 2:
[B]B = 49 mi/hr[/B]
Substitute this into either equation and solve for S.
B + S = 55
49 + S = 55
To solve this, we [URL='https://www.mathcelebrity.com/1unk.php?num=49%2Bs%3D55&pl=Solve']type it in our search engine[/URL] and we get:
S = [B]6 mi/hr[/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 pennies splits into 4 equal groupsa number of pennies splits into 4 equal groups
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
We take x and divide it by 4 to get 4 equal groups:
[B]x/4[/B]
A parking garage charges $5 plus $2 per hour. You have $16 to spend for parking. How many hours canA parking garage charges $5 plus $2 per hour. You have $16 to spend for parking. How many hours can you park?
Subtract the flat rate to get the amount you have for hourly parking:
16 - 5 = 11
So we divide 11 dollars to park by 2 dollars per hour to get:
11/2
[B]5.5 hours[/B]
A parking meter contains 27.05 in quarters and dimes. All together there are 146 coins. How many ofA parking meter contains 27.05 in quarters and dimes. All together there are 146 coins. How many of each coin are there?
Let d = the number of dimes and q = the number of quarters. We have two equations:
(1) d + q = 146
(2) 0.1d + 0.25q = 27.05
Rearrange (1) into (3) solving for d
(3) d = 146 - q
Substitute (3) into (2)
0.1(146 - q) + 0.25q = 27.05
14.6 - 0.1q + 0.25q = 27.05
Combine q's
0.15q + 14.6 = 27.05
Subtract 14.6 from each side
0.15q = 12.45
Divide each side by 0.15
[B]q = 83[/B]
Plugging that into (3), we have:
d = 146 - 83
[B]d = 63[/B]
A passenger train left station A at 6:00 P.M. Moving with the average speed 45 mph, it arrived at stA passenger train left station A at 6:00 P.M. Moving with the average speed 45 mph, it arrived at station B at 10:00 p.m. A transit train left from station A 1 hour later than the passenger train, but it arrived at the station B at the same time with the passenger train. What was the average speed of the transit train?
[U]Passenger Train[/U]
[LIST]
[*]45 miles per hour and it got there in 4 hours.
[/LIST]
Using our formula D = rt where:
[LIST]
[*]D = Distance
[*]r = rate
[*]t = time
[/LIST]
[LIST]
[*]D = rt
[*]D = 45(4)
[*]D = 180 miles from Station A to Station B
[/LIST]
Transit Train
[LIST]
[*]It has to go the same distance, 180 miles, so D = 180
[*]It made it there in 3 hours. This is r
[*]We want to solve for t
[/LIST]
D = rt
180 = 3r
Divide each side by 3
[B]r = 60 miles per hour[/B]
A penny has a diameter of 19 millimeters. What is the radius of the penny.A penny has a diameter of 19 millimeters. What is the radius of the penny.
D = 2r
To solve for r, we divide each side by 2:
r = D/2
Plugging in D = 19, we get:
r = [B]19/2 or 9.5[/B]
a piece of ribbon is y cm long. Ashley cut it into 4 equal pieces to do her 4 sisters hair. write ana piece of ribbon is y cm long. Ashley cut it into 4 equal pieces to do her 4 sisters hair. write an expression for the amount of ribbon used for each sister
We take y cm and divide it equal among 4 sisters:
[B]y/4[/B]
A pile of coins, consisting of quarters and half dollars, is worth 11.75. If there are 2 more quarteA pile of coins, consisting of quarters and half dollars, is worth 11.75. If there are 2 more quarters than half dollars, how many of each are there?
Let h be the number of half-dollars and q be the number of quarters. Set up two equations:
(1) q = h + 2
(2) 0.25q + 0.5h = 11.75
[U]Substitute (1) into (2)[/U]
0.25(h + 2) + 0.5h = 11.75
0.25h + 0.5 + 0.5h = 11.75
[U]Group h terms[/U]
0.75h + 0.5 = 11.75
[U]Subtract 0.5 from each side[/U]
0.75h = 11.25
[U]Divide each side by h[/U]
[B]h = 15[/B]
[U]Substitute h = 15 into (1)[/U]
q = 15 + 2
[B]q = 17[/B]
A population grows at 6% per year. How many years does it take to triple in size?A population grows at 6% per year. How many years does it take to triple in size?
With a starting population of P, and triple in size means 3 times the original, we want to know t for:
P(1.06)^t = 3P
Divide each side by P, and we have:
1.06^t = 3
Typing this equation into our search engine to solve for t, we get:
t = [B]18.85 years[/B]
Note: if you need an integer answer, we round up to 19 years
A private jet flies the same distance in 4 hours that a commercial jet flies in 2 hours. If the speeA private jet flies the same distance in 4 hours that a commercial jet flies in 2 hours. If the speed of the commercial jet was 154 mph less than 3 times the speed of the private jet, find the speed of each jet.
Let p = private jet speed and c = commercial jet speed. We have two equations:
(1) c = 3p - 154
(2) 4p =2c
Plug (1) into (2):
4p = 2(3p - 154)
4p = 6p - 308
Subtract 4p from each side:
2p - 308 = 0
Add 308 to each side:
2p = 308
Divide each side by 2:
[B]p = 154[/B]
Substitute this into (1)
c = 3(154) - 154
c = 462 - 154
[B]c = 308[/B]
A realtor makes an annual salary of $25000 plus a 3% commission on sales. If a realtor's salary is $A realtor makes an annual salary of $25000 plus a 3% commission on sales. If a realtor's salary is $67000, what was the amount of her sales?
Total post-salary pay = $67,000 - $25,000 = $42,000
Let Sales be s.
So 0.03s = $42,000
Divide each side by 0.03
s = $1,400,000
a recipe of 20 bread rolls requires 5 tablespoons of butter. How many tablespoons of butter are needa 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 twice as long as it is wide. If the perimeter is 360 what are the dimensions?A rectangular field is twice as long as it is wide. If the perimeter is 360 what are the dimensions?
We are given or know the following about the rectangle
[LIST]
[*]l = 2w
[*]P = 2l + 2w
[*]Since P = 360, we have 2l + 2w = 360
[/LIST]
Since l = 2w, we have 2l + (l) = 360
3l = 360
Divide by 3, we get [B]l = 120[/B]
Which means w = 120/2
[B]w = 60[/B]
A restaurant earns $1073 on Friday and $1108 on Saturday. Write and solve an equation to find the amA 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 retired couple invested $8000 in bonds. At the end of one year, they received an interest paymentA 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 receives a weekly salary of $100 plus a 5.5% commission on sales. Her salary last weekA Salesperson receives a weekly salary of $100 plus a 5.5% commission on sales. Her salary last week was $1090. What were her sales that week?
$1,090 - 100 = $990.
This is her commission.
Let s = Sales.
So 0.055s = $990
Divide each side by 0.055.
s = $18,000
A section of land measuring 3 & 3/6 acres is divided equally among 7 people. How many acres will eacA section of land measuring 3 & 3/6 acres is divided equally among 7 people. How many acres will each person get?
We want 3&3/6 /7
[URL='https://www.mathcelebrity.com/fraction.php?frac1=3%263%2F6&frac2=7&pl=Divide']Using our fraction calculator[/URL], we get:
[B]1/2 acre per person[/B]
A skier is trying to decide whether or not to buy a season ski pass. A daily pass costs $75. A seasA skier is trying to decide whether or not to buy a season ski pass. A daily pass costs $75. A season ski pass costs $350. The skier would have to rent skis with either pass for $20 per day. How many days would the skier have to go skiing in order to make the season pass less expensive than the daily passes?
Let d be the number of days:
Daily Plan cost: 75d + 20d = 95d
Season Pass: 350 + 20d
We want to find d such that
350 + 20d < 95d
Subtract 20d from each side
75d > 350
Divide each side by 75
d > 4.66667
[B]d = 5[/B]
A skydiver falls 144 feet in three seconds. How far does the skydiver fall per second?A skydiver falls 144 feet in three seconds. How far does the skydiver fall per second?
144 feet/3 seconds
Divide top and bottom by 3 to get feet per second
[B]48 feet per second[/B]
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 spinner is divided into 4 equal sections numbered 1 to 4. The theoretical probability of the spinnA spinner is divided into 4 equal sections numbered 1 to 4. The theoretical probability of the spinner stopping on 3 is 25%. Which of the following is most likely the number of 3s spun in 10,000 spins?
We want Expected Value of s spins. Set up the expected value formula for any number 1-4
E(s) = 0.25 * n where n is the number of spins.
Using s = 3, n = 10,000, we have:
E(10,000) = 0.25 * 10,000
E(10,000) = [B]2,500[/B]
A stack of boards is 21 inches high. Each board is 1¾ inches thick. How many boards are there?A stack of boards is 21 inches high. Each board is 1¾ inches thick. How many boards are there?
We want 21 / 1 & 3/4
Using our [URL='https://www.mathcelebrity.com/fraction.php?frac1=21&frac2=1%263%2F4&pl=Divide']fraction operation calculator[/URL], we get:
[B]12 boards[/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 student walks 1500 meters to school in 30 minutes. What is their average speed in meters per minutA student walks 1500 meters to school in 30 minutes. What is their average speed in meters per minute?
1500 meters / 30 minutes
Divide top and bottom by 30
[B]50 meters / minute[/B]
A suitcase contains nickels, dimes and quarters. There are 2&1/2 times as many dimes as nickels andA suitcase contains nickels, dimes and quarters. There are 2&1/2 times as many dimes as nickels and 5 times the number of quarters as the number of nickels. If the coins have a value of $24.80, how many nickels are there in the suitcase?
Setup number of coins:
[LIST]
[*]Number of nickels = n
[*]Number of dimes = 2.5n
[*]Number of quarters = 5n
[/LIST]
Setup value of coins:
[LIST]
[*]Value of nickels = 0.05n
[*]Value of dimes = 2.5 * 0.1n = 0.25n
[*]Value of quarters = 5 * 0.25n = 1.25n
[/LIST]
Add them up:
0.05n + 0.25n + 1.25n = 24.80
Solve for [I]n[/I] in the equation 0.05n + 0.25n + 1.25n = 24.80
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(0.05 + 0.25 + 1.25)n = 1.55n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
1.55n = + 24.8
[SIZE=5][B]Step 3: Divide each side of the equation by 1.55[/B][/SIZE]
1.55n/1.55 = 24.80/1.55
n = [B]16[/B]
[B]
[URL='https://www.mathcelebrity.com/1unk.php?num=0.05n%2B0.25n%2B1.25n%3D24.80&pl=Solve']Source[/URL][/B]
A sum of money doubles in 20 years on simple interest. It will get triple at the same rate in: a.A sum of money doubles in 20 years on simple interest. It will get triple at the same rate in: a. 40 years b. 50 years c. 30 years d. 60 years e. 80 years
Simple interest formula if we start with 1 dollar and double to 2 dollars:
1(1 + i(20)) = 2
1 + 20i = 2
Subtract 1 from each side:
20i = 1
Divide each side by 20
i = 0.05
Now setup the same simple interest equation, but instead of 2, we use 3:
1(1 + 0.05(t)) = 3
1 + 0.05t = 3
Subtract 1 from each side:
0.05t = 2
Divide each side by 0.05
[B]t = 40 years[/B]
A super deadly strain of bacteria is causing the zombie population to double every day. Currently, tA super deadly strain of bacteria is causing the zombie population to double every day. Currently, there are 25 zombies. After how many days will there be over 25,000 zombies?
We set up our exponential function where n is the number of days after today:
Z(n) = 25 * 2^n
We want to know n where Z(n) = 25,000.
25 * 2^n = 25,000
Divide each side of the equation by 25, to isolate 2^n:
25 * 2^n / 25 = 25,000 / 25
The 25's cancel on the left side, so we have:
2^n = 1,000
Take the natural log of each side to isolate n:
Ln(2^n) = Ln(1000)
There exists a logarithmic identity which states: Ln(a^n) = n * Ln(a). In this case, a = 2, so we have:
n * Ln(2) = Ln(1,000)
0.69315n = 6.9077
[URL='https://www.mathcelebrity.com/1unk.php?num=0.69315n%3D6.9077&pl=Solve']Type this equation into our search engine[/URL], we get:
[B]n = 9.9657 days ~ 10 days[/B]
A taxi charges a flat rate of 1.75, plus an additional 0.65 per mile. If Erica has at most 10 to speA taxi charges a flat rate of 1.75, plus an additional 0.65 per mile. If Erica has at most 10 to spend on the cab ride, how far could she travel?
Setup an equation where x is the number of miles traveled:
0.65x + 1.75 = 10
Subtract 1.75 from each side:
0.65x = 8.25
Divide each side by 0.65
[B]x = 12.69 miles[/B]
If we do full miles, we round down to 12.
[MEDIA=youtube]mFqUe2mjX-w[/MEDIA]
A test has 90 questions and you answered 69 correctly. What fraction did you get correct?A test has 90 questions and you answered 69 correctly. What fraction did you get correct?
69/90
Divide top and bottom by 3 to simplify:
[B]23/30[/B]
a textbook store sold a combined total of 296 sociology and history text books in a week. the numbera textbook store sold a combined total of 296 sociology and history text books in a week. the number of history textbooks sold was 42 less than the number of sociology textbooks sold. how many text books of each type were sold?
Let h = history book and s = sociology books. We have 2 equations:
(1) h = s - 42
(2) h + s = 296
Substitute (1) to (2)
s - 42 + s = 296
Combine variables
2s - 42 = 296
Add 42 to each side
2s = 338
Divide each side by 2
s = 169
So h = 169 - 42 = 127
A thermometer has a range of 1.5 degrees of the temperature, what is the maximum and minimum at 87.4A thermometer has a range of 1.5 degrees of the temperature, what is the maximum and minimum at 87.4 degrees
Range = Max - Min
Divide this by 2 to get the lesser half and larger half:
Half-Range = 1.5/2
Half-Range = 0.75
[U]Our Maximum temperature is:[/U]
Max Temp = Current Temp + Half-Range
Max Temp = 87.4 + 0.75
Max Temp = [B]88.15
[/B]
[U]Our Minimum temperature is:[/U]
Min Temp = Current Temp - Half-Range
Min Temp = 87.4 - 0.75
Min Temp = [B][B]86.65[/B][/B]
a times b divided by the quantity a minus ba times b divided by the quantity a minus b
a times b:
ab
a minus b:
a - b
Now divide a times b by a minus b:
[B]ab/(a - b)[/B]
A tortoise is walking in the desert. It walks at a speed of 5 meters per minute for 12.5 meters. ForA tortoise is walking in the desert. It walks at a speed of 5 meters per minute for 12.5 meters. For how many minutes does it walk?
Distance formula (d) for a rate (r) and time (t) is:
d = rt
We're given d = 12.5 and r = 5
12.5 = 5t
5t = 12.5
Solve for t. Divide each side of the equation by 5:
5t/5 = 12.5/5
Cancel the 5's on left side and we get:
t = [B]2.5[/B]
A tow truck charges a service fee of $50 and an additional fee of $1.75 per mile. What distance wasA tow truck charges a service fee of $50 and an additional fee of $1.75 per mile. What distance was Marcos car towed if he received a bill for $71
Set up a cost equation C(m) where m is the number of miles:
C(m) = Cost per mile * m + Service Fee
Plugging in the service fee of 50 and cost per mile of 1.75, we get:
C(m) = 1.75m + 50
The question asks for what m is C(m) = 71. So we set C(m) = 71 and solve for m:
1.75m + 50 = 71
Solve for [I]m[/I] in the equation 1.75m + 50 = 71
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 50 and 71. To do that, we subtract 50 from both sides
1.75m + 50 - 50 = 71 - 50
[SIZE=5][B]Step 2: Cancel 50 on the left side:[/B][/SIZE]
1.75m = 21
[SIZE=5][B]Step 3: Divide each side of the equation by 1.75[/B][/SIZE]
1.75m/1.75 = 21/1.75
m = [B]12[/B]
A trapezoid has one base that is 120% of the length of the other base. The two sides are each 1/2 thA trapezoid has one base that is 120% of the length of the other base. The two sides are each 1/2 the length of the smaller base. If the perimeter of the trapezoid is 54.4 inches, what is the length of the smaller base of the trapezoid?
Setup measurements:
[LIST]
[*]Small base = n
[*]Large base = 1.2n
[*]sides = n/2
[*]Perimeter = n + 1.2n + 0.5n + 0.5n = 54.4
[/LIST]
Solve for [I]n[/I] in the equation n + 1.2n + 0.5n + 0.5n = 54.4
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 1.2 + 0.5 + 0.5)n = 3.2n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
3.2n = + 54.4
[SIZE=5][B]Step 3: Divide each side of the equation by 3.2[/B][/SIZE]
3.2n/3.2 = 54.4/3.2
n = [B]17[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=n%2B1.2n%2B0.5n%2B0.5n%3D54.4&pl=Solve']Source[/URL]
A trench is 40 feet long and the trencher goes 2 feet per minute. How long does it take to dig theA trench is 40 feet long and the trencher goes 2 feet per minute. How long does it take to dig the trench?
2 feet per minute * x minutes = 40 feet
Divide each side by 2
[B]x = 20 minutes[/B]
a triangle has side lengths of 12,16, and 20 centimeters. is it a right triangle?a triangle has side lengths of 12,16, and 20 centimeters. is it a right triangle?
First, we see if we can simplify. So we [URL='https://www.mathcelebrity.com/gcflcm.php?num1=12&num2=16&num3=20&pl=GCF']type GCF(12,16,20) [/URL]and we get 4.
We divide the 3 side lengths by 4:
12/4 = 3
16/4 = 4
20/4 = 5
And lo and behold, we get a Pythagorean Triple of 3, 4, 5. So [B]yes, this is a right triangle[/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 ca 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 varies inversely with b, c and da varies inversely with b, c and d
Varies inversely means we divide. Given a constant, k, we have:
[B]a = k/bcd[/B]
A vendor sells h hot dogs and s sodas. If a hot dog costs twice as much as a soda, and if the vendorA vendor sells h hot dogs and s sodas. If a hot dog costs twice as much as a soda, and if the vendor takes in a total of d dollars, how many cents does a soda cost?
Let the cost of the soda be p. So the cost of a hot dog is 2p.
The total cost of hot dogs:
2hp
The total cost of sodas:
ps
The total cost of both equals d. So we set the total cost of hots dogs plus sodas equal to d:
2hp + ps = d
We want to know the cost of a soda (p). So we have a literal equation. We factor out p from the left side:
p(2h + s) = d
Divide each side of the equation by (2h + s)
p(2h + s)/(2h + s) = d/(2h + s)
Cancel the (2h + s) on the left side, we get:
p = [B]d/(2h + s[/B])
A woman walked for 5 hours, first along a level road, then up a hill, and then she turned around andA 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 writer can write a novel at a rate of 3 pages per 5 hour work. if he wants to finish the novel ina writer can write a novel at a rate of 3 pages per 5 hour work. if he wants to finish the novel in x number of pages, determine a function model that will represent the accumulated writing hours to finish his novel
if 3 pages = 5 hours, then we divide each side by 3 to get:
1 page = 5/3 hours per page
So x pages takes:
5x/3 hours
Our function for number of pages x is:
[B]f(x) = 5x/3[/B]
A zoo has 15 Emperor penguins. The Emperor penguins make up 30% percent of all the penguins at the zA zoo has 15 Emperor penguins. The Emperor penguins make up 30% percent of all the penguins at the zoo. How many penguins live at the zoo?
Let p be the total number penguins at the zoo.
We're told:
30% of p = 15
Since 30% = 0.3, we have:
0.3p = 15
Solve for [I]p[/I] in the equation 0.3p = 15
[SIZE=5][B]Step 1: Divide each side of the equation by 0.3[/B][/SIZE]
0.3p/0.3 = 15/0.3
p = [B]50[/B]
a/m - b = c for ma/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=0.5(bh), for hA=0.5(bh), for h
Divide each side by 0.5b
[B]h = A/0.5b[/B]
A=2(l+w) for lMultiply 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 wMultiply 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
[MEDIA=youtube]Nm-tYD4aEY4[/MEDIA]
ab/d + c = e for dab/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 dab/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]
Absolute ValueFree Absolute Value Calculator - Add, subtract, multiply or divide any two numbers with absolute value signs. Positive Difference.
acw+cz=y for aacw+cz=y for a
Solve this literal equation:
Subtract cz from each side:
acw + cz - cz = y - cz
Cancel the cz on the left side:
acw = y - cz
Divide each side by cw to isolate a:
acw/cw = (y - cz)/cw
Cancel cw on the left side:
[B]a = (y - cz)/cw[/B]
Adam drove the 10 miles to school at a speed of 60 mph. On his way home, due to traffic, his speed wAdam drove the 10 miles to school at a speed of 60 mph. On his way home, due to traffic, his speed was 30 mph. What was his average speed for the round trip to school and back?
D = rt
To school:
60 miles in 60 minutes = 10 miles in 10 minutes
To home:
30 miles in 60 minutes = 10 miles in 20 minutes
Total time:
10 + 20 = 30 minutes or 0.5 hours
With a speed of s, we have:
d = st
20 = 0.5s
Divide each side by 2:
s = [B]40 mph[/B]
Add 5 to p, then divide the sum by 4Add 5 to p, then divide the sum by 4
Add 5 to p:
p + 5
Divide the sum by 4:
[B](p + 5)/4
[/B]
note: B[I]ecause this is a sum, we wrap it in parentheses to divide the sum by a number[/I]
Add 7 to a, and divide the sum by bAdd 7 to a, and divide the sum by b
Add 7 to a:
a + 7
Divide the sum by b:
[B](a + 7)/b[/B]
add 8 and 10 then divide uadd 8 and 10 then divide u
Add 8 and 10
8 + 10
Divide by u
(8 + 10)/u
Simplified, it is 18/u
add c to d, multiply a by the result, then divide what you have by badd 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 q to p, add a to the result, then divide r by what you haveAdd q to p, add a to the result, then divide r by what you have
Add q to p:
p + q
Add a to the result:
p + q + a
Then divide r by what you have:
[B]r/(p + q + a)[/B]
add r and q, divide the result by s, then triple what you haveadd 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 haveadd 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 u and t divide s by the result then triple what you haveadd 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 w to t, add u to the result, then divide what you have by vadd w to t, add u to the result, then divide what you have by v
Take this algebraic expression in parts:
[LIST]
[*]Add w to t: t + w
[*]Add u to the result: t + w + u
[*]Divide what you have by v:
[/LIST]
([B]t + w + u)/v[/B]
Age now problemsLet f be the age of the father and d be the age of the daughter and s be the age of the son. We have:
[LIST=1]
[*]f = 3s
[*]d = s - 3
[*]d - 3 + f - 3 + s - 3 = 63
[/LIST]
Simplify (3)
d + f + s - 9 = 63
d + f + s = 72
Now, substitute (1) and (2) into the modified (3)
(s - 3) + 3s + s = 72
Combine like terms:
5s - 3 = 72
Add 3 to each side
5s = 75
Divide each side by 5
s = 15
We want f, so we substitute s = 15 into (1)
f = 3(15)
[B]f = 45[/B]
Alex sells cars at Keith Palmer ford. He earns 400 dollars a week plus 150 dollars per car he sells.Alex sells cars at Keith Palmer ford. He earns 400 dollars a week plus 150 dollars per car he sells. If he earned 1450 dollars last week, how many cars did he sell?
Subtract the base salary of $400
$1,450 - 400 =$1,050
Divide this by 150 per car
$1,050/$150 = [B]7 cars[/B]
Algebraic ExpressionsFree Algebraic Expressions Calculator - This calculator builds algebraic expressions based on word representations of numbers using the four operators and the words that represent them(increased,product,decreased,divided,times)
Also known as Mathematical phrases
Alvin is 12 years younger than Elga. The sum of their ages is 60 . What is Elgas age?Alvin is 12 years younger than Elga. The sum of their ages is 60 . What is Elgas age?
Let a be Alvin's age and e be Elga's age. We have the following equations:
[LIST=1]
[*]a = e - 12
[*]a + e = 60
[/LIST]
Plugging in (1) to (2), we get:
(e - 12) + e = 60
Grouping like terms:
2e - 12 = 60
Add 12 to each side:
2e = 72
Divide each side by 2
[B]e = 36[/B]
An elevator can hold less than 2700 pounds of extra weight. If an average person weighs 150 pounds,An elevator can hold less than 2700 pounds of extra weight. If an average person weighs 150 pounds, what is the maximum number of people (p) that can be on the elevator at one time?
Total weight = average weight per person * Number of people
Total weight = 150p
We know from the problem that:
150p < 2700
We want to solve this inequality for p. Divide each side of the inequality by 150:
150p/150 < 2700/150
Cancel the 150's on the left side and we get:
p < [B]18[/B]
An employee earns $7.00 an hour for the first 35 hours worked in a week and $10.50 for any hours oveAn employee earns $7.00 an hour for the first 35 hours worked in a week and $10.50 for any hours over 35. One weeks paycheck (before deductions) was for $308.00. How many hours did the employee work?
Let's first check to see if the employee worked overtime:
Regular Hours: 35 * 7 = 245
Since the employee made $308, they worked overtime. Let's determine how much overtime money was made:
308 - 245 = 63
Now, to calculate the overtime hours, we divide overtime pay by overtime rate
63/10.50 = 6
Now figure out the total hours worked in the week:
Total Hours = Regular Pay Hours + Overtime Hours
Total Hours = 35 + 6
[B]Total Hours = 41[/B]
An experienced accountant can balance the books twice as fast as a new accountant. Working togetherAn 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]
Ana has 24 carrots, 12 cucumbers, and 36 radishes. She wants to make identical vegetable baskets usiAna has 24 carrots, 12 cucumbers, and 36 radishes. She wants to make identical vegetable baskets using all of the vegetables. What is the greatest number of baskets she can make
The key to solving this problem is asking what is the common factor between the 3 numbers. We want the greatest common factor or GCF
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=12&num2=24&num3=36&pl=GCF']GCF(12, 24, 36) [/URL]= [B]12[/B]
We divide up our 12 baskets into carrots, cucumbers, and radishes. Each basket of the 12 baskets has the following:
[LIST=1]
[*]12 cucumbers / GCF of 12 = [B]1 cucumber per basket[/B]
[*]24 carrots / GCF of 12 = [B]2 carrots per basket[/B]
[*]36 radishes / GCF of 12 = [B]3 radishes per basket[/B]
[/LIST]
[B][MEDIA=youtube]D1KTOP0h2P4[/MEDIA][/B]
Anita read 150 pages in 5 hours. What is her reading rate in pages per minute?Anita read 150 pages in 5 hours. What is her reading rate in pages per minute?
150 pages / 5 hours
Divide top and bottom by 5:
150/5 = 30
5/5 = 1
So we have 30 pages per hour
And 1 hour is 60 minutes, so we have:
(30 pages / 1 hour) * (1 hour / 60 minutes)
30 pages / 60 minutes
[B]0.5 pages per minute[/B]
anne is building bookcases that are 3.4 feet long. How many complete shelves can be cut from a 12-foanne is building bookcases that are 3.4 feet long. How many complete shelves can be cut from a 12-foot board?
We divide 12 feet of board by 3.4 per bookcase and we get:
12/3.4 = 3.52
So complete boards = [B]3[/B]
Anne wants to make a platform that is 7 feet wide and 10 feet long. If she uses boards that measureAnne wants to make a platform that is 7 feet wide and 10 feet long. If she uses boards that measure 6 inches wide by 2 feet long, how many boards will she need to complete the job?
Area of platform which is a rectangle:
A = lw
A = 10 * 7
A = 70
Area of boards which are rectangles:
A = lw
A = 2 * 6
A = 12
We divide our platform area by our board area to get the number of boards needed:
Boards needed = Platform Area / Board Area
Boards needed = 70/12
Boards needed = 5.83333
We round up if we want full boards to be [B]6[/B]
April, May and June have 90 sweets between them. May has three-quarters of the number of sweets thatApril, 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]
Arnie bought some bagels at 20 cents each. He ate 4, and sold the rest at 30 cents each. His profitArnie bought some bagels at 20 cents each. He ate 4, and sold the rest at 30 cents each. His profit was $2.40. How many bagels did he buy?
Let x be the number of bagels Arnie sold. We have the following equation:
0.30(x - 4) - 0.20(4) = 2.40
Distribute and simplify:
0.30x - 1.20 - 0.8 = 2.40
Combine like terms:
0.30x - 2 = 2.40
Add 2 to each side:
0.30x = 4.40
Divide each side by 0.3
[B]x = 14.67 ~ 15[/B]
As a salesperson, you are paid $50 per week plus $2 per sale. This week you want your pay to be at lAs a salesperson, you are paid $50 per week plus $2 per sale. This week you want your pay to be at least $100. What is the minimum number of sales you must make to earn at least $100?
Set up the inequality where s is the amount of sales you make:
50 + 2s >= 100
We use >= because the phrase [I]at least[/I] 100 means 100 or more
Subtract 50 from each side:
2s >= 50
Divide each side by 2
[B]s >= 25[/B]
At a local fitness center, members pay a $10 membership fee and $3 for each aerobics class. NonmeAt a local fitness center, members pay a $10 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost for members and nonmembers be the same?
Set up the cost functions where x is the number of aerobics classes:
[LIST]
[*]Members: C(x) = 10 + 3x
[*]Non-members: C(x) = 5x
[/LIST]
Set them equal to each other
10 + 3x = 5x
Subtract 3x from both sides:
2x = 10
Divide each side by 2
[B]x = 5 classes[/B]
At a local fitness center, members pay an $8 membership fee and $3 for each aerobics class. NonmembeAt a local fitness center, members pay an $8 membership fee and $3 for each aerobics class. Nonmembers pay $5 for each aerobics class. For what number of aerobics classes will the cost for members be equal to nonmembers?
Set up two cost equations C(x):
[LIST=1]
[*]Members: C(x) = 8 + 3x
[*]Nonmembers: C(x) = 5x
[/LIST]
Set the two cost equations equal to each other:
8 + 3x = 5x
Subtract 3x from each side
2x = 8
Divide each side by 2
[B]x = 4[/B]
at a party, there are 72 people. The ratio of men to ladies to kids is 4 to 3 to 2.at a party, there are 72 people. The ratio of men to ladies to kids is 4 to 3 to 2.
[LIST]
[*]How many men at the party?
[*]How many ladies at the party?
[*]How many kids at the party?
[/LIST]
Our total ratio denominator is 4 + 3 + 2 = 9. To find the number of each type of person, we take their ratio divided by their ratio numerator times 72 people at the party
[U]Calculate ratios:[/U]
[LIST]
[*]Men: [URL='https://www.mathcelebrity.com/fraction.php?frac1=4%2F9&frac2=72&pl=Multiply']4/9 * 72[/URL] = [B]32[/B]
[*]Ladies: [URL='https://www.mathcelebrity.com/fraction.php?frac1=3%2F9&frac2=72&pl=Multiply']3/9 * 72[/URL] = [B]24[/B]
[*]Kids: [URL='https://www.mathcelebrity.com/fraction.php?frac1=2%2F9&frac2=72&pl=Multiply']2/9 * 72[/URL] = [B]16[/B]
[/LIST]
[U]Check our work:[/U]
Men + Ladies + Kids = 32 + 24 + 16
Men + Ladies + Kids = 72 <-- This checks out!
At the end of the day, a bakery had 1/2 of a pie left over. The 4 employees each took home the sameAt the end of the day, a bakery had 1/2 of a pie left over. The 4 employees each took home the same amount of leftover pie. How much pie did each employee take home?
We have 1/2 of the pie eaten, if 1/2 was left over.
So 1/2 of a pie divided by 4 employees = [B]1/8 of a pie per person[/B].
To check our work, we have 4 * 1/8 = 4/8 = 1/2 of pie eaten.
At what simple interest rate will 4500$ 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]
At Zabowood’s Gadget Store, some items are paid on instalment basis through credit cards. Clariza wa[B]A[/B]t Zabowood’s Gadget Store, some items are paid on instalment basis through credit cards. Clariza was able to sell 10 cellphones costing Php 18,000.00 each. Each transaction is payable in 6 months equally divided into 6 equal instalments without interest. Clariza gets 2% commission on the first month for each of the 10 cellphones. Commission decreases by 0.30% every month thereafter and computed on the outstanding balance for the month. How much commission does Clariza receive on the third month?
Calculate Total Sales Amount:
Calculate Total Sales Amount = 10 cellphones * 18000 per cellphone
Calculate Total Sales Amount = 180000
Calculate monthly sales amount installment:
monthly sales amount installment = Total Sales Amount / 6
monthly sales amount installment = 180000/6
monthly sales amount installment = 30000 per month
Calculate Third Month Commission:
Third month commission = First Month Commission - 0.30% - 0.30%
Third month Commission = 2% - 0.30% - 0.30% = 1.4%
Calculate 3rd month commission amount:
3rd month Commission amount = 1.4% * 30000
3rd month Commission amount = [B]420[/B]
Ava is 4 times as old as Peter. What equation can be used to find Peter’s age?Ava is 4 times as old as Peter. What equation can be used to find Peter’s age?
[U]Assumptions[/U]
Let a be Ava's age
Let p be Peter's age
We're given:
a = 4p
To find Peter's age, we divide each side of the equation by 4 to get:
a/4 = 4p/4
p = [B]a/4[/B]
ax + b = cx - dWe are solving for x:
Subtract b from each side:
ax = cx - d - b
Subtract cx from each side:
ax - cx = -d - b
Factor out x from the left side:
x(a - c) = -d - b
Divide each side by (a - c)
x = (-d - b)/(a - c)
ax - mn = mn + bx for xax - mn = mn + bx for x
Add mn to each side:
ax - mn + mn = mn + bx + mn
Cancel the mn terms on the left side and we get:
ax = bx + 2mn
Subtract bx from each side:
ax - bx = bx - bx + 2mn
Cancel the bx terms on the right side:
ax - bx = 2mn
Factor out x on the left side:
x (a - b) = 2mn
Divide each side of the equation by (a - b):
x (a - b)/(a - b) = 2mn/(a - b)
Cancel the (a - b) on the left side and we get:
x = [B]2mn/(a - b)[/B]
B is the midpoint of AC and BC=5B is the midpoint of AC and BC=5
Since the midpoint divides a segment into two equal segments, we know that:
AB = BC
So AB =[B] 5[/B]
And AC = 5 + 5 = [B]10[/B]
B+c =10/a for aB+c =10/a for a
Cross multiply:
a(B + c) = 10
Divide each side by a
[B]a = 10/(B + c)[/B]
Barney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks willBarney has $450 and spends $3 each week. Betty has $120 and saves $8 each week. How many weeks will it take for them to have the same amount of money?
Let w be the number of weeks that go by for saving/spending.
Set up Barney's balance equation, B(w). Spending means we [U]subtract[/U]
B(w) = Initial Amount - spend per week * w weeks
B(w) = 450 - 3w
Set up Betty's balance equation, B(w). Saving means we [U]add[/U]
B(w) = Initial Amount + savings per week * w weeks
B(w) = 120 + 8w
The same amount of money means both of their balance equations B(w) are equal. So we set Barney's balance equal to Betty's balance and solve for w:
450 - 3w = 120 + 8w
Add 3w to each side to isolate w:
450 - 3w + 3w = 120 + 8w + 3w
Cancelling the 3w on the left side, we get:
450 = 120 + 11w
Rewrite to have constant on the right side:
11w + 120 = 450
Subtract 120 from each side:
11w + 120 - 120 = 450 - 120
Cancelling the 120's on the left side, we get:
11w = 330
To solve for w, we divide each side by 11
11w/11 = 330/11
Cancelling the 11's on the left side, we get:
w = [B]30
[MEDIA=youtube]ifG_q-utgJI[/MEDIA][/B]
Base Conversion OperationsFree Base Conversion Operations Calculator - This calculator allows you to add, subtract, multiply, and divide two numbers with different bases.
Basic Math OperationsFree 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
Bawi solves a problem that has an answer of x = -4. He first added 7 to both sides of the equal signBawi solves a problem that has an answer of x = -4. He first added 7 to both sides of the equal sign, then divided by 3. What was the original equation
[LIST=1]
[*]If we added 7 to both sides, that means we had a minus 7 (-7) to start with as a constant. Since subtraction undoes addition.
[*]If we divided by 3, this means we multiplied x by 3 to begin with. Since division undoes multiplication
[/LIST]
So we have the start equation:
3x - 7
If the answer was x = -4, then we plug this in to get our number on the right side of the equation:
3(-4) - 7
-12 - 7
-19
This means our original equation was:
[B]3x - 7 = -19[/B]
And if we want to solve this to prove our answer, we [URL='https://www.mathcelebrity.com/1unk.php?num=3x-7%3D-19&pl=Solve']type the equation into our search engine [/URL]and we get:
x = -4
Ben has $4.50 in quarters(Q) and dimes(D). a)Write an equation expressing the total amount of moneyBen has $4.50 in quarters(Q) and dimes(D). a)Write an equation expressing the total amount of money in terms of the number of quarters and dimes. b)Rearrange the equation to isolate for the number of dimes (D)
a) The equation is:
[B]0.1d + 0.25q = 4.5[/B]
b) Isolate the equation for d. We subtract 0.25q from each side of the equation:
0.1d + 0.25q - 0.25q = 4.5 - 0.25q
Cancel the 0.25q on the left side, and we get:
0.1d = 4.5 - 0.25q
Divide each side of the equation by 0.1 to isolate d:
0.1d/0.1 = (4.5 - 0.25q)/0.1
d = [B]45 - 2.5q[/B]
Beth made a trip to the train station and back. On the trip there she traveled 45 km/h and on the reBeth made a trip to the train station and back. On the trip there she traveled 45 km/h and on the return trip she went 30 km/h. How long did the trip there take if the return trip took six hours?
We use the distance formula: D = rt where D = distance, r = rate, and t = time.
Start with the return trip:
D = 45(6)
D = 270
The initial trip is:
270= 30t
Divide each side by 30
[B]t = 9 hours[/B]
Blake writes 4 pages per hour. How many hours will Blake have to spend writing this week in order toBlake writes 4 pages per hour. How many hours will Blake have to spend writing this week in order to have written a total of 16 pages?
[U]Let x = the number of hours Blake needs to write[/U]
4 pages per hour * x hours = 16
[U]Divide each side by 4[/U]
[B]x = 4 hours[/B]
Bob can complete 35 math problems in 5 minutes how many can he complete in 1 minuteBob can complete 35 math problems in 5 minutes how many can he complete in 1 minute
35 math problems / 5 minutes
Divide the top and bottom of the fraction by 5:
35 math problems / 5 minutes =[B] 7 math problems per minute[/B]
Bob has 4/5 of a pound of fudge. He wants to share it with Sue. How much of a pound of fudge will boBob has 4/5 of a pound of fudge. He wants to share it with Sue. How much of a pound of fudge will both of them get?
If Bob shares the fudge with Sue, we assume they split equal parts. This means:
We take 4/5 total and divide into 2 for 2 people:
4/5/2
This is the same as 4/5 * 1/2
4/10
This fraction is not simplified.
Factor of 4 = {1, [U]2[/U], 4}
Factors of 10 = {1, [U]2[/U], 5, 10}
In both of these lists, we see the greatest common factor is 2.
So we divide top and bottom of 4/10 by 2:
4/2 / 10 / 2
[B]2/5
Bob gets 2/5 of a pound of fudge and Sue gets [B]2/5 of a pound of fudge[/B][/B]
Brice has 1200 in the bank. He wants to save a total of 3000 by depositing 40 per week from his paycBrice has 1200 in the bank. He wants to save a total of 3000 by depositing 40 per week from his paycheck. How many weeks will it take until he saves 3000?
Remaining Savings = 3,000 - 1,200 = 1,800
40 per week * x weeks = 1,800
40x = 1800
Divide each side of the equation by 40
[B]x = 45 weeks[/B]
Brighthouse charges $120 a month for their basic plan, plus $2.99 for each on demand movie you buy.Brighthouse charges $120 a month for their basic plan, plus $2.99 for each on demand movie you buy. Write and solve and inequality to find how many on demand movies could you buy if you want your bill to be less than $150 for the month.
Let x equal to the number room movie rentals per month. Our inequality is:
120 + 2.99x < 150
To solve for the number of movies, Add 120 to each side
2.99x < 30
Divide each side by 2.99
x < 10.03, which means 10 since you cannot buy a fraction of a movie
by + 2/3 = c for yby + 2/3 = c for y
Subtract 2/3 from each side of the literal equation:
by + 2/3 - 2/3 = c - 2/3
Cancel the 2/3 on the left side to get:
by = c - 2/3
Divide each side by b to isolate y:
by/b = (c - 2/3)/b
Cancel the b's on the left side to get:
y = [B](c - 2/3)/b[/B]
by + 2/3 = c, for yby + 2/3 = c, for y
Subtract 2/3 from each side:
by = c - 2/3
Divide each side by b
y = [B](c - 2/3)/b[/B]
b^2 - 6 = 5an for ab^2 - 6 = 5an for a
Divide each side of the equation by 5n to isolate a:
(b^2 - 6)/5n = 5an/5n
Cancel the 5n on the right side and we get:
a = [B](b^2 - 6)/5n[/B]
C varies directly as the cube of a and inversely as the 4th power of BC 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 ac/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]
c=59f-288 for fc=59f-288 for f
Add 288 to each side:
c + 288 = 59f - 288 + 288
Cancel the 288 on the right side, we get:
59f = c + 288
Divide each side by 59 to isolate f:
59f/59 = (c + 288)/59
Cancel the 59 on the left side, we get:
f = [B](c + 288)/59[/B]
calculate cos(x) given tan(x)=8/15calculate 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]
Casey is 26 years old. Her daughter Chloe is 4 years old. In how many years will Casey be double herCasey is 26 years old. Her daughter Chloe is 4 years old. In how many years will Casey be double her daughter's age
Declare variables for each age:
[LIST]
[*]Let Casey's age be c
[*]Let her daughter's age be d
[*]Let n be the number of years from now where Casey will be double her daughter's age
[/LIST]
We're told that:
26 + n = 2(4 + n)
26 + n = 8 + 2n
Solve for [I]n[/I] in the equation 26 + n = 8 + 2n
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables n and 2n. To do that, we subtract 2n from both sides
n + 26 - 2n = 2n + 8 - 2n
[SIZE=5][B]Step 2: Cancel 2n on the right side:[/B][/SIZE]
-n + 26 = 8
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 26 and 8. To do that, we subtract 26 from both sides
-n + 26 - 26 = 8 - 26
[SIZE=5][B]Step 4: Cancel 26 on the left side:[/B][/SIZE]
-n = -18
[SIZE=5][B]Step 5: Divide each side of the equation by -1[/B][/SIZE]
-1n/-1 = -18/-1
n = [B]18[/B]
Check our work for n = 18:
26 + 18 ? 8 + 2(18)
44 ? 8 + 36
44 = 44
Charles LawFree Charles Law Calculator - This will solve for any of the 4 items in Charles Law assuming constant pressure
V1 ÷ T1 = V2 ÷ T2
Charrie found a piece of 8 meters rope. She cuts it into equal length. She made three cuts. How longCharrie found a piece of 8 meters rope. She cuts it into equal length. She made three cuts. How long is each piece of the rope?
Equal length means we divide the length of the rope by the number of equal cuts
[B]8/3 or 2 & 2/3 meters[/B]
Chris walks 12 blocks north and then 16 blocks East. How far is his home from the parkChris walks 12 blocks north and then 16 blocks East. How far is his home from the park
We've got a right triangle. If we divide 12 and 16 by 4, we get:
12/4 = 3
16/4 = 4
Since the hypotenuse is the distance from the home to the park, we have a classic 3-4-5 right triangle.
So our hypotenuse is 5*4 = [B]20[/B]
Chris, Alex and Jesse are all siblings in the same family. Alex is 5 years older than chris. Jesse iChris, Alex and Jesse are all siblings in the same family. Alex is 5 years older than chris. Jesse is 6 years older than Alex. The sum of their ages is 31 years. How old is each one of them?
Set up the relational equations where a = Alex's age, c = Chris's aged and j = Jesse's age
[LIST=1]
[*]a = c + 5
[*]j = a + 6
[*]a + c + j = 31
[*]Rearrange (1) in terms of c: c = a - 5
[/LIST]
[U]Plug in (4) and (2) into (3)[/U]
a + (a - 5) + (a + 6) = 31
[U]Combine like terms:[/U]
3a + 1 = 31
[U]Subtract 1 from each side[/U]
3a = 30
[U]Divide each side by 3[/U]
[B]a = 10[/B]
[U]Plug in 1 = 10 into Equation (4)[/U]
c = 10 - 5
[B]c = 5[/B]
Now plug 1 = 10 into equation (2)
j = 10 + 6
[B]j = 16[/B]
Cindy is c years old. Cindy is 5 years younger than half Jennifer's age ( j )Cindy is c years old. Cindy is 5 years younger than half Jennifer's age ( j )
Build an algebraic expression:
[B]c = j/2 - 5[/B] <-- Half means we divide by 2 and [I]younger[/I] means we subtract
Claire bought 180 candies and 140 pens for goody bags for her birthday. What is the largest number oClaire bought 180 candies and 140 pens for goody bags for her birthday. What is the largest number of goody bags that Claire can make so that each goody bag has the same number of candies and the same number of pens? (All candies and pens should be used.)
We want the greatest common factor of 180 and 140. When we [URL='https://www.mathcelebrity.com/gcflcm.php?num1=140&num2=180&num3=&pl=GCF+and+LCM']run GCF(180,140) in our calculator[/URL], we get 20.
We divide our total candies and total pens by our GCF. So each bag has the following:
Candies: 180/20 = [B]9 candies[/B]
Pens: 140/20 = [B]7 pens[/B]
cody takes about 24,040 breaths a day. how many breaths is that in an hour?cody takes about 24,040 breaths a day. how many breaths is that in an hour?
There are 24 hours in a day, so we divide 24,040 / 24 to get breaths per hour:
24,040 / 24 = [B]1001.67 [/B]
Colin was thinking of a number. Colin divides by 8, then adds 1 to get an answer of 2. What was theColin was thinking of a number. Colin divides by 8, then adds 1 to get an answer of 2. What was the original number?
Let the number be n.
Divide by 8:
n/8
Then add 1:
n/8 + 1
The phrase [I]get an answer[/I] of means an equation, so we set n/8 + 1 equal to 2:
n/8 + 1 = 2
To solve for n, we subtract 1 from each side to isolate the n term:
n/8 + 1 - 1 = 2 - 1
Cancel the 1's on the left side, we get:
n/8 = 1
Cross multiply:
n = 8*1
n = [B]8[/B]
Company a charges $25 plus $0.10 a mile. Company b charges $20 plus $0.15 per mile. How far would yoCompany a charges $25 plus $0.10 a mile. Company b charges $20 plus $0.15 per mile. How far would you need to travel to get each charge to be the same?
Let x be the number of miles traveled
Company A charge: C = 25 + 0.10x
Company B charge: C = 20 + 0.15x
Set up an equation find out when the charges are the same.
25 + 0.10x = 20 + 0.15x
Combine terms and simplify
0.05x = 5
Divide each side of the equation by 0.05 to isolate x
x = [B]100[/B]
Complex Number OperationsFree Complex Number Operations Calculator - Given two numbers in complex number notation, this calculator:
1) Adds (complex number addition), Subtracts (complex number subtraction), Multiplies (complex number multiplication), or Divides (complex number division) any 2 complex numbers in the form a + bi and c + di where i = √-1.
2) Determines the Square Root of a complex number denoted as √a + bi
3) Absolute Value of a Complex Number |a + bi|
4) Conjugate of a complex number a + bi
Connie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. How many chiConnie’s kindergarten class has 24 children. She wants them to get into 4 equal groups. How many children will she put in each group?
We take 24 children divided by 4 equal groups = 24/4
24/4 = [B]6 children per group[/B]
Consider the formula for the area of a trapezoid: A=12h(a+b) . Is it mathematically simpler to solveConsider the formula for the area of a trapezoid: A=12h(a+b) . Is it mathematically simpler to solve for a, b, or h? Why? Solve for each of these variables to demonstrate.
The variable "h" is the easiest to solve for. Because you only have one step. Let's review:
Divide each side of the equation by 12(a + b)
h = 12(a + b)/A
Solving for "a", we two steps. Divide each side by 12h:
A/12h = a + b
Subtract b from each side
a = A/12h - b
Solving for "b" takes two steps as well. Divide each side by 12h:
A/12h = a + b
Subtract a from each side
b = A/12h - a
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]
Country A produces about 7 times the amount of diamonds in carats produce in Country B. If the totalCountry A produces about 7 times the amount of diamonds in carats produce in Country B. If the total produced in both countries is 40,000,000 carats, find the amount produced in each country.
Set up our two given equations:
[LIST=1]
[*]A = 7B
[*]A + B = 40,000,000
[/LIST]
Substitute (1) into (2)
(7B) + B = 40,000,000
Combine like terms
8B = 40,000,000
Divide each side by 8
[B]B = 5,000,000[/B]
Substitute this into (1)
A = 7(5,000,000)
[B]A = 35,000,000[/B]
d - f^3 = 4a for ad - f^3 = 4a for a
Solve this literal equation for a:
Divide each side of the equation by 4:
(d - f^3)/4 = 4a/4
Cancel the 4's on the right side, and rewrite with our variable to solve for on the left side:
a = [B](d - f^3)/4[/B]
Dan makes 11 an hour working at the local grocery store. Over the past year he has saved 137.50 towaDan makes 11 an hour working at the local grocery store. Over the past year he has saved 137.50 toward a new pair of retro sneakers. If sneakers cost 240, how many hours will he need to be able to buy the sneakers?
Figure out his remaining savings target:
240 - 137.50 = 102.50
Let x equal the number of remaining hours Dan needs to work
11x = 102.50
Divide each side by 11
x = 9.318
We round up for a half-hour to 9.5, or a full hour to 10.
Dan needs 309 programs for the school play on Thursday. How many boxes of programs will he need, givDan needs 309 programs for the school play on Thursday. How many boxes of programs will he need, given that each box contains 41 programs?
Each box contains 41 programs, so we divide 309 programs by 41 programs per box to get our boxes:
309/41 using our [URL='https://www.mathcelebrity.com/longdiv.php?num1=309&num2=41&pl=Long%20Division%20%28Decimals%29']division calculator[/URL] is 7.5365.
Since we don't have fractional boxes, we round up to the next highest integer. [B]8 boxes[/B]
Danny's mom ate 1/6 of an ice cream cake. Danny and his sister want to split the remainder of it. HoDanny'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]
Debra buys candy that costs 4 per pound. She will spend less than 20 on candy. What are the possibleDebra buys candy that costs 4 per pound. She will spend less than 20 on candy. What are the possible numbers of pounds she will buy?
Set up an inequality using less than < and p for pounds:
4p < 20
Divide each side by 4:
4p/4 < 20/4
[B]p < 5[/B]
Del and his 5 friends eat 4 pizzas. If each has the same amount and there is 1/4 of a pizza left oveDel 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]
Dennis was getting in shape for a marathon. The first day of the week he ran n miles. Dennis then adDennis was getting in shape for a marathon. The first day of the week he ran n miles. Dennis then added a mile to his run each day. By the end of the week (7 days), he had run a total of 70 miles. How many miles did Dennis run the first day?
Setup distance ran for the 7 days:
[LIST=1]
[*]n
[*]n + 1
[*]n + 2
[*]n + 3
[*]n + 4
[*]n + 5
[*]n + 6
[/LIST]
Add them all up:
7n + 21 = 70
Solve for [I]n[/I] in the equation 7n + 21 = 70
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 21 and 70. To do that, we subtract 21 from both sides
7n + 21 - 21 = 70 - 21
[SIZE=5][B]Step 2: Cancel 21 on the left side:[/B][/SIZE]
7n = 49
[SIZE=5][B]Step 3: Divide each side of the equation by 7[/B][/SIZE]
7n/7 = 49/7
n =[B] 7
[URL='https://www.mathcelebrity.com/1unk.php?num=7n%2B21%3D70&pl=Solve']Source[/URL][/B]
Deon opened his account starting with $650 and he is going to take out $40 per month. Mai opened upDeon opened his account starting with $650 and he is going to take out $40 per month. Mai opened up her account with a starting amount of $850 and is going to take out $65 per month. When would the two accounts have the same amount of money?
We set up a balance equation B(m) where m is the number of months.
[U]Set up Deon's Balance equation:[/U]
Withdrawals mean we subtract from our current balance
B(m) = Starting Balance - Withdrawal Amount * m
B(m) = 650 - 40m
[U]Set up Mai's Balance equation:[/U]
Withdrawals mean we subtract from our current balance
B(m) = Starting Balance - Withdrawal Amount * m
B(m) = 850 - 65m
When the two accounts have the same amount of money, we can set both balance equations equal to each other and solve for m:
650 - 40m = 850 - 65m
Solve for [I]m[/I] in the equation 650 - 40m = 850 - 65m
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables -40m and -65m. To do that, we add 65m to both sides
-40m + 650 + 65m = -65m + 850 + 65m
[SIZE=5][B]Step 2: Cancel -65m on the right side:[/B][/SIZE]
25m + 650 = 850
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 650 and 850. To do that, we subtract 650 from both sides
25m + 650 - 650 = 850 - 650
[SIZE=5][B]Step 4: Cancel 650 on the left side:[/B][/SIZE]
25m = 200
[SIZE=5][B]Step 5: Divide each side of the equation by 25[/B][/SIZE]
25m/25 = 200/25
m = [B]8[/B]
Determine whether the statement is true or false. You can always divide by e^xDetermine whether the statement is true or false. You can always divide by e^x
[B]True. As x --> infinity, 1/e^x approaches 0 but never touches it.[/B]
Diana invested $3000 in a savings account for 3 years. She earned $450 in interest over that time peDiana invested $3000 in a savings account for 3 years. She earned $450 in interest over that time period. What interest rate did she earn? Use the formula I=Prt to find your answer, where I is interest, P is principal, r is rate and t is time. Enter your solution in decimal form rounded to the nearest hundredth. For example, if your solution is 12%, you would enter 0.12.
Our givens are:
[LIST]
[*]I = 450
[*]P = 3000
[*]t = 3
[*]We want r
[/LIST]
450 = 3000(r)(3)
450 = 9000r
Divide each side by 9000
[B]r = 0.05[/B]
Difference of a and b, divided by 2Difference of a and b, divided by 2.
The difference of a and b is written as:
a - b
We divide this by 2:
[B](a - b)/2[/B]
Divide 17 by g. Then, subtract 9.Divide 17 by g. Then, subtract 9.
Divide 17 by g
17/g
Subtract 9
[B]17/g - 9[/B]
Divide 73 into two parts whose product is 402Divide 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 8 by 9, then subtract tdivide 8 by 9, then subtract t
Divide 8 by 9
8/9
Then subtract t
[B]8/9 - t[/B]
divide 8 by t, raise the result to the 7th powerdivide 8 by t, raise the result to the 7th power.
We take this algebraic expression in two parts:
1. Divide 8 by t
8/t
2. Raise the result to the 7th power. (This means we use an exponent of 7)
[B](8/t)^7[/B]
divide a by 8, triple the result, then add 7divide 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 haveDivide 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 bdivide 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 a number by 10. Then, add 10.Divide a number by 10. Then, add 10.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
Divide the number by 10 mean we have a quotient, of x over 10
x / 10
Then, add 10:
[B](x / 10) + 10[/B]
divide b by a, subtract the result from c, then add what you have to ddivide b by a, subtract the result from c, then add what you have to d
Take this algebraic expression in 3 parts:
[U]1) Divide b by a:[/U]
b/a
[U]2) Subtract the result from c:[/U]
c - b/a
[U]3) Then add what you have to d:[/U]
[B]c - b/a + d[/B]
divide d by a, add the result to b, then add cdivide d by a, add the result to b, then add c
[LIST]
[*]Divide d by a: d/a
[*]add the result to b: b + d/a
[*]Then add c
[/LIST]
[B]b + d/a + c[/B]
Divide m by 3 and then add 10Divide m by 3 and then add 10
Divide m by 3:
m/3
Then add 10:
[B]m/3 + 10[/B]
Divide the difference of 4 and r by 10Divide the difference of 4 and r by 10
The difference of 4 and r, mean we subtract r from 4:
4 - r
Now we divide this expression by 10:
[B](4 - r)/10 [/B]
divide the difference of q and s by the sum of p and rdivide the difference of q and s by the sum of p and r
Take this algebraic expression in pieces:
[LIST]
[*]The difference of q and s: q - s
[*]The sum of p and r: p + r
[*]The word [I]divide[/I] means we divide q - s by p + r
[/LIST]
[B](q - s)/(p + r)[/B]
Divide the sum of a and b by the square of cDivide the sum of a and b by the square of c
The sum of a and b:
a + b
The square of c means we raise c to the power of 2:
c^2
Divide means we have a quotient, with a + b on top, and c^2 on the bottom:
[B](a + b)/c^2[/B]
divide the sum of the square of a and b by thrice cdivide 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 the sum of the squares of a and b by the square of cDivide the sum of the squares of a and b by the square of c
square of a:
a^2
square of b:
b^2
Sum of the squares of a and b:
a^2 + b^2
square of c:
c^2
Divide the Sum of the squares of a and b by the square of c:
[B](a^2 + b^2)/c^2[/B]
Divide the sum x and y by the difference of subtracting a from bDivide the sum x and y by the difference of subtracting a from b
The sum x and y is written as:
x + y
The difference of subtracting a from b is written as:
b - a
We divide and get the algebraic expression:
[B](x + y)/(b - a)[/B]
divide u by s multiply the result by vdivide u by s multiply the result by v
Divide u by s:
u/s
Multiply the result by v:
[B]uv/s[/B]
divide u by s, then subtract the result from tdivide u by s, then subtract the result from t
Divide u by s:
u/s
Subtract the result from t:
[B]t - u/s[/B]
divide u by w add the result to vdivide u by w add the result to v
Divide u by w:
u/w
Add the result to v:
[B]v + u/w[/B]
Divide v by the sum of 4 and wDivide v by the sum of 4 and w
The sum of 4 and w means we add w to 4:
4 + w
Next, we divide v by this sum to get our final algebraic expression:
[B]v/(4 + w)[/B]
Divide x by 2.2, and then add 2.2 to the quotient.Divide x by 2.2, and then add 2.2 to the quotient.
Divide x by 2.2 (This is a quotient):
x/2.2
Then add 2.2 to the quotient
[B]x/2.2 + 2.2[/B]
Divide x cubed by the quantity x minus 7Divide x cubed by the quantity x minus 7
x cubed means we raise x to the power of 3:
x^3
We divide this by x - 7:
[B]x^3/(x - 7)[/B]
Dividend Discount ModelFree Dividend Discount Model Calculator - This calculator determines the present value of dividends using the Dividend Discount Model.
double 6 , divide the result by y ,then raise what you have to the 10th powerdouble 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 v, add u, then divide t by what you havedouble 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]
Dunder Mifflin will print business cards for $0.10 each plus setup charge of $15. Werham Hogg offersDunder Mifflin will print business cards for $0.10 each plus setup charge of $15. Werham Hogg offers business cards for $0.15 each with a setup charge of $10. What numbers of business cards cost the same from either company
Declare variables:
[LIST]
[*]Let b be the number of business cards.
[/LIST]
[U]Set up the cost function C(b) for Dunder Mifflin:[/U]
C(b) = Cost to print each business card * b + Setup Charge
C(b) = 0.1b + 15
[U]Set up the cost function C(b) for Werham Hogg:[/U]
C(b) = Cost to print each business card * b + Setup Charge
C(b) = 0.15b + 10
The phrase [I]cost the same[/I] means we set both C(b)'s equal to each other and solve for b:
0.1b + 15 = 0.15b + 10
Solve for [I]b[/I] in the equation 0.1b + 15 = 0.15b + 10
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 0.1b and 0.15b. To do that, we subtract 0.15b from both sides
0.1b + 15 - 0.15b = 0.15b + 10 - 0.15b
[SIZE=5][B]Step 2: Cancel 0.15b on the right side:[/B][/SIZE]
-0.05b + 15 = 10
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 15 and 10. To do that, we subtract 15 from both sides
-0.05b + 15 - 15 = 10 - 15
[SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE]
-0.05b = -5
[SIZE=5][B]Step 5: Divide each side of the equation by -0.05[/B][/SIZE]
-0.05b/-0.05 = -5/-0.05
b = [B]100[/B]
Each of 6 students reported the number of movies they saw in the past year. Here is what they reporEach of 6 students reported the number of movies they saw in the past year. Here is what they reported. 19, 9, 14, 10, 16, 17. Find the mean number of movies that the students saw. If necessary, round your answer to the nearest tenth.
The mean is the average, so we add up the 6 movie scores, and divide by 6.
[URL='https://www.mathcelebrity.com/statbasic.php?num1=19%2C+9%2C+14%2C+10%2C+16%2C+17&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']Mean (Average)[/URL] = Sum of 6 Movie Scores / 6
[URL='https://www.mathcelebrity.com/statbasic.php?num1=19%2C+9%2C+14%2C+10%2C+16%2C+17&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']Mean (Average)[/URL] = 84 / 6
[URL='https://www.mathcelebrity.com/statbasic.php?num1=19%2C+9%2C+14%2C+10%2C+16%2C+17&num2=+0.2%2C0.4%2C0.6%2C0.8%2C0.9&pl=Number+Set+Basics']Mean (Average)[/URL] = 14.16666667
The problem asks us to round to the nearest tenth, which is the first decimal place.
Since the 2nd decimal place, 6 is more than 5, we round the first decimal place up one and remove the rest.
[B]14.2[/B]
Eight times the quantity y plus two divided by fourthe quantity y plus two
y + 2
the quantity y plus two divided by four
(y +2)/4
Eight times the quantity y plus two divided by four
8(y +2)/4
8/4 = 2, so we have:
[B]2(y +2) or 2y + 4
[MEDIA=youtube]xzwaXi6N1uI[/MEDIA][/B]
Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30.Emily is three years older than twice her sister Mary’s age. The sum of their ages is less than 30. What is the greatest age Mary could be?
Let e = Emily's age and m = Mary's age.
We have the equation e = 2m + 3 and the inequality e + m < 30
Substitute the equation for e into the inequality:
2m + 3 + m < 30
Add the m terms
3m + 3 < 30
Subtract 3 from each side of the inequality
3m < 27
Divide each side of the inequality by 3 to isolate m
m < 9
Therefore, the [B]greatest age[/B] Mary could be is 8, since less than 9 [U]does not include[/U] 9.
Equation 2y+5x=40. Interprt the interceptsEquation 2y+5x=40. Interprt the intercepts
Y intercept is when X = 0
2y + 5(0) = 40
2y = 40
Divide each side by 2
[B]y = 20
[/B]
X intercept is when Y = 0
2(0) + 5x = 40
5x = 40
Divide each side by 5
[B]x = 8[/B]
Erin has 72 stamps in her stamp drawer along with a quarter, two dimes and seven pennies. She has 3Erin has 72 stamps in her stamp drawer along with a quarter, two dimes and seven pennies. She has 3 times as many 3-cent stamps as 37-cent stamps and half the number of 5-cent stamps as 37-cent stamps. The value of the stamps and coins is $8.28. How many 37-cent stamps does Erin have?
Number of stamps:
[LIST]
[*]Number of 37 cent stamps = s
[*]Number of 3-cent stamps = 3s
[*]Number of 5-cent stamps = 0.5s
[/LIST]
Value of stamps and coins:
[LIST]
[*]37 cent stamps = 0.37s
[*]3-cent stamps = 3 * 0.03 = 0.09s
[*]5-cent stamps = 0.5 * 0.05s = 0.025s
[*]Quarter, 2 dime, 7 pennies = 0.52
[/LIST]
Add them up:
0.37s + 0.09s + 0.025s + 0.52 = 8.28
Solve for [I]s[/I] in the equation 0.37s + 0.09s + 0.025s + 0.52 = 8.28
[SIZE=5][B]Step 1: Group the s terms on the left hand side:[/B][/SIZE]
(0.37 + 0.09 + 0.025)s = 0.485s
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
0.485s + 0.52 = + 8.28
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 0.52 and 8.28. To do that, we subtract 0.52 from both sides
0.485s + 0.52 - 0.52 = 8.28 - 0.52
[SIZE=5][B]Step 4: Cancel 0.52 on the left side:[/B][/SIZE]
0.485s = 7.76
[SIZE=5][B]Step 5: Divide each side of the equation by 0.485[/B][/SIZE]
0.485s/0.485 = 7.76/0.485
s = [B]16[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=0.37s%2B0.09s%2B0.025s%2B0.52%3D8.28&pl=Solve']Source[/URL]
Explain the steps you would take to find an equation for the line perpendicular to 4x - 5y = 20 andExplain 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]
ey/n + k = t for yey/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 varies directly as g and inversely as r^2F 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 bf(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/B=(M-N*L)/D for LF/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]
FactorialsFree Factorials Calculator - Calculates the following factorial items:
* A factorial of one number such as n!
* A factorial of a numerator divided by a factorial of a denominator such as n!m!/a!b!
* Double Factorials such as n!!
* Stirlings Approximation for n!
Farmer Yumi has too many plants in her garden. If she starts out with 150 plants and is losing themFarmer Yumi has too many plants in her garden. If she starts out with 150 plants and is losing them at a rate of 4% each day, how long will it take for her to have 20 plants left? Round UP to the nearest day.
We set up the function P(d) where d is the number of days sine she started losing plants:
P(d) = Initial plants * (1 - Loss percent / 100)^d
Plugging in our numbers, we get:
20 = 150 * (1 - 4/100)^d
20 = 150 * (1 - 0.04)^d
Read left to right so it's easier to read:
150 * 0.96^d = 20
Divide each side by 150, and we get:
0.96^d = 0.13333333333
To solve this logarithmic equation for d, we [URL='https://www.mathcelebrity.com/natlog.php?num=0.96%5Ed%3D0.13333333333&pl=Calculate']type it in our search engine[/URL] and we get:
d = 49.35
The problem tells us to round up, so we round up to [B]50 days[/B]
Find the greatest number which divides 845 and 1250Find the greatest number which divides 845 and 1250
This is the greatest common factor. We [URL='https://www.mathcelebrity.com/gcflcm.php?num1=845&num2=1250&num3=&pl=GCF+and+LCM']type GCF(845,1250) into our search engine [/URL]and we get:
[B]5[/B]
Find the last digit of 2 raised to the 2020 no calculatorCheck out this pattern:
2^1= 2
2^2= 4
2^3 = 8
2^4= 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
The last digit repeats itself in blocks of 4
2, 4, 8, 6
We want to know what is the largest number in 1, 2, 3, 4 that divides 2020 without a remainder.
LEt's start with 4 and work backwards.
2020/4 = 505
Ever power of 2^4(n) ends in 6, so our answer is [B]6
[MEDIA=youtube]6uX5gwb1jdY[/MEDIA][/B]
Find the odd number less than 100 that is divisible by 9, and when divided by 10 has a remainder ofFind the odd number less than 100 that is divisible by 9, and when divided by 10 has a remainder of 7.
From our [URL='http://www.mathcelebrity.com/divisibility.php?num=120&pl=Divisibility']divisibility calculator[/URL], we see a number is divisible by 9 if the sum of its digits is divisible by 9.
Starting from 1 to 99, we find all numbers with a digit sum of 9.
This would be digits with 0 and 9, 1 and 8, 2 and 7, 3 and 6, and 4 and 5.
9
18
27
36
45
54
63
72
81
90
Now remove even numbers since the problem asks for odd numbers
9
27
45
63
81
Now, divide each number by 10, and find the remainder
9/10 = 0
[URL='http://www.mathcelebrity.com/modulus.php?num=27mod10&pl=Calculate+Modulus']27/10[/URL] = 2 R 7
We stop here. [B]27[/B] is an odd number, less than 100, with a remainder of 7 when divided by 10.
Find the velocity of a cheetah that runs 100m in 4 secondsFind the velocity of a cheetah that runs 100m in 4 seconds
100m / 4 seconds
Divide top and bottom by 4
[B]25m/second[/B]
Finding a 20% tip no calculatorFinding a 20% tip no calculator
We have 2 methods to calculate a 20% tip.
[LIST=1]
[*]Divide by 5
[*]Shift one decimal place left and take the value. Multiply by 2
[/LIST]
Example: 180 tip, find a 20% tip:
Method 1:
180/5 = 36
Method 2:
Move decimal place left = 18
Multiply this value by 2: 18 * 2 = 36
[MEDIA=youtube]UW4GYWfMhsE[/MEDIA]
Fixed cost 500 marginal cost 8 item sells for 30fixed cost 500 marginal cost 8 item sells for 30.
Set up Cost Function C(x) where x is the number of items sold:
C(x) = Marginal Cost * x + Fixed Cost
C(x) = 8x + 500
Set up Revenue Function R(x) where x is the number of items sold:
R(x) = Revenue per item * items sold
R(x) = 30x
Set up break even function (Cost Equals Revenue)
C(x) = R(x)
8x + 500 = 30x
Subtract 8x from each side:
22x = 500
Divide each side by 22:
x = 22.727272 ~ 23 units for breakeven
Foster is centering a photo that is 9/1/2 inches wide on a scrapbook pages that is 10 inches wide. HFoster is centering a photo that is 9/1/2 inches wide on a scrapbook pages that is 10 inches wide. How far from each side of the pages should he put the picture? Enter your answer as a mixed number.
First, determine your margins, which is the difference between the width and photo width, divided by 2.
10 - 9 & 1/2 = 1/2
1/2 / 2 = [B]1/4[/B]
Fred earns $420 a month. If his monthly car payment is one quarter of his pay, how much is his car pFred earns $420 a month. If his monthly car payment is one quarter of his pay, how much is his car payment?
1/4 means divided by 4, so we have:
Monthly Payment = Earnings/4
Monthly Payment =420/4
Monthly Payment = [B]$105[/B]
Fred had 156 dollars to spend on 9 books. After buying them he had 12 dollars. How much did each booFred had 156 dollars to spend on 9 books. After buying them he had 12 dollars. How much did each book cost?
Subtract the 12 dollars left over from the $156 starting amount:
$156 - $12 = $144
Now divide $144 / 9 books to get the cost per book:
$144/9 = [B]$16 per book[/B]
f^2+5g=3md for df^2+5g = 3md for d
Divide each side by 3m to isolate d:
(f^2+5g)/3m = 3md/3md
Cancel the 3m on the right side and we get:
d = [B](f^2+5g)/3m[/B]
Gary has three less pets than Abe. If together they own 15 pets, how many pets does Gary own?Let g = Gary's pets and a = Abe's pets.
We are given two equations:
(1) g = a - 3
(2) a + g = 15
Substitute (1) into (2)
a + (a - 3) = 15
Combine Like Terms:
2a - 3 = 15
Add 3 to each side:
2a = 18
Divide each side by 2 to isolate a:
a = 9 --> Abe has 9 pets
Substitute a = 9 into Equation (1)
g = 9 - 3
g = 6 --> Gary has 6 pets
Gino has 7 whole pineapples. He cuts each whole into 4 equal parts. Write an improper fraction for tGino has 7 whole pineapples. He cuts each whole into 4 equal parts. Write an improper fraction for the cut parts of pineapples.
Take our whole pineapples divided by the number of equal parts:
[B]7/4[/B]
Giovanni is thinking of a number. If he adds 2 to it, then divides that sum by 3, he gets 7. What isGiovanni 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: 9 - 4x = -19 Prove: x = 7Given: 9 - 4x = -19 Prove: x = 7
Solve for [I]x[/I] in the equation 9 - 4x = - 19
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 9 and -19. To do that, we subtract 9 from both sides
-4x + 9 - 9 = -19 - 9
[SIZE=5][B]Step 2: Cancel 9 on the left side:[/B][/SIZE]
-4x = -28
[SIZE=5][B]Step 3: Divide each side of the equation by -4[/B][/SIZE]
-4x/-4 = -28/-4
x = [B]7[/B]
gy=-g/v+w for ggy=-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]
Half of abHalf of ab
Half means we divide by 2:
[B]ab/2[/B]
half of c subtracted from the sum of a and bhalf of c subtracted from the sum of a and b
The sum of a and b:
a + b
half of c means we divide c by 2:
c/2
half of c subtracted from the sum of a and b:
[B]a + b - c/2[/B]
Half of the difference of a and bHalf of the difference of a and b
The difference of a and b is written as:
a - b
Half of the difference means we divide (a - b) by 2:
[B](a - b)/2[/B]
half of the sum of 2p and qhalf of the sum of 2p and q
The sum of 2p and q means we add q to 2p:
2p + q
Half of this means we divide the sum by 2:
[B](2p + q)/2[/B]
half of z increased by 10half of z increased by 10
Half of z (means we divide z by 2)
z/2
Increased by 10 means we add 10
[B]z/2 + 10[/B]
half the difference of x and 3half the difference of x and 3
The difference of x and 3 means we subtract 3 from x:
x - 3
half of the difference means we divide the difference by 2:
[B](x - 3)/2[/B]
half the sum of the numbers s, t, and uhalf the sum of the numbers s, t, and u
The [I]sum [/I]of s, t, and u means we add all 3:
s + t + u
[I]Half[/I] the sum means we divide the sum by 2:
[B](s + t + u)/2[/B]
Happy Paws charges $16.00 plus $1.50 per hour to keep a dog during the day. Woof Watchers charges $1Happy Paws charges $16.00 plus $1.50 per hour to keep a dog during the day. Woof Watchers charges $11.00 plus $2.75 per hour. Complete the equation and solve it to find for how many hours the total cost of the services is equal. Use the variable h to represent the number of hours.
Happy Paws Cost: C = 16 + 1.5h
Woof Watchers: C = 11 + 2.75h
Setup the equation where there costs are equal
16 + 1.5h = 11 + 2.75h
Subtract 11 from each side:
5 + 1.5h = 2.75h
Subtract 1.5h from each side
1.25h = 5
Divide each side by 1.25
[B]h = 4[/B]
How many 1/4 sheets are there in 5 sheetsHow 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 bHow 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 kobo are there in y naira?How many kobo are there in y naira?
One naira is divided into [B]100 kobo[/B].
So we have [B]100y kobo[/B]
How much do 10 pieces of candy cost if 1000 pieces cost 100.00?How much do 10 pieces of candy cost if 1000 pieces cost 100.00?
Set up a proportion of pieces to cost
10/x = 1000/100
Divide the right side by 100 on top and bottom
10/x = 10/1
[B]x = 1[/B]
How much would you need to deposit in an account now in order to have $6000 in the account in 10 yeaHow much would you need to deposit in an account now in order to have $6000 in the account in 10 years? Assume the account earns 6% interest compounded monthly.
We start with a balance of B. We want to know:
B(1.06)^10 = 6000
B(1.79084769654) = 6000
Divide each side of the equation by 1.79084769654 to solve for B
B = [B]3,350.37[/B]
How much would you need to deposit in an account now in order to have $6000 in the account in 15 yeaHow much would you need to deposit in an account now in order to have $6000 in the account in 15 years? Assume the account earns 8% interest compounded monthly.
8% compounded monthly = 8/12 = 0.6667% per month.
15 years = 15*12 = 180 months
We want to know an initial balance B such that:
B(1.00667)^180 = $6,000
3.306921B = $6,000
Divide each side by 3.306921
[B]B = $1,814.38[/B]
I grade 160 tests in 5 hours. How many tests do I grade per hour?I grade 160 tests in 5 hours. How many tests do I grade per hour?
160 tests / 5 hours
Divide top and bottom by 5:
[B]32 tests per hour[/B]
I have 6 cakes and I want to divide them between 8 people how much does each person get?6 cakes for 8 people. Divide by 8 people to get the cakes for each person.
6/8 cake per person. However, this fraction can be simplified.
Divide the top and bottom by 2.
We get 3/4, or 0.75 cake for each person.
I invest $3000 at 5% interest a year. So far I have made $600 with simple interest. How many years hI invest $3000 at 5% interest a year. So far I have made $600 with simple interest. How many years have I been investing?
Simple interest is calculated using interest * principal.
We have 5% * 3000 = $150 interest per year
We take our $600 of total interest and divide it by our interest per year to get the total years:
$600 / $150 = [B]4 years[/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]
If $9000 grows to $9720 in 2 years find the simple interest rate.If $9000 grows to $9720 in 2 years find the simple interest rate.
Simple interest formula is Initial Balance * (1 + tn) = Current Balance
We have
[LIST]
[*]Initial Balance = 9000
[*]Current Balance = 9720
[*]n = 2
[/LIST]
Plugging in these values, we get:
9000 * (1 + 2t) = 9720
Divide each side by 9000
1 + 2t = 1.08
Subtract 1 from each sdie
2t = 0.08
Divide each side by 2
t = [B]0.04 or 4%[/B]
If 100 runners went with 4 bicyclists and 5 walkers, how many bicyclists would go with 20 runners anIf 100 runners went with 4 bicyclists and 5 walkers, how many bicyclists would go with 20 runners and 2 walkers?
[U]Set up a joint variation equation, for the 100 runners, 4 bicyclists, and 5 walkers:[/U]
100 = 4 * 5 * k
100 = 20k
[U]Divide each side by 20[/U]
k = 5 <-- Coefficient of Variation
[U]Now, take scenario 2 to determine the bicyclists with 20 runners and 2 walkers[/U]
20 = 2 * 5 * b
20 = 10b
[U]Divide each side by 10[/U]
[B]b = 2[/B]
if 200 is divided in the ratio of 1:3:4 , what is the greatest numberif 200 is divided in the ratio of 1:3:4 , what is the greatest number
Determine the ratio denominator by adding up the ratio amounts:
1 + 3 + 4 = 8
So we have the following ratios and ratio amounts with our greatest number in bold:
[LIST]
[*]1/8 * 200 = 25
[*]3/8 * 200 = 75
[*]4/8 * 200 = [B]100[/B]
[/LIST]
If 2x + y = 7 and y + 2z = 23, what is the average of x, y, and z?If 2x + y = 7 and y + 2z = 23, what is the average of x, y, and z?
A. 5
B. 7.5
C. 15
D. 12.25
Add both equations to get all variables together:
2x + y + y + 2z = 23 + 7
2x + 2y + 2z = 30
We can divide both sides by 2 to simplify:
(2x + 2y + 2z)/2= 30/2
x + y + z = 15
Notice: the average of x, y, and z is:
(x + y + z)/3
But x + y + z = 15, so we have:
15/3 = [B]5, answer A[/B]
[MEDIA=youtube]tOCAhhfMCLI[/MEDIA]
If 3 times a number added to 2 is divided by the number plus 4 the result is 4/3If 3 times a number added to 2 is divided by the number plus 4 the result is 4/3
Take this in pieces, where "a number" means an arbitrary variable, let's call it "x".
[LIST=1]
[*]3 times a number --> 3x
[*]3 times a number added to 2 --> 3x + 2
[*]The number plus 4 --> x + 4
[*]is divided by --> (3x + 2)/(x + 4)
[*]the result is 4/3 --> (3x + 2)/(x + 4) = 4/3
[/LIST]
If 3(c + d) = 5, what is the value of c + d?If 3(c + d) = 5, what is the value of c + d?
A) 3/5
B) 5/3
C) 3
D) 5
Divide each side of the equation by 3 to [U]isolate[/U] c + d
3(c + d)/3 = 5/3
Cancel the 3's on the left side, we get:
c + d = [B]5/3, or answer B[/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 4x+7=xy-6, then what is the value of x, in terms of yIf 4x+7=xy-6, then what is the value of x, in terms of y
Subtract xy from each side:
4x + 7 - xy = -6
Add 7 to each side:
4x - xy = -6 - 7
4x - xy = -13
Factor out x:
x(4 - y) = -13
Divide each side of the equation by (4 - y)
[B]x = -13/(4 - y)[/B]
if a + b = 2 and a2 - b2 = -4, what is the value of a - b?if a+b=2 and a2-b2=-4, what is the value of a-b?
a^2 - b^2 = -4
Factor this:
(a + b)(a - b) = -4
We know from above, (a +b) = 2, so substitute:
2(a - b) = -4
Divide each side by 2
[B](a - b) = -2[/B]
if a divides b, then a divides bcif a divides b, then a divides bc
Suppose a divides b. Then there exists an integer q such that b = aq, so that bc = a(qc) and a divides bc.
Suppose that a divides c. Then there exists an integer k such that c = ak, so that bc = a(kb) and a divides bc.
If a number is increased by 16 and then divided by 3, the result is 8If 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 pound of coffee costs $4, how many ounces can be bought for $1.80If a pound of coffee costs $4, how many ounces can be bought for $1.80
Using our conversion calculator, we find
[URL='https://www.mathcelebrity.com/weightcon.php?quant=1&pl=Calculate&type=pound']1 pound [/URL]= 16 ounces
$4 per 16 ounces = 4/16
[URL='https://www.mathcelebrity.com/perc.php?num=4&den=16&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']We can simplify this fraction[/URL] to 0.25 per ounce
We take our 1.80 divided by 0.25 per ounce
1.80/0.25 = [B]7.2 ounces of coffee
[MEDIA=youtube]5eZAav1drX0[/MEDIA][/B]
If c=3 and d=4 evaluate cd divided by 2If c=3 and d=4 evaluate cd divided by 2
cd = 3(4)
cd = 12
Divide this by 2:
12/2
[B]6[/B]
If Distance equals Speed times Time (D = S x T), then what does time equal in terms of speed and disIf Distance equals Speed times Time (D = S x T), then what does time equal in terms of speed and distance?
Divide each side by S to isolate T:
D/S = S x T/S
Cancel the S's on the right side:
[B]T = D/S[/B]
If EF = 9x - 17, FG = 17x - 14, and EG = 20x + 17, what is FG?If EF = 9x - 17, FG = 17x - 14, and EG = 20x + 17, what is FG?
By segment addition, we know that:
EF + FG = EG
Substituting in our values for the 3 segments, we get:
9x - 17 + 17x - 14 = 20x + 17
Group like terms and simplify:
(9 + 17)x + (-17 - 14) = 20x - 17
26x - 31 = 20x - 17
Solve for [I]x[/I] in the equation 26x - 31 = 20x - 17
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 26x and 20x. To do that, we subtract 20x from both sides
26x - 31 - 20x = 20x - 17 - 20x
[SIZE=5][B]Step 2: Cancel 20x on the right side:[/B][/SIZE]
6x - 31 = -17
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -31 and -17. To do that, we add 31 to both sides
6x - 31 + 31 = -17 + 31
[SIZE=5][B]Step 4: Cancel 31 on the left side:[/B][/SIZE]
6x = 14
[SIZE=5][B]Step 5: Divide each side of the equation by 6[/B][/SIZE]
6x/6 = 14/6
x = [B]2.3333333333333[/B]
If f(x) = 3x + 1 and g(x) = x^2 + 2x, find x when f(g(x)) = 10If f(x) = 3x + 1 and g(x) = x^2 + 2x, find x when f(g(x)) = 10
[U]Evaluate f(g(x))[/U]
f(g(x)) = 3(x^2 + 2x) + 1
f(g(x)) = 3x^2 + 6x + 1
[U]When f(g(x)) = 10, we have[/U]
10 = 3x^2 + 6x + 1
[U]Subtract 10 from each side:[/U]
3x^2 + 6x - 9 = 0
Divide each side of the equation by 3
x^2 + 2x - 3 = 0
Factor, we have: (x + 3)(x - 1) = 0
So x is either [B]1 or -3[/B]
If Mr hernandez has 5 times as many students as Mr daniels and together they have 150 students how mIf Mr hernandez has 5 times as many students as Mr daniels and together they have 150 students how many students do each have?
Let h = Mr. Hernandez's students and d = Mr. Daniels students.
We are given two equations:
(1) h = 5d
(2) d + h = 150
Substitute equation (1) into equation (2)
d + (5d) = 150
Combine like terms:
6d = 150
Divide each side of the equation by 6 to isolate d
d = 25 <-- Mr. Daniels Students
Now, plug the value for d into equation (1)
h = 5(25)
h = 125 <-- Mr. Hernandez students
If p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equIf p is inversely proportional to the square of q, and p is 2 when q is 4, determine p when q is equal to 2.
We set up the variation equation with a constant k such that:
p = k/q^2 [I](inversely proportional means we divide)
[/I]
When q is 4 and p is 2, we have:
2 = k/4^2
2 = k/16
Cross multiply:
k = 2 * 16
k = 32
Now, the problem asks for p when q = 2:
p = 32/2^2
p = 32/4
p = [B]8
[MEDIA=youtube]Mro0j-LxUGE[/MEDIA][/B]
If 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 circumference of a circular rug is 16? feet, then what is the area of the rug in terms of piIf the circumference of a circular rug is 16? feet, then what is the area of the rug in terms of pi
C = 2pir, so we have:
C = 16?
16? = 2?r
Divide each side by 2?:
r = 16?/2?
r = 8
Now, the area of a circle A is denoted below:
A = ?r^2
Given r = 8 from above, we have:
A = ?(8)^2
A = [B]64?[/B]
If the cost of a bat and a baseball combined is $1.10 and the bat cost $1.00 more than the ball howLet a be the cost of the ball and b be the cost of the bat:
We're given 2 equations:
[LIST=1]
[*]a + b = 1.10
[*]b = a + 1
[/LIST]
Substitute equation (2) into equation (1) for b:
a + a + 1 = 1.10
Combine like terms:
2a + 1 = 1.10
Subtract 1 from each side:
2a + 1 - 1 = 1.10 - 1
2a = 0.10
Divide each side by 2:
2a/2 = 0.10/2
a = [B]0.05[/B]
[MEDIA=youtube]79q346Hy7R8[/MEDIA]
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 thIf 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 twice a number is divided by 7, the result is -28If twice a number is divided by 7, the result is -28.
The phrase [I]a number[/I] means an arbitrary variable, let's call it "x".
Twice x means we multiply x by 2: 2x
Divide this by 7: 2x/7
We set this equal to -28, and we have our algebraic expression:
[B]2x/7 = -28 [/B]
If two consecutive even numbers are added, the sum is equal to 226. What is the smaller of the two nIf two consecutive even numbers are added, the sum is equal to 226. What is the smaller of the two numbers?
Let the smaller number be n.
The next consecutive even number is n + 2.
Add them together to equal 226:
n + n + 2 = 226
Solve for [I]n[/I] in the equation n + n + 2 = 226
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 1)n = 2n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2n + 2 = + 226
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 2 and 226. To do that, we subtract 2 from both sides
2n + 2 - 2 = 226 - 2
[SIZE=5][B]Step 4: Cancel 2 on the left side:[/B][/SIZE]
2n = 224
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2n/2 = 224/2
n = [B]112
[URL='https://www.mathcelebrity.com/1unk.php?num=n%2Bn%2B2%3D226&pl=Solve']Source[/URL][/B]
If x 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 x^2=y^3, for what value of z does x^{3z}= y^9if x^2=y^3, for what value of z does x^{3z}= y^9
y^9 = y^3 * y^3, so if we square the right side, we must square the left side for equivalence:
x^2 * x^2 = x^4
Therefore,
x^4 = y^9
Going back to our problem, x^{3z}= y^9, so 3z = 4
Divide each side by 3 to isolate z, and we have:
3z/3 = 4/3
z = [B]4/3[/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 can buy 1?3 of a box of chocolates for 6 dollars, how much can you purchase for 4 dollars? WrIf 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 brIf 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
In 16 years, Ben will be 3 times as old as he is right now.In 16 years, Ben will be 3 times as old as he is right now.
Let Ben's age right now be b.
We have, in 16 years, Ben's age will be 3 times what his age is now:
b + 16 = 3b
Subtract b from each side:
2b = 16
Divide each side by 2
[B]b = 8[/B]
Check our work:
16 years from now, Ben's age is 8 + 16 = 24
And, 8 x 3 = 24
In 45 years, Gabriela will be 4 times as old as she is right now.In 45 years, Gabriela will be 4 times as old as she is right now.
Let a be Gabriela's age. we have:
a + 45 = 4a
Subtract a from each side:
3a = 45
Divide each side by a
[B]a = 15[/B]
In a class there are 5 more boys than girls. There are 13 students in all. How many boys are there iIn a class there are 5 more boys than girls. There are 13 students in all. How many boys are there in the class?
We start by declaring variables for boys and girls:
[LIST]
[*]Let b be the number of boys
[*]Let g be the number of girls
[/LIST]
We're given two equations:
[LIST=1]
[*]b = g + 5
[*]b + g = 13
[/LIST]
Substitute equation (1) for b into equation (2):
g + 5 + g = 13
Grouping like terms, we get:
2g + 5 = 13
Subtract 5 from each side:
2g + 5 - 5 = 13 - 5
Cancel the 5's on the left side and we get:
2g = 8
Divide each side of the equation by 2 to isolate g:
2g/2 = 8/2
Cancel the 2's on the left side and we get:
g = 4
Substitute g = 4 into equation (1) to solve for b:
b = 4 + 5
b = [B]9[/B]
In a newspaper, it was reported that yearly robberies in Springfield were up 40% to 77 in 2012 fromIn a newspaper, it was reported that yearly robberies in Springfield were up 40% to 77 in 2012 from 2011. How many robberies were there in Springfield in 2011?
Let r be the number of robberies in 2011. We have:
Robberies in 2012 = Robberies in 2011 * 1.4
77 = r * 1.4
Divide each side by 1.4
[B]r = 55[/B]
In Super Bowl XXXV, the total number of points scored was 41. The winning team outscored the losingIn Super Bowl XXXV, the total number of points scored was 41. The winning team outscored the losing team by 27 points. What was the final score of the game? In Super Bowl XXXV, the total number of points scored was 41. The winning team outscored the losing team by 27 points. What was the final score of the game?
Let w be the winning team's points, and l be the losing team's points. We have two equations:
[LIST=1]
[*]w + l = 41
[*]w - l = 27
[/LIST]
Add the two equations:
2w = 68
Divide each side by 2
[B]w = 34[/B]
Substitute this into (1)
34 + l = 41
Subtract 34 from each side
[B]l = 7[/B]
Check your work:
[LIST=1]
[*]34 + 7 = 41 <-- check
[*]34 - 7 = 27 <-- check
[/LIST]
The final score of the game was [B]34 to 7[/B].
You could also use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=w+%2B+l+%3D+41&term2=w+-+l+%3D+27&pl=Cramers+Method']simultaneous equation solver[/URL].
In the year 1980, Rick was twice as old as Nancy who was twice as old as Michael. In the year 1992 RIn the year 1980, Rick was twice as old as Nancy who was twice as old as Michael. In the year 1992 Ric, Nancy, and Michael ages added up to 78 years. How old was Ric in 1980?
Age in 1980:
[LIST]
[*]Let Michael's age be m
[*]Nancy's age is 2m
[*]Rick's age is 2 * 2m = 4m
[/LIST]
Age in 1992:
[LIST]
[*]Michael's age = m + 12
[*]Nancy's age is 2m + 12
[*]Rick's age is 2 * 2m = 4m + 12
[/LIST]
Total them up:
m + 12 + 2m + 12 + 4m + 12 = 78
Solve for [I]m[/I] in the equation m + 12 + 2m + 12 + 4m + 12 = 78
[SIZE=5][B]Step 1: Group the m terms on the left hand side:[/B][/SIZE]
(1 + 2 + 4)m = 7m
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
12 + 12 + 12 = 36
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
7m + 36 = + 78
[SIZE=5][B]Step 4: Group constants:[/B][/SIZE]
We need to group our constants 36 and 78. To do that, we subtract 36 from both sides
7m + 36 - 36 = 78 - 36
[SIZE=5][B]Step 5: Cancel 36 on the left side:[/B][/SIZE]
7m = 42
[SIZE=5][B]Step 6: Divide each side of the equation by 7[/B][/SIZE]
7m/7 = 42/7
m = 6
Rick's age = 6 * 4 = [B]24
[URL='https://www.mathcelebrity.com/1unk.php?num=m%2B12%2B2m%2B12%2B4m%2B12%3D78&pl=Solve']Source[/URL]
[/B]
In this class of 4/5 students are right handed. if there are 20 right handed students, what is the tIn 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]
It takes 60 minutes for 7 people to paint 5 walls. How many minutes does it take 10 people to paintIt 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]
Jack bought 7 tickets for a movie. He paid $7 for each adult ticket and $4 for each child ticket. JaJack bought 7 tickets for a movie. He paid $7 for each adult ticket and $4 for each child ticket. Jack spent $40 for the tickets
Let a = Number of adult tickets and c be the number of child tickets.
[LIST=1]
[*]7a + 4c = 40
[*]a + c = 7
[*]Rearrange (2): a = 7 - c
[/LIST]
Now substitute a in (3) into (1):
7(7 - c) + 4c = 40
49 - 7c + 4c = 40
49 - 3c = 40
Add 3c to each side and subtract 40:
3c = 9
Divide each side by 3:
[B]c = 3
[/B]
Substitute c = 3 into Equation (2)
a + 3 = 7
Subtract 3 from each side:
[B]a = 4[/B]
Jack is making snack bags. He has 18 baby carrots and 42 pretzels. He wants to divide them equally aJack is making snack bags. He has 18 baby carrots and 42 pretzels. He wants to divide them equally among the bags. What is the greatest number of snack bags he can make?
Find the [URL='http://www.mathcelebrity.com/gcflcm.php?num1=18&num2=42&num3=&pl=GCF']Greatest Common Factor[/URL] of (18, 42) = 6
6 bags for 18 carrots = 3 carrots per bag
6 bags for 42 pretzels = 7 pretzels per bag
[B]6 bags is the answer[/B]
Jake used 5 boxes to pack 43.5 kg of books. If the boxes each weighed the same and held 8 books, whJake used 5 boxes to pack 43.5 kg of books. If the boxes each weighed the same and held 8 books, what did each book weigh?
[U]Set up equations were w is the weight of each book:[/U]
[LIST=1]
[*]5 boxes * 8 books * w = 43.5
[*]40w = 43.5
[/LIST]
[U]Divide each side by 40[/U]
[B]w = 1.0875 kg[/B]
James is ordering action figures online. Each action figure is $9. Shipping cost $10 per order. JameJames is ordering action figures online. Each action figure is $9. Shipping cost $10 per order. James does not want to spend over $154. How many action figures can he order?
Step 1: Subtract the cost of shipping from the spend
$154 - $10 = $144
Step 2: Divide $144 to spend after shipping by $9 action figures
144/9 = [B]$16 action figures[/B]
Jamie spent $15.36 on several items at the store. he spent an equal amount on each item. if jamie spJamie spent $15.36 on several items at the store. he spent an equal amount on each item. if jamie spent $1.92 on each item, how many items did he buy?
Let x equal the number of items Jamie bought. We have:
1.92x = 15.36
Divide each side by 1.92
[B]x = 8[/B]
jane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an inejane has 55$ to spend at cedar point. the admission price is 42$ and each soda is 4.25. write an inequality to show how many sodas he can buy.
Let s be the number of sodas.
Cost for the day is:
Price per soda * s + Admission Price
4.25s + 42
We're told that Jane has 55, which means Jane cannot spend more than 55. Jane can spend up to or less than 55. We write this as an inequality using <= 55
[B]4.25s + 42 <= 55[/B]
[B][/B]
If the problems asks you to solve for s, we type it in our math engine and we get:
Solve for [I]s[/I] in the inequality 4.25s + 42 ? 55
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 42 and 55. To do that, we subtract 42 from both sides
4.25s + 42 - 42 ? 55 - 42
[SIZE=5][B]Step 2: Cancel 42 on the left side:[/B][/SIZE]
4.25s ? 13
[SIZE=5][B]Step 3: Divide each side of the inequality by 4.25[/B][/SIZE]
4.25s/4.25 ? 13/4.25
[B]s ? 3.06[/B]
Jane is twice a old as Joel. If their ages total 63, how old is Joel?Jane is twice a old as Joel. If their ages total 63, how old is Joel?
Joel = j
Jane = 2j
j + 2j = 63
3j = 63
Divide each side by 3:
3j/3 = 63/j
Cancel the 3's on the left side:
j = [B]21[/B]
Janet drove 395 kilometers and the trip took 5 hours. How fast was Janet traveling?Janet drove 395 kilometers and the trip took 5 hours. How fast was Janet traveling?
Distance = Rate * Time
We're given D = 395 and t = 5
We want Rate. We divide each side of the equation by time:
Distance / Time = Rate * Time / Time
Cancel the Time's on each side and we get:
Rate = Distance / Time
Plugging our numbers in, we get:
Rate = 395/5
Rate = [B]79 kilometers[/B]
Jason has an equal number of nickels and dimes. The total value of his nickels and dimes is $2.25. HJason has an equal number of nickels and dimes. The total value of his nickels and dimes is $2.25. How many nickels does Jason have?
Let the number of nickels be n
Let the number of dimes be d
We're given two equations:
[LIST=1]
[*]d = n
[*]0.05n + 0.1d = 2.25
[/LIST]
Substitute equation (1) for d into equation (2):
0.05n + 0.1n = 2.25
Solve for [I]n[/I] in the equation 0.05n + 0.1n = 2.25
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(0.05 + 0.1)n = 0.15n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
0.15n = + 2.25
[SIZE=5][B]Step 3: Divide each side of the equation by 0.15[/B][/SIZE]
0.15n/0.15 = 2.25/0.15
n = [B]15[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=0.05n%2B0.1n%3D2.25&pl=Solve']Source[/URL]
Jason is 9 miles ahead of Joe running at 5.5 miles per hour and Joe is running at the speed of 7 mil[SIZE=6]Jason is 9 miles ahead of Joe running at 5.5 miles per hour and Joe is running at the speed of 7 miles per hour. How long does it take Joe to catch Jason?
A. 3 hours
B. 4 hours
C. 6 hours
D. 8 hours
Distance formula is d = rt
Jason's formula (Add 9 since he's ahead 9 miles):
d = 5.5t + 9
Joe's formula:
d = 7t
Set both distance formulas equal to each other:
5.5t + 9 = 7t
Subtract 5.5t from each side:
5.5t - 5.5t + 9 = 7t - 5.5t
1.5t = 9
Divide each side by 1.5:
1.5t/1.5 = 9/1.5
t = [B]6 hours[/B]
[U]Check our work with t = 6[/U]
Joe = 7(6) = 42
Jason = 5.5(6) + 9= 33 + 9 = 42
[MEDIA=youtube]qae3WCq9wzM[/MEDIA]
[/SIZE]
Jay purchased tickets for a concert. To place the order, a handling charge of $7 per ticket was chaJay purchased tickets for a concert. To place the order, a handling charge of $7 per ticket was charged. A sales tax of 4% was also charged on the ticket price and the handling charges. If the total charge for four tickets was $407.68, what was the ticket price? Round to the nearest dollar.
with a ticket price of t, we have the total cost written as:
1.04 * (7*4 + 4t)= 407.68
Divide each side by 1.04
1.04 * (7*4 + 4t)/1.04= 407.68/1.04
Cancel the 1.04 on the left side and we get:
7*4 + 4t = 392
28 + 4t = 392
To solve this equation for t, we [URL='https://www.mathcelebrity.com/1unk.php?num=28%2B4t%3D392&pl=Solve']type it in our math engine[/URL] and we get:
t = [B]91[/B]
Jenny went shoe shopping. Now she has 5 more pairs than her brother. Together they have 25 pairs. HoJenny went shoe shopping. Now she has 5 more pairs than her brother. Together they have 25 pairs. How many pairs does Jenny have and how many pairs does her brother have?
[U]Let j be the number of shoes Jenny has and b be the number of s hoes her brother has. Set up 2 equations:[/U]
(1) b + j = 25
(2) j = b + 5
[U]Substitute (2) into (1)[/U]
b + (b + 5) = 25
[U]Group the b terms[/U]
2b + 5 = 25
[U]Subtract 5 from each side[/U]
2b = 20
[U]Divide each side by b[/U]
[B]b = 10
[/B]
[U]Substitute b = 10 into (2)[/U]
j = 10 + 5
[B]j = 15[/B]
Jill made 122 muffins. She put them into 3 boxes and has two muffins left. How many are in each boxJill made 122 muffins. She put them into 3 boxes and has two muffins left. How many are in each box if they all contain the same amount of muffins?
Let m equal the number of muffins per box.
We're told that we have 3 boxes and 2 muffins left after filling up all 3 boxes.
3m + 2 = 122
To solve for m, we subtract 2 from each side:
3m + 2 - 2 = 122 - 2
Cancel the 2's on the left side and we get:
3m = 120
Divide each side by 3 to isolate m:
3m/3 = 120/3
Cancel the 3's on the left side and we get:
m = [B]40[/B]
Jim was thinking of a number. Jim adds 20 to it, then doubles it and gets an answer of 99.2. What waJim was thinking of a number. Jim adds 20 to it, then doubles it and gets an answer of 99.2. What was the original number?
Start with x.
Add 20 to it
x + 20
Double it
2(x + 20)
Set this equal to 99.2
2(x + 20) = 99.2
Divide each side by 2:
x + 20 = 49.6
Subtract 20 from each side:
x = [B]29.6[/B]
John has 30 marbles, 18 of which are red and 12 of which are blue. Jane has 20 marbles, all of themJohn has 30 marbles, 18 of which are red and 12 of which are blue. Jane has 20 marbles, all of them either red or blue. If the ratio of the red marbles to the blue marbles is the same for both John and Jane, then John has how many more blue marbles than Jane?
John's red ratio = 18/30
Using a [URL='https://www.mathcelebrity.com/gcflcm.php?num1=18&num2=30&num3=&pl=GCF+and+LCM']GCF for (18, 30)[/URL], we get 6.
Divide top and bottom of 18/30 by 6, we get 3/5
John's blue ratio is 12/30
Using a [URL='https://www.mathcelebrity.com/gcflcm.php?num1=12&num2=30&num3=&pl=GCF']GCF of (12, 30)[/URL], we get 6.
Divide top and bottom of 12/30 by 6, we get 2/5
Use these same ratios for Jane, we get:
Red: 3(20)/5 = 12
Blue: 20 - 12 = 8
Now the problem asks how many more blue marbles John has then Jane. We have 12 - 8 = [B]4[/B].
John need 250 hours of community service. He volunteers 2 days a week for 4 hours. How long will itJohn need 250 hours of community service. He volunteers 2 days a week for 4 hours. How long will it take John to reach 250 hours?
Each week, John serves 2 days * 4 hours per day = 8 hours.
We divide 250/8 to get [B]31.25 weeks[/B].
Jonathan earns a base salary of $1500 plus 10% of his sales each month. Raymond earns $1200 plus 15%Jonathan earns a base salary of $1500 plus 10% of his sales each month. Raymond earns $1200 plus 15% of his sales each month. How much will Jonathan and Raymond have to sell in order to earn the same amount each month?
[U]Step 1: Set up Jonathan's sales equation S(m) where m is the amount of sales made each month:[/U]
S(m) = Commission percentage * m + Base Salary
10% written as a decimal is 0.1. We want decimals to solve equations easier.
S(m) = 0.1m + 1500
[U]Step 2: Set up Raymond's sales equation S(m) where m is the amount of sales made each month:[/U]
S(m) = Commission percentage * m + Base Salary
15% written as a decimal is 0.15. We want decimals to solve equations easier.
S(m) = 0.15m + 1200
[U]The question asks what is m when both S(m)'s equal each other[/U]:
The phrase [I]earn the same amount [/I]means we set Jonathan's and Raymond's sales equations equal to each other
0.1m + 1500 = 0.15m + 1200
We want to isolate m terms on one side of the equation.
Subtract 1200 from each side:
0.1m + 1500 - 1200 = 0.15m + 1200 - 1200
Cancel the 1200's on the right side and we get:
0.1m - 300 = 0.15m
Next, we subtract 0.1m from each side of the equation to isolate m
0.1m - 0.1m + 300 = 0.15m - 0.1m
Cancel the 0.1m terms on the left side and we get:
300 = 0.05m
Flip the statement since it's an equal sign to get the variable on the left side:
0.05m = 300
To solve for m, we divide each side of the equation by 0.05:
0.05m/0.05 = 300/0.05
Cancelling the 0.05 on the left side, we get:
m = [B]6000[/B]
joseph buys 3 1/2 pounds of hamburger. how many quarter -pound can he make?joseph buys 3 1/2 pounds of hamburger. how many quarter -pound can he make?
A quarter pound is 1/4
[URL='https://www.mathcelebrity.com/fraction.php?frac1=3%261%2F2&frac2=3%2F8&pl=Simplify']3 & 1/2[/URL] = 7/2
[URL='https://www.mathcelebrity.com/fraction.php?frac1=7%2F2&frac2=1%2F4&pl=Divide']7/2 / 1/4[/URL] = [B]14 quarter pounders[/B]
Julia owes 18.20 for the month of November. Her plan costs 9.00 for the first 600 text messages andJulia owes 18.20 for the month of November. Her plan costs 9.00 for the first 600 text messages and .10 cents for additional texts. How many texts did she send out?
Let m be the number of messages. We have a cost function of:
C(m) = 9 + 0.1(m - 600)
We are given C(m) = 18.20
18.20 = 9 + 0.1(m - 600)
18.20 = 9 + 0.1m - 60
Combine like terms:
18.20 = 0.1m - 51
Add 51 to each side
0.1m = 69.20
Divide each side by 0.1
[B]m = 692[/B]
Juliana answered 42 times correctly on an 80-item quiz, while her sister Angela answered 21 items coJuliana answered 42 times correctly on an 80-item quiz, while her sister Angela answered 21 items correctly on a 40-item quiz. Do they have the same portion of correct answers?
Let's compare based on correct answers to questions:
Juliana = 42/80 = 0.525
Angela = 21/40 = 0.525
So yes, they do have the same portion of correct answers.
But there's another way to solve this:
[LIST=1]
[*]Divide Juliana's the top and bottom of Juliana's fraction by 2.
[*]We picked 2 as a GCF shown in our calculator.
[*]Type [URL='https://www.mathcelebrity.com/gcflcm.php?num1=42&num2=80&num3=&pl=GCF']GCF of 42 and 80[/URL].
[/LIST]
Divide top and bottom of Juliana's fraction by the GCF of 2
42/2 = 80/2 = 21/40
This ratio equals Angela's.
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]
Karin has 3 to spend in the arcade. The game she likes costs 50c per play. What are the possible numKarin has 3 to spend in the arcade. The game she likes costs 50c per play. What are the possible numbers of times that she can play?
[U]Let x = the number of games Karin can play with her money[/U]
0.5x = 3
[U]Divide each side by 0.5[/U]
[B]x = 6[/B]
Katie is twice as old as her sister Mara. The sum of their age is 24.Let k = Katie's age and m = Mara's age.
We have 2 equations:
(1) k = 2m
(2) k + m = 24
Substitute (1) into (2)
(2m) + m = 24
Combine like terms:
3m = 24
Divide each side of the equation by 3 to isolate m
m = 8
If m = 8, substituting into (1) or (2), we get k = 16.
[MEDIA=youtube]Cu7gSgNkQPg[/MEDIA]
Kerry asked a bank teller to cash 390 check using 20 bills and 50 bills. If the teller gave her a toKerry asked a bank teller to cash 390 check using 20 bills and 50 bills. If the teller gave her a total of 15 bills, how many of each type of bill did she receive?
Let t = number of 20 bills and f = number of 50 bills. We have two equations.
(1) 20t + 50f = 390
(2) t + f = 15
[U]Rearrange (2) into (3) for t, by subtracting f from each side:[/U]
(3) t = 15 - f
[U]Now substitute (3) into (1)[/U]
20(15 - f) + 50f = 390
300 - 20f + 50f = 390
[U]Combine f terms[/U]
300 + 30f = 390
[U]Subtract 300 from each side[/U]
30f = 90
[U]Divide each side by 30[/U]
[B]f = 3[/B]
[U]Substitute f = 3 into (3)[/U]
t = 15 - 3
[B]t = 12[/B]
Kevin is 4 times old as Daniel and is also 6 years older than DanielKevin is 4 times old as Daniel and is also 6 years older than Daniel.
Let k be Kevin's age and d be Daniel's age. We have 2 equations:
[LIST=1]
[*]k = 4d
[*]k = d + 6
[/LIST]
Plug (1) into (2):
4d = d + 6
Subtract d from each side:
4d - d = d - d + 6
Cancel the d terms on the right side and simplify:
3d = 6
Divide each side by 3:
3d/3 = 6/3
Cancel the 3 on the left side:
d = 2
Plug this back into equation (1):
k = 4(2)
k = 8
So Daniel is 2 years old and Kevin is 8 years old
Kimberly is taking three online classes during the summer. She spends 10 hours each week studying foKimberly is taking three online classes during the summer. She spends 10 hours each week studying for her marketing class, 12 hours studying for her statistics class, and 8 hours studying for her business law class. What percent of her study time does she spend for her statistics class?
The percentage equals hours spent on statistics divided by total hours spent studying for everything.
[U]Calculate total study hours:[/U]
Total Study Hours = Marketing Class Study Hours + Statistics Class Study Hours + Business Law Study Hours
Total Study Hours = 10 + 8 + 12
Total Study Hours = [B]30[/B]
[U]Calculate Statistics Study Hours Percentage:[/U]
Statistics Study Hours Percentage = Statistics Class Study Hours / Total Study Hours
Statistics Class Study Hours = 8/30
Using our [URL='https://www.mathcelebrity.com/perc.php?num=8&den=30&pcheck=1&num1=16&pct1=80&pct2=70&den1=80&idpct1=10&hltype=1&idpct2=90&pct=82&decimal=+65.236&astart=12&aend=20&wp1=20&wp2=30&pl=Calculate']fraction to decimal calculator[/URL], we get
Statistics Class Study Hours = [B]26.67%[/B]
larger of 2 numbers is 12 more than the smaller number. if the sum of the 2 numbers is 74 find the 2larger of 2 numbers is 12 more than the smaller number. if the sum of the 2 numbers is 74 find the 2 numbers
Declare Variables for each number:
[LIST]
[*]Let l be the larger number
[*]Let s be the smaller number
[/LIST]
We're given two equations:
[LIST=1]
[*]l = s + 12
[*]l + s = 74
[/LIST]
Equation (1) already has l solved for. Substitute equation (1) into equation (2) for l:
s + 12 + s = 74
Solve for [I]s[/I] in the equation s + 12 + s = 74
[SIZE=5][B]Step 1: Group the s terms on the left hand side:[/B][/SIZE]
(1 + 1)s = 2s
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2s + 12 = + 74
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 12 and 74. To do that, we subtract 12 from both sides
2s + 12 - 12 = 74 - 12
[SIZE=5][B]Step 4: Cancel 12 on the left side:[/B][/SIZE]
2s = 62
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2s/2 = 62/2
s = [B]31[/B]
To solve for l, we substitute in s = 31 into equation (1):
l = 31 + 12
l = [B]43[/B]
larger of 2 numbers is 4 more than the smaller. the sum of the 2 is 40. what is the larger number?larger of 2 numbers is 4 more than the smaller. the sum of the 2 is 40. what is the larger number?
Declare variables for the 2 numbers:
[LIST]
[*]Let l be the larger number
[*]Let s be the smaller number
[/LIST]
We're given two equations:
[LIST=1]
[*]l = s + 4
[*]l + s = 40
[/LIST]
To get this problem in terms of the larger number l, we rearrange equation (1) in terms of l.
Subtract 4 from each side in equation (1)
l - 4 = s + 4 - 4
Cancel the 4's and we get:
s = l - 4
Our given equations are now:
[LIST=1]
[*]s = l - 4
[*]l + s = 40
[/LIST]
Substitute equation (1) into equation (2) for s:
l + l - 4 = 40
Grouping like terms for l, we get:
2l - 4 = 40
Add 4 to each side:
2l - 4 + 4 = 40 + 4
Cancelling the 4's on the left side, we get
2l = 44
Divide each side of the equation by 2 to isolate l:
2l/2 = 44/2
Cancel the 2's on the left side and we get:
l = [B]22[/B]
Larry Mitchell invested part of his $31,000 advance at 6% annual simple interest and the rest at 7%Larry Mitchell invested part of his $31,000 advance at 6% annual simple interest and the rest at 7% annual simple interest. If the total yearly interest from both accounts was $2,090, find the amount invested at each rate.
Let x be the amount invested at 6%. Then 31000 - x is invested at 7%.
We have the following equation:
0.06x + (31000 - x)0.07 = 2090
Simplify:
0.06x + 2170 - 0.07x = 2090
Combine like Terms
-0.01x + 2170 = 2090
Subtract 2170 from each side
-0.01x = -80
Divide each side by -0.01
x = [B]8000 [/B]at 6%
Which means at 7%, we have:
31000 - 8000 = [B]23,000[/B]
Laura found a roll of fencing in her garage. She couldn't decide whether to fence in a square gardenLaura found a roll of fencing in her garage. She couldn't decide whether to fence in a square garden or a round garden with the fencing.
Laura did some calculations and found that a circular garden would give her 1380 more square feet than a square garden. How many feet of fencing were in the roll that Laura found? (Round to the nearest foot.)
Feet of fencing = n
Perimeter of square garden = n
Each side of square = n/4
Square garden's area = (n/4)^2 = n^2/16
Area of circle garden with circumference = n is:
Circumference = pi * d
n = pi * d
Divide body tissues by pi:
d = n/pi
Radius = n/2pi
Area = pi * n/2pi * n/2pi
Area = pin^2/4pi^2
Reduce by canceling pi:
n^2/4pi
n^2/4 * 3.14
n^2/12.56
The problem says that the difference between the square's area and the circle's area is equal to 1380 square feet.
Area of Circle - Area of Square = 1380
n^2/12.56 - n^2/16 = 1380
Common denominator = 200.96
(16n^2 - 12.56n^2)/200.96 = 1380
3.44n^2/200.96 = 1380
Cross multiply:
3.44n^2 = 277,324.8
n^2 = 80,617.7
n = 283.9
Nearest foot = [B]284[/B]
Laura spent half of her money on a necklace. She spent 14.60 of what was left having dinner with carLaura spent half of her money on a necklace. She spent 14.60 of what was left having dinner with carolyn. if she had 3.90 left, how much money did she start out with?
Let x equal Laura's starting money
1/2x = 14.60 + 3.90
1/2x = 18.5
Divide each side by 1/2
[B]x = $37[/B]
Laura weighs 45 pounds more than her pet dog. When they are on the scale together, they weigh 85 pouLaura weighs 45 pounds more than her pet dog. When they are on the scale together, they weigh 85 pounds. How much does Laura weigh?
Let Laura weigh l and her dog weigh d. WE have:
[LIST=1]
[*]l = d + 45
[*]d + l = 85
[/LIST]
Substitute equation (1) into Equation (2) for l:
d + d + 45 = 85
Solve for [I]d[/I] in the equation d + d + 45 = 85
[SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE]
(1 + 1)d = 2d
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2d + 45 = + 85
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 45 and 85. To do that, we subtract 45 from both sides
2d + 45 - 45 = 85 - 45
[SIZE=5][B]Step 4: Cancel 45 on the left side:[/B][/SIZE]
2d = 40
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2d/2 = 40/2
d = 20
From equation (1), we substitute d = 20:
l = d + 45
l = 20 + 45
l = [B]65 pounds
[URL='https://www.mathcelebrity.com/1unk.php?num=d%2Bd%2B45%3D85&pl=Solve']Source[/URL][/B]
Lauren wrote a total of 6 pages over 2 hours. How many hours will Lauren have to spend writing thisLauren wrote a total of 6 pages over 2 hours. How many hours will Lauren have to spend writing this week in order to have written a total of 9 pages? Solve using unit rates.
6 pages per 2 hours equals 6/2 = 3 pages per hour as a unit rate
Set up equation using h hours:
3h = 9
Divide each side by 3
[B]h = 3[/B]
Let P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. ForLet P(n) and S(n) denote the product and the sum, respectively, of the digits of the integer n. For example, P(23) = 6 and S(23) = 5. Suppose N is a two-digit number such that N = P(N) + S(N). What could N be? Is there more than one answer?
For example, for 23 P(23) = 6 and S(23) = 5, but 23 could not be the N that we want since 23 <> 5 + 6
Let t = tens digit and o = ones digit
P(n) = to
S(n) = t + o
P(n) + S(n) = to + t + o
N = 10t + o
Set them equal to each other N = P(N) + S(N)
10t + o = to + t + o
o's cancel, so we have
10t = to + t
Subtract t from each side, we have
9t = to
Divide each side by t
o = 9
So any two-digit number with 9 as the ones digit will work:
[B]{19,29,39,49,59,69,79,89,99}[/B]
Liz harold has a jar in her office that contains 47 coins. Some are pennies and the rest are dimes.Liz harold has a jar in her office that contains 47 coins. Some are pennies and the rest are dimes. If the total value of the coins is 2.18, how many of each denomination does she have?
[U]Set up two equations where p is the number of pennies and d is the number of dimes:[/U]
(1) d + p = 47
(2) 0.1d + 0.01p = 2.18
[U]Rearrange (1) into (3) by solving for d[/U]
(3) d = 47 - p
[U]Substitute (3) into (2)[/U]
0.1(47 - p) + 0.01p = 2.18
4.7 - 0.1p + 0.01p = 2.18
[U]Group p terms[/U]
4.7 - 0.09p = 2.18
[U]Add 0.09p to both sides[/U]
0.09p + 2.18 = 4.7
[U]Subtract 2.18 from both sides[/U]
0.09p = 2.52
[U]Divide each side by 0.09[/U]
[B]p = 28[/B]
[U]Now substitute that back into (3)[/U]
d =47 - 28
[B]d = 19[/B]
Luke and Dan's total debt is $72. If Luke's debt is three times Dan's debt, what is Dan's debt?Luke and Dan's total debt is $72. If Luke's debt is three times Dan's debt, what is Dan's debt?
Let Dan's debt be d.
Let Luke's debt be l.
We're given two equations:
[LIST=1]
[*]d + l = 72
[*]l = 3d
[/LIST]
Substitute equation (2) for l into equation (1):
d + 3d = 72
Solve for [I]d[/I] in the equation d + 3d = 72
[SIZE=5][B]Step 1: Group the d terms on the left hand side:[/B][/SIZE]
(1 + 3)d = 4d
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
4d = + 72
[SIZE=5][B]Step 3: Divide each side of the equation by 4[/B][/SIZE]
4d/4 = 72/4
d = [B]18[/B]
M is halved, then 7 is addedM is halved, then 7 is added
Take this algebraic expression in parts:
[LIST]
[*]M is halved. This means we divide M by 2: M/2
[*]Then 7 is added. We add 7 to M/2
[/LIST]
[B]M/2 + 7[/B]
M is the sum of a and its reciprocalM is the sum of a and its reciprocal
The reciprocal of a variable is 1 divided by the variable
1/a
The sum of a and its reciprocal means we add:
a + 1/a
The phrase [I]is[/I] means an equation, so we set M equal to the sum of a + 1/a:
[B]M = 1 + 1/a[/B]
m=u/k-r/k for km=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]
Marco orders a large pizza, with a diameter of 14 inches. It is cut into 8 congruent pieces. what isMarco orders a large pizza, with a diameter of 14 inches. It is cut into 8 congruent pieces. what is the area of one piece?
A pizza is a circle. If the diameter of the pizza is 14 inches, the radius is 14/2 = 7 inches.
Area of a circle is pi(r^2). With r = 7, we have:
A =7^2(pi)
A = 49pi
Area of a slice of pizza is the area of the full pizza divided by 8
A(Slice) = [B]49pi/8[/B]
Margaret earns 15 an hour at her job. Each week she earns at least 450 and no more than 600. How manMargaret earns 15 an hour at her job. Each week she earns at least 450 and no more than 600. How many hours does Margaret work each week?
Let h be the hours worked
We know that hourly rate * h equals total earnings.
The phrases at least and no more than signify inequalities, so we have:
450 <= 15h <= 600
Divide each entry by 15:
[B]30 <= h <= 40[/B]
This means Margaret works at least 30 hours a week and no more than 40
Maria bought seven boxes. A week later half of all her boxes were destroyed in a fire. There are nowMaria 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]
Marty is 3 years younger than 6 times his friend Warrens age. The sum of their ages is greater thanMarty is 3 years younger than 6 times his friend Warrens age. The sum of their ages is greater than 11. What is the youngest age Warren can be?
Let m be Marty's age and w be Warren's age. We have two equations:
(1) m = 6w - 3
(2) m + w > 11
Plug (1) into (2)
6w - 3 + w > 11
Combine w terms
7w - 3 > 11
Add 3 to each side
7w > 14
Divide each side by 7
w > 2 which means [B]w = 3[/B] as the youngest age.
Math Problem Solving (Help Please)Volume of rectangular prism is:
V = lwh
Plugging in the numbers you gave:
195 = (6)(5)h
195 = 30h
Divide each side by 30
h = 6.5
6.5 feet is 6 feet, 6 inches. This is 2 inches more than your actor, so [B]yes[/B], he will fit in the box standing up.
Matilda needs at least $112 to buy an new dress. She has already saved $40. She earns $9 an hour babMatilda needs at least $112 to buy an new dress. She has already saved $40. She earns $9 an hour babysitting. Write and solve and inequality to find how many hours she will need to babysit to buy the dress.
Subtract remaining amount needed after savings:
112 - 40 = 72
Let h be her hourly wages for babysitting. We have the equation:
[B]9h = 72[/B]
Divide each side by 9
[B]h = 8[/B]
Matthew's cat weighs 10 pounds more than his pet hamster. His dog weighs the same as his cat. If theMatthew's cat weighs 10 pounds more than his pet hamster. His dog weighs the same as his cat. If the weight of all three pets is 35 pounds, ow much does his hamster weigh?
Setup weights and relations:
[LIST]
[*]Hamster weight: w
[*]Cat weight: w + 10
[*]Dog weight:w + 10
[/LIST]
Add all the weights up:
w + w + 10 + w + 10 = 35
Solve for [I]w[/I] in the equation w + w + 10 + w + 10 = 35
[SIZE=5][B]Step 1: Group the w terms on the left hand side:[/B][/SIZE]
(1 + 1 + 1)w = 3w
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
10 + 10 = 20
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
3w + 20 = + 35
[SIZE=5][B]Step 4: Group constants:[/B][/SIZE]
We need to group our constants 20 and 35. To do that, we subtract 20 from both sides
3w + 20 - 20 = 35 - 20
[SIZE=5][B]Step 5: Cancel 20 on the left side:[/B][/SIZE]
3w = 15
[SIZE=5][B]Step 6: Divide each side of the equation by 3[/B][/SIZE]
3w/3 = 15/3
w =[B] 5[/B]
[B]
[URL='https://www.mathcelebrity.com/1unk.php?num=w%2Bw%2B10%2Bw%2B10%3D35&pl=Solve']Source[/URL][/B]
Max and Bob went to the store. Max bought 2 burgers and 2 drinks for $5.00 bob bought 3 burgers andMax and Bob went to the store. Max bought 2 burgers and 2 drinks for $5.00. Bob bought 3 burgers and 1 drink for $5.50. How much is each burger and drink?
[U]Set up the givens where b is the cost of a burger and d is the cost of a drink:[/U]
Max: 2b + 2d = 5
Bob: 3b + d = 5.50
[U]Rearrange Bob's equation by subtracting 3b from each side[/U]
(3) d = 5.50 - 3b
[U]Now substitute that d equation back into Max's Equation[/U]
2b + 2(5.50 - 3b) = 5
2b + 11 - 6b = 5
[U]Combine b terms:[/U]
-4b + 11 = 5
[U]Subtract 11 from each side[/U]
-4b = -6
[U]Divide each side by -4[/U]
b = 3/2
[B]b = $1.50[/B]
[U]Now plug that back into equation (3):[/U]
d = 5.50 - 3(1.50)
d = 5.50 - 4.50
[B]d = $1.00[/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]
Michael is riding his bicycle. He rides 25.6 kilometers in 4 hours. What is his speed?We need the speed of KM per hour.
25.6 km / 4 hours
[U]Divide top and bottom by 4 to get km per hour[/U]
[B]6.4km per hour[/B]
Michelle and Julie sold 65 cupcakes. If Julie sold 9 more cupcakes than Michelle, how many cupcakesMichelle and Julie sold 65 cupcakes. If Julie sold 9 more cupcakes than Michelle, how many cupcakes did each of them sell?
Let m = Michelle's cupcakes and j = Julie's cupcakes.
We have two equations:
m + j = 65
j = m + 9
Substituting, we get:
m + (m + 9) = 65
Combine like terms, we get:
2m + 9 = 65
Subtract 9 from each side:
2m = 56
Divide each side by 2 to isolate m
m = 28
If m = 28, then j = 28 + 9 = 37
So (m, j) = (28, 37)
MonomialsFree Monomials Calculator - This calculator will raise a monomial to a power,multiply monomials, or divide monomials.
Mr. Vukovic is making pasta. He makes 3 ½ cups of pasta. How many ¾ cup servings can Mr. Vukovic sMr. Vukovic is making pasta. He makes 3 ½ cups of pasta. How many ¾ cup servings can Mr. Vukovic serve
3 & 1/2 = 7/2
[URL='https://www.mathcelebrity.com/fraction.php?frac1=3%261%2F2&frac2=3%2F4&pl=Divide']7/2 /3/4[/URL] = 14/3 = 4 & 2/3
Mr. Winkle downloaded 34 more songs than Mrs. Winkle downloaded. Together they downloaded 220 songMr. Winkle downloaded 34 more songs than Mrs. Winkle downloaded. Together they downloaded 220 songs. How many songs did each download?
Let x = Mr. Winkle downloads and y = Mrs. Winkle downloads.
We then have x = y + 34 and x + y = 220.
Substitute equation 1 into equation 2, we have:
(y + 34) + y = 220
2y + 34 = 220
Subtract 34 from each side:
2y = 186
Divide each side by 2:
y = 93 (Mrs. Winkle)
x = 93 + 34
x = 127 (Mr. Winkle)
Multiply Even Numbers by 5 No Calculator ShortcutTake the number being multiplied by 5.
Divide it in half
Add a zero
14 * 5
Divide 14/2 = 7
Add a 0 --> 70
[MEDIA=youtube]lOJmx0Ygpz8[/MEDIA]
mx=ac/np for nmx=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 + .07n = $90.95n + .07n = $90.95
Group like terms:
1.07n = $90.95
Solve for [I]n[/I] in the equation 1.07n = 90.95
[SIZE=5][B]Step 1: Divide each side of the equation by 1.07[/B][/SIZE]
1.07n/1.07 = 90.95/1.07
n = [B]85
[URL='https://www.mathcelebrity.com/1unk.php?num=1.07n%3D90.95&pl=Solve']Source[/URL][/B]
n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6
Solve for [I]n[/I] in the equation n + 2n + 3n + 4n = 2 + 3 + 4 + 5 + 6
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 2 + 3 + 4)n = 10n
[SIZE=5][B]Step 2: Group the constant terms on the right hand side:[/B][/SIZE]
2 + 3 + 4 + 5 + 6 = 20
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
10n = + 20
[SIZE=5][B]Step 4: Divide each side of the equation by 10[/B][/SIZE]
10n/10 = 20/10
n = [B]2[/B]
n + 9n - 8 - 5 = 2n + 3n + 9n - 8 - 5 = 2n + 3
Solve for [I]n[/I] in the equation n + 9n - 8 - 5 = 2n + 3
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 9)n = 10n
[SIZE=5][B]Step 2: Group the constant terms on the left hand side:[/B][/SIZE]
-8 - 5 = -13
[SIZE=5][B]Step 3: Form modified equation[/B][/SIZE]
10n - 13 = 2n + 3
[SIZE=5][B]Step 4: Group variables:[/B][/SIZE]
We need to group our variables 10n and 2n. To do that, we subtract 2n from both sides
10n - 13 - 2n = 2n + 3 - 2n
[SIZE=5][B]Step 5: Cancel 2n on the right side:[/B][/SIZE]
8n - 13 = 3
[SIZE=5][B]Step 6: Group constants:[/B][/SIZE]
We need to group our constants -13 and 3. To do that, we add 13 to both sides
8n - 13 + 13 = 3 + 13
[SIZE=5][B]Step 7: Cancel 13 on the left side:[/B][/SIZE]
8n = 16
[SIZE=5][B]Step 8: Divide each side of the equation by 8[/B][/SIZE]
8n/8 = 16/8
n = [B]2[/B]
n + 9n - 90 = 0n + 9n - 90 = 0
Solve for [I]n[/I] in the equation n + 9n - 90 = 0
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(1 + 9)n = 10n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
10n - 90 =
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -90 and 0. To do that, we add 90 to both sides
10n - 90 + 90 = 0 + 90
[SIZE=5][B]Step 4: Cancel 90 on the left side:[/B][/SIZE]
10n = 90
[SIZE=5][B]Step 5: Divide each side of the equation by 10[/B][/SIZE]
10n/10 = 90/10
n = [B]9[/B]
n + n/2 + n/4 + n/8 + n/16 = 19,375n + 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 = 3n - 1/2n = 3n - 1/2
Solve for [I]n[/I] in the equation n = 3n - 1/2
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables n and 3n. To do that, we subtract 3n from both sides
n - 3n = 3n - 0.5 - 3n
[SIZE=5][B]Step 2: Cancel 3n on the right side:[/B][/SIZE]
-2n = -0.5
[SIZE=5][B]Step 3: Divide each side of the equation by -2[/B][/SIZE]
-2n/-2 = -0.5/-2
n = [B]0.25 or 1/4[/B]
n = 5m^2d for dn = 5m^2d for d
Divide each side by 5m^2 to isolate d:
n/5m^2 = 5m^2d/5m^2
Cancel the 5m^2 on the right side and we get:
d = [B]n/5m^2[/B]
n = b + d^2a for an = b + d^2a for a
Let's start by isolating the one term with the a variable.
Subtract b from each side:
n - b = b - b + d^2a
Cancel the b terms on the right side and we get:
n - b = d^2a
With the a term isolated, let's divide each side of the equation by d^2:
(n - b)/d^2 = d^2a/d^2
Cancel the d^2 on the right side, and we'll display this with the variable to solve on the left side:
a = [B](n - b)/d^2
[MEDIA=youtube]BCEVsZmoKoQ[/MEDIA][/B]
n=i*x+y for in=i*x+y for i
This is a literal equation.
Subtract y from each side of the equation:
n - y = i*x + y - y
The y's cancel on the right side, so we have:
n - y = ix
Divide each side of the equation by x, to isolate i
(n - y)/x = ix/x
The x's cancel on the right side, so we have:
i = [B](n - y)/x[/B]
Nancy is 10 years less than 3 times her daughters age. If Nancy is 41 years old, how old is her daugNancy is 10 years less than 3 times her daughters age. If Nancy is 41 years old, how old is her daughter?
Declare variables for each age:
[LIST]
[*]Let Nancy's age be n
[*]Let her daughter's age be d
[/LIST]
We're given two equations:
[LIST=1]
[*]n = 3d - 10
[*]n = 41
[/LIST]
We set 3d - 10 = 41 and solve for d:
Solve for [I]d[/I] in the equation 3d - 10 = 41
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants -10 and 41. To do that, we add 10 to both sides
3d - 10 + 10 = 41 + 10
[SIZE=5][B]Step 2: Cancel 10 on the left side:[/B][/SIZE]
3d = 51
[SIZE=5][B]Step 3: Divide each side of the equation by 3[/B][/SIZE]
3d/3 = 51/3
d = [B]17[/B]
Nate jars 2 liters of jam everyday. How many days did Nate spend making jam if he jarred 8 liters ofNate 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].
Nervous Speaker #1 says the word "um" 8 times each minute. Nervous Speaker #2 says the word "um" 10[SIZE=5]Nervous Speaker #1 says the word "um" 8 times each minute. Nervous Speaker #2 says the word "um" 10 times each minute. Working together, how many minutes will it take them to say the word "um" 270 times?
[/SIZE]
[SIZE=4]In one minute, Nervous speaker 1 and 2 say "um" 8 + 10 = 18 times per minute.
We want to know how many minutes it takes for both of them to say 270 "um"s.
We divide 270/18 to get [B]15 minutes[/B][/SIZE]
Notebooks cost $1.39 each. What are the possible numbers of notebooks that can be purchased with $10Notebooks cost $1.39 each. What are the possible numbers of notebooks that can be purchased with $10?
Let n be the number of notebooks you can purchase. We have the following inequality:
1.39n <= 10
Divide each side by 1.39
n <= 7.194
We want whole notebooks, we cannot buy fractions of notebooks, so we have:
n <= 7
The question asks for the possible numbers of notebooks we can buy. This implies we buy at least 1, but our inequality says not more than 7. So our number set is:
[B]N = {1, 2, 3, 4, 5, 6, 7}[/B]
N^2=5qd for dN^2=5qd for d
Divide each side by 5q to isolate d:
N^2/5q = 5qd/5q
Cancel 5q on the right side and we get:
d = [B]N^2/5q[/B]
Of the 800 students of a school, 600 have traveled. What percentage of the students went on a trip?Of the 800 students of a school, 600 have traveled. What percentage of the students went on a trip?
Our percentage is found as 600/800.
Simplifying by dividing top and bottom by 100, we have:
6/8
Divide top and bottom by 2, we get:
3/4 or [B]75%
[/B]
You can also type in the [URL='http://www.mathcelebrity.com/perc.php?num=600&den=800&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']search engine[/URL]: 600/800 as percent.
Oliver invests $1,000 at a fixed rate of 7% compounded monthly, when will his account reach $10,000?Oliver invests $1,000 at a fixed rate of 7% compounded monthly, when will his account reach $10,000?
7% monthly is:
0.07/12 = .00583
So we have:
1000(1 + .00583)^m = 10000
divide each side by 1000;
(1.00583)^m = 10
Take the natural log of both sides;
LN (1.00583)^m = LN(10)
Use the identity for natural logs and exponents:
m * LN (1.00583) = 2.30258509299
0.00252458479m = 2.30258509299
m = 912.064867899
Round up to [B]913 months[/B]
Olivia bought 20 notebooks. Her cost for all of the notebooks was 2.00$. If each notebook cost the sOlivia bought 20 notebooks. Her cost for all of the notebooks was 2.00$. If each notebook cost the same amount then how much did she pay for one
20 notebooks / 2
Divide top and bottom by 20:
1 notebook = 2/20
1 notebook = 1/10
1 notebook = [B]10 cents[/B]
On a trip, a family drove 270 kilometers in 3 hours. how many kilometers were traveled in one hour.On a trip, a family drove 270 kilometers in 3 hours. how many kilometers were traveled in one hour. Express this as a rate per hour.
270 kilometers per 3 hours
270/3
Divide top and bottom by 3 to get km/hr
[B]90 kilometers per hour[/B]
On your first draw, what is the probability of drawing a red card, without looking, from a shuffledOn your first draw, what is the probability of drawing a red card, without looking, from a shuffled deck containing 6 red cards, 6 blue cards, and 8 black cards?
P(Red) = Total Red / Total Cards
P(Red) = 6 red/(6 red + 6 blue + 8 black)
P(Red) = 6/20
This fraction can be simplified.
The [URL='https://www.mathcelebrity.com/gcflcm.php?num1=6&num2=20&num3=&pl=GCF+and+LCM']greatest common factor of 6 and 20[/URL] is 2.
So we divide top and bottom of our probability by 2:
P(Red) = 6/2 / 20 / 2
P(Red) = [B]3/10[/B]
One fifth of the square of a numberOne fifth of the square of a number
We have an algebraic expression. Let's break this into parts.
[LIST=1]
[*]The phrase [I]a number[/I] means an arbitrary variable, let's call it x
[*]The square of a number means we raise it to the power of 2. So we have x^2
[*]One-fifth means we have a fraction, where we divide our x^2 in Step 2 by 5. So we get our final answer below:
[/LIST]
[B]x^2/5[/B]
One-fourth the sum of m and pOne-fourth the sum of m and p
Take this algebraic expression in parts:
[LIST]
[*]The sum of m and p means we add p to m: m + p
[*]1/4 of the sum mean we divide m + p by 4
[/LIST]
[B](m + p)/4[/B]
One-half a number is fiftyThe phrase [I]a number[/I] means an arbitrary variable. We can pick any letter a-z except for i and e.
Let's choose x.
One-half a number means we divide x by2:
x/2
The word [I]is[/I] means equal to. We set x/2 equal to 50 for our algebraic expression
[B]x/2 = 50
[/B]
If the problem asks us to solve for x, we cross multiply:
x = 2 * 50
x = [B]100[/B]
p = i^2r for rp = i^2r for r
Divide each side of the equation by i^2 to isolate r:
p/i^2 = i^2/ri^2
Cancel the i^2 on the right side and we get:
r = [B]p/i^2[/B]
p is halved and 4 is addedp is halved and 4 is added
[U]p is halved means we divide p by 2:[/U]
p/2
[U]4 is added:[/U]
[B]p/2 + 4[/B]
P varies directly as q and the square of r and inversely as sP 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 = 6p 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 fp/q=f/q-f for f
To solve this literal equation for f, let's factor out f on the right side:
p/q=f(1/q-1)
Divide each side by (1/q - 1)
p/(q(1/q - 1)) = f(1/q-1)/(1/q - 1)
Cancelling the (1/q - 1) on the right side, we get:
f = p/(1/q - 1)
Rewriting this since (1/q -1) = (1 - q)/q since q/q = 1 we have:
f = [B]pq/(1 - q)[/B]
P/v=nr/t for rP/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= 4/q what kind of variation is this?p= 4/q what kind of variation is this?
[B]Inverse Variation [/B]since we divide by q
P=15+5d/11 for dSubtract 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 cP=ab/c, for c
Cross multiply:
cP = ab
Divide each side by P
[B]c = (ab)/P[/B]
Partial QuotientFree Partial Quotient Calculator - Divides 2 numbers using the Partial Quotient
Perimeter of a rectangle is 372 yards. If the length is 99 yards, what is the width?Perimeter of a rectangle is 372 yards. If the length is 99 yards, what is the width?
The perimeter P of a rectangle with length l and width w is:
2l + 2w = P
We're given P = 372 and l = 99, so we have:
2(99) + 2w = 372
2w + 198 = 372
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 198 and 372. To do that, we subtract 198 from both sides
2w + 198 - 198 = 372 - 198
[SIZE=5][B]Step 2: Cancel 198 on the left side:[/B][/SIZE]
2w = 174
[SIZE=5][B]Step 3: Divide each side of the equation by 2[/B][/SIZE]
2w/2 = 174/2
w = [B]87[/B]
Pet supplies makes a profit of $5.50 per bag, if the store wants to make a profit of no less than $5Pet supplies makes a profit of $5.50 per bag, if the store wants to make a profit of no less than $5225, how many bags does it need to sell?
5.5ob >= $5,225
Divide each side of the inequality by $5.50
b >=9.5 bags, so round up to a whole number of 10 bags.
Phyllis's mother has 6 pounds of candy to divide evenly among her 8 children. This is an average ofPhyllis's mother has 6 pounds of candy to divide evenly among her 8 children. This is an average of how many pounds per child?
6 pounds divide among 8 children can be represented as a fraction. We want to simplify this. So we use our [URL='https://www.mathcelebrity.com/fraction.php?frac1=6%2F8&frac2=3%2F8&pl=Simplify']fraction simplify calculator[/URL], and we get:
3 pounds per 4 children, or 0.75 pounds per child.
please answer my second word problemDistance = Rate x Time
6.4 meters = 4 meters/minute * t
Divide each side by 4
[B]t = 1.6 minutes[/B]
please answer this word problemTime 1, distance apart is 105 + 85 = 190
So every hour, the distance between them is 190 * t where t is the number of hours. Set up our distance function:
D(t) = 190t
We want D(t) = 494
190t = 494
Divide each side by 190
[B]t = 2.6 hours[/B]
please solve the fifth word problemFind what was used:
Used Money = Prepaid original cost - Remaining Credit
Used Money = 20 - 17.47
Used Money = 2.53
Using (m) as the number of minutes, we have the following cost equation:
C(m) = 0.11m
C(m) = 2.53, so we have:
0.11m = 2.53
Divide each side by 0.11
[B]m = 23[/B]
please solve the third word problemA Web music store offers two versions of a popular song. The size of the standard version is
2.7
megabytes (MB). The size of the high-quality version is
4.7
MB. Yesterday, the high-quality version was downloaded three times as often as the standard version. The total size downloaded for the two versions was
4200
MB. How many downloads of the standard version were there?
Let s be the standard version downloads and h be the high quality downloads. We have two equations:
[LIST=1]
[*]h = 3s
[*]2.7s + 4.7h = 4200
[/LIST]
Substitute (1) into (2)
2.7s + 4.7(3s) = 4200
2.7s + 14.1s = 4200
Combine like terms:
16.8s = 4200
Divide each side by 16.8
[B]s = 250[/B]
Pound of strawberries for $4.00. What is the price, in dollars, per ounce of strawberries?Pound of strawberries for $4.00. What is the price, in dollars, per ounce of strawberries?
1 pound equals 16 ounces. So the pounds per ounce equals:
$4.00/16 ounces
Divide top and bottom by 16, we get:
[B]$0.25 per ounce[/B]
pr=xf/y for rpr=xf/y for r
So for this literal equation, we divide each side of the equation by p to isolate r.
pr/p = xf/yp
Cancel the p's on the left side and we get:
r = [B]xf/yp
[MEDIA=youtube]6ekuN4H3mM4[/MEDIA][/B]
Pressure LawFree Pressure Law Calculator - This will solve for any of the 4 items in the Pressure Law equation, also known as Gay-Lussacs Law assuming constant volume
P1 ÷ T1 = P2 ÷ T2
Prove the following statement for non-zero integers a, b, c, If a divides b and b divides c, then aProve the following statement for non-zero integers a, b, c,
If a divides b and b divides c, then a divides c.
If an integer a divides an integer b, then we have:
b = ax for some non-zero integer x
If an integer b divides an integer c, then we have:
c = by for some non-zero integer y
Since b = ax, we substitute this into c = by for b:
c = axy
We can write this as:
c = a(xy)
[LIST]
[*]Since x and y are integers, then xy is also an integer.
[*]Therefore, c is the product of some integer multiplied by a
[*]This means a divides c
[/LIST]
[MEDIA=youtube]VUIUFAFFVU4[/MEDIA]
Prove there is no integer that is both even and oddLet us take an integer x which is both even [I]and[/I] odd.
[LIST]
[*]As an even integer, we write x in the form 2m for some integer m
[*]As an odd integer, we write x in the form 2n + 1 for some integer n
[/LIST]
Since both the even and odd integers are the same number, we set them equal to each other
2m = 2n + 1
Subtract 2n from each side:
2m - 2n = 1
Factor out a 2 on the left side:
2(m - n) = 1
By definition of divisibility, this means that 2 divides 1.
But we know that the only two numbers which divide 1 are 1 and -1.
Therefore, our original assumption that x was both even and odd must be false.
[MEDIA=youtube]SMM9ubEVcLE[/MEDIA]
pv/t = ab/c for cpv/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]
quotient of the sum of 17 and x and yquotient of the sum of 17 and x and y
The sum of 17 and x means we add x to 17:
17 + x
quotient of the sum of 17 and x and y means we divide 17 + x by y
[B](17 + x)/y[/B]
r varies directly with s and inversely with the square root of tr 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 wr=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]
Rachel buys some scarves that cost $10 each and 2 purses that cost $16 each. The cost of Rachel's toRachel buys some scarves that cost $10 each and 2 purses that cost $16 each. The cost of Rachel's total purchase is $62. What equation can be used to find n, the number of scarves that Rebecca buys
Scarves Cost + Purses Cost = Total Cost
[U]Calculate Scarves Cost[/U]
Scarves cost = Cost per scarf * number of scarves
Scarves cost = 10n
[U]Calculate Purses Cost[/U]
Purses cost = Cost per purse * number of purses
Purses cost = 16 * 2
Purses cost = 32
Total Cost = 62. Plug in our numbers and values to the Total Cost equation :
10n + 32 = 62
Solve for [I]n[/I] in the equation 10n + 32 = 62
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 32 and 62. To do that, we subtract 32 from both sides
10n + 32 - 32 = 62 - 32
[SIZE=5][B]Step 2: Cancel 32 on the left side:[/B][/SIZE]
10n = 30
[SIZE=5][B]Step 3: Divide each side of the equation by 10[/B][/SIZE]
10n/10 = 30/10
n = [B]3[/B]
raise 2 to the 10th power and divide k by the resultraise 2 to the 10th power and divide k by the result
Raise 2 to the 10th power:
2^10
Divide k by the result:
k / 2^10
raise 3 to the 4th power, subtract w from the result, then divide v by what you haveraise 3 to the 4th power, subtract w from the result, then divide v by what you have
Raise 3 to the 4th power:
3^4
Simplified, this is
81
Subtract w from the result. We subtract w from 81:
81 - w
Then divide v by what you have. We divide v by (81 -w)
[B]v/(81 - w)[/B]
raise 3 to the 8th power, then divide the result by traise 3 to the 8th power, then divide the result by t
3 to the 8th power
3^8
Divide the result by t
3^8/t
Now, if they want you to evaluate 3 to the 8th, you have:
6,561/t
Raise 9 to the 3rd power, subtract d from the result, then divide what you have by cRaise 9 to the 3rd power, subtract d from the result, then divide what you have by c.
This is an algebraic expression, let's take in parts (or chunks).
Raise 9 to the 3rd power. This means we take 9, and raise it to an exponent of 3
9^3
Subtract d from the result, means we subtract d from 9^3
9^3 - d
Now we divide 9^3 - d by c
[B](9^3 - d) / c[/B]
Raise c to the 7th power, divide the result by 4, then triple what you haveRaise 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 3rd power, then find the quotient of the result and graise f to the 3rd power, then find the quotient of the result and g
Take this algebraic expression in two parts:
[LIST=1]
[*]Raise f to the 3rd power means we take f, and write it with an exponent of 3: f^3
[*]Find the quotient of the result and g. We take f^3, and divide it by g
[/LIST]
[B]f^3/g[/B]
Raise f to the 8th power, divide the result by 5, then multiply 10Raise 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 F to the second power then divide G by the resultRaise F to the second power then divide G by the result
F to the second power:
F^2
Divide G by the result:
[B]G/F^2[/B]
Raise p to the 9th power, multiply the result by q, then divide what you have by rRaise 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
[MEDIA=youtube]I5PShTfas4Y[/MEDIA][/B]
raise q to the 5th power add the result to p then divide what you have by rraise q to the 5th power add the result to p then divide what you have by r
Take this algebraic expression in parts:
[LIST]
[*]Raise q to the 5th power: q^5
[*]Add the result to p: p + q^5
[*]Divide what you have by r. This means we take our result above and divide it by r:
[/LIST]
[B](p + q^5)/r[/B]
raise v to the 9th power, then dividethe result by uV to the 9th power means we use an exponent:
v^9
Divide that result by u
[B]v^9/u[/B]
raise x to the 10th power, then divide b by the resultraise x to the 10th power, then divide b by the result
x to the 10th power
x^10
Divide b by the result:
[B]b/x^10[/B]
Rates of ReturnFree Rates of Return Calculator - Given a set of stock prices and dividends if applicable, this calculates the periodic rate of return and the logarithmic rate of return
ratio of the squares of t and uratio of the squares of t and u
Ratio is also known as quotient in algebraic expression problems.
The square of t means we raise t to the power of 2:
t^2
The square of u means we raise u to the power of 2:
u^2
ratio of the squares of t and u means we divide t^2 by u^2:
[B]t^2/u^2[/B]
Rearrange the following equation to make x the subject, and select the correct rearrangement from thRearrange 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]
Relative ErrorFree Relative Error Calculator - Relative error is the absolute error divided by quantity
Robert has 45 dollars. He buys 6 tshirts and has 7 dollars left over. How much did each tshirt cost?Let x be the price of one t-shirt. Set up an equation:
6 times the number of t-shirts plus 7 dollars left over get him to a total of 45
6x = 45 - 7
6x = 38
Divide each side by 6
[B]x = 6.33[/B]
Ronald scored 4 goals in his first soccer game. He then scored the same amount of goals in his nextRonald scored 4 goals in his first soccer game. He then scored the same amount of goals in his next 3 games. If Ronald has 10 goals total, how many did he score in each game?
For the remaining 3 games, he scored 10 - 4 = 6 goals.
6 goals divided by 3 games = [B]2 goals in each game[/B].
Roster form of: A = {3x-2/x are integers between 0 and 8}Roster form of: A = {3x-2/x are integers between 0 and 8}
x = 0 = Undefined since we divide by 0
x = 1: 3*1 + 2/1 = 5
x = 2: 3*2 + 2/2 = 7
x = 3: 3*3 + 2/3 = 9.66666666666667
x = 4: 3*4 + 2/4 = 12.5
x = 5: 3*5 + 2/5 = 15.4
x = 6: 3*6 + 2/6 = 18.3333333333333
x = 7: 3*7 + 2/7 = 21.2857142857143
x = 8: 3*8 + 2/8 = 24.25
[B]A = {(0, undefined), (1, 5), (2, 7), (3, 9.6667), (4, 12.5), (5, 15.4), (6, 18.3333), (7, 21.2857142857143), (8, 24.25)}[/B]
s = tu^2 for us = tu^2 for u
Divide each side by t
u^2 = s/t
Take the square root of each side
[LIST]
[*]u = sqrt(s/t)
[*]u = -sqrt(s/t)
[/LIST]
We have two answers due to negative number squared is positive
s=u^2t for ts=u^2t for t
Divide each side by u^2 to isolate t:
u^2t/u^2 = s/u^2
Cancel the u^2 on the left side, we get:
t = [B]s/u^2[/B]
Salma purchased a prepaid phone card for 30. Long distance calls cost 9 cents a minute using this caSalma purchased a prepaid phone card for 30. Long distance calls cost 9 cents a minute using this card. Salma used her card only once to make a long distance call. If the remaining credit on her card is 28.38, how many minutes did her call last?
[U]Set up the equation where m is the number of minutes used:[/U]
0.09m = 30 - 28.38
0.09m = 1.62
[U]Divide each side by 0.09[/U]
[B]m = 18[/B]
Sarah has 12 apples she divided them in 4 groups. How many are in each group?Sarah has 12 apples she divided them in 4 groups. How many are in each group?
12 apples per group divided by 4 groups is written as:
12/4
So we have [B]3 groups[/B].
Scientists are studying a cell that divides in half every 15 minutes. How many cells will there by aScientists are studying a cell that divides in half every 15 minutes. How many cells will there by after 2.5 hours?
Divide 2.5 hours into 15 minute blocks.
2.5 hours = 2(60) + 0.5(60) minutes
2.5 hours = 120 + 30 minutes
2.5 hours = 150 minutes
Now determine the amount of 15 minute blocks
150 minutes/15 minutes = 10 blocks or divisions
[LIST]
[*]We start with 1 cell at time 0, and double it every 15 minutes
[*]We have A(0) = 1, we want A(10).
[*]Our accumulation function is A(t) = A(0) * 2^t
[/LIST]
A(10) = 1 * 2^10
A(10) = [B]1024[/B]
Scott and Dylan both leave the park at the same time, but in opposite directions. If Scott travels 1Scott and Dylan both leave the park at the same time, but in opposite directions. If Scott travels 12 mph and Dylan travels 19 mph, how long until they are 186 miles apart?
Hour 1, they are 19 + 12 = 31 miles apart. So each hour, they get 31 miles more apart.
When they are [URL='https://www.mathcelebrity.com/fraction.php?frac1=186%2F31&frac2=3%2F8&pl=Simplify']186 miles apart[/URL], we divide this by 31 miles apart per hour:
186/31 = [B]6 hours[/B]
Set C is the set of two-digit even numbers less than 56 that are divisible by 5[U]Two digit Numbers less than 56:[/U]
{10, 11, 12, ..., 55}
[U]Two Digit Even Numbers of that Set:[/U]
{10, 12, 14, ..., 54}
[U]Two Digit Even numbers Divisible by 5[/U]
[B]C = {10, 20, 30, 40, 50}[/B]
[I]Note: Even means you can divide it by 2 with no remainder. Divisible by 5 means the number ends in 5 or 0. Since it is even numbers only, end in 0.
[MEDIA=youtube]aQKLVxIB-p4[/MEDIA][/I]
Shalini gave 0.4 of her plums to her brother and 20% to her sister. She kept 16 for herself how manyShalini gave 0.4 of her plums to her brother and 20% to her sister. She kept 16 for herself how many plums did she have at first?
Let p be the number of plums Shalini started with. We have:
[LIST]
[*]0.4 given to her brother
[*]20% which is 0.2 given away to her sister
[*]What this means is she kept 1 - (0.4 + 0.2) = 1 - 0.6 = 0.4 for herself
[/LIST]
0.4p = 16
Divide each side by 0.4
[B]p = 40[/B]
Shalini gave 0.4 of her plums to her brother and 20% to her sister. She kept 16 for herself. How manShalini gave 0.4 of her plums to her brother and 20% to her sister. She kept 16 for herself. How many plums did she have first?
Let's convert everything to decimals. 20% = 0.2
So Shalini gave 0.4 + 0.2 = 0.6 of the plums away. Which means she has 1 = 0.6 = 0.4 or 40% left over.
40% represents 16 plums
So our equation is 0.4p = 16 where p is the number of plums to start with
Divide each side by 0.4
[B]p = 40[/B]
She ordered 6 large pizzas. Luckily, she had a coupon for 3 off each pizza. If the bill came to 38.9She ordered 6 large pizzas. Luckily, she had a coupon for 3 off each pizza. If the bill came to 38.94, what was the price for a large pizza?
[U]Determine additional amount the pizzas would have cost without the coupon[/U]
6 pizzas * 3 per pizza = 18
[U]Add 18 to our discount price of 38.94[/U]
Full price for 6 large pizzas = 38.94 + 18
Full price for 6 large pizzas = 56.94
Let x = full price per pizza before the discount. Set up our equation:
6x = 56.94
Divide each side by 6
[B]x = $9.49[/B]
Sheila wants build a rectangular play space for her dog. She has 100 feet of fencing and she wants iSheila wants build a rectangular play space for her dog. She has 100 feet of fencing and she wants it to be 5 times as long as it is wide. What dimensions should the play area be?
Sheila wants:
[LIST=1]
[*]l =5w
[*]2l + 2w = 100 <-- Perimeter
[/LIST]
Substitute (1) into (2)
2(5w) + 2w = 100
10w + 2w = 100
12w = 100
Divide each side by 12
[B]w = 8.3333[/B]
Which means l = 5(8.3333) -->[B] l = 41.6667[/B]
Since pounds are smaller than tons, i need to ______ the number of pounds by _____Since pounds are smaller than tons, i need to ______ the number of pounds by _____
[B]Divide[/B] the number of pounds by [B]2,000[/B]
Solve mgh=1/2mv^2+1/2(2/5)mr^2(v^2/r^2) for vSolve 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]
Sonia visited a park in California that had redwood trees. When Sonia asked how tall a certain largeSonia visited a park in California that had redwood trees. When Sonia asked how tall a certain large redwood tree was, the ranger said that he wouldn't tell its height, but would give Sonia a clue. How tall is the redwood tree Sonia asked about?
Sonia said the tree is 64 times my height. The tree is also 112 feet taller than the tree next to it. The two trees plus my height total 597.5 feet.
[LIST]
[*]Rangers's height = n
[*]Tree height = 64n
[*]Smaller tree height = 64n - 112
[*]Total height = 64n - 112 + 64n = 597.5
[/LIST]
Solve for [I]n[/I] in the equation 64n - 112 + 64n = 597.5
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(64 + 64)n = 128n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
128n - 112 = + 597.5
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -112 and 597.5. To do that, we add 112 to both sides
128n - 112 + 112 = 597.5 + 112
[SIZE=5][B]Step 4: Cancel 112 on the left side:[/B][/SIZE]
128n = 709.5
[SIZE=5][B]Step 5: Divide each side of the equation by 128[/B][/SIZE]
128n/128 = 709.5/128
n = 5.54296875
Tree height = 64 * ranger height
Tree height = 64 * 5.54296875
Tree height = [B]354.75 feet[/B]
Sophie and Claire are having a foot race. Claire is given a 100-foot head-start. If Sophie is runnSophie and Claire are having a foot race. Claire is given a 100-foot head-start. If Sophie is running at 5 feet per second and Claire is running at 3 feet per second.
i. After how many seconds will Sophie catch Claire?
ii. If the race is 500 feet, who wins?
i.
Sophie's distance formula is given as D = 5s
Claire's distance formula is given as D = 3s + 100
Set them equal to each other
5s = 3s + 100
Subtract 3s from both sides:
2s = 100
Divide each side by 2
[B]s = 50[/B]
ii. [B]Sophie since after 50 seconds, she takes the lead and never gives it back.[/B]
Square root of 9136 divided by 43Square root of 9136 divided by 43
First, [URL='https://www.mathcelebrity.com/powersq.php?num=sqrt%289136%29&pl=Calculate']take the square root of 9136 in our calculator[/URL]:
4 * sqrt(571)
Now divide this by 43:
[B]4 * sqrt(571) / 43[/B]
standard deviation of 545 dollars. Find the sample size needed to have a confidence level of 95% andStandard 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]
Stanley bought a ruler and a yardstick for $1.25. If the yardstick cost 45 cents more than the rulerStanley bought a ruler and a yardstick for $1.25. If the yardstick cost 45 cents more than the ruler, what was the cost of the yardstick?
Let r be the cost of the ruler
Let y be the cost of the yardstick
We're given 2 equations:
[LIST=1]
[*]r + y = 1.25
[*]y = r + 0.45
[/LIST]
Substitute equation (2) into equation (1) for y
r + r + 0.45 = 1.25
Solve for [I]r[/I] in the equation r + r + 0.45 = 1.25
[SIZE=5][B]Step 1: Group the r terms on the left hand side:[/B][/SIZE]
(1 + 1)r = 2r
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2r + 0.45 = + 1.25
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 0.45 and 1.25. To do that, we subtract 0.45 from both sides
2r + 0.45 - 0.45 = 1.25 - 0.45
[SIZE=5][B]Step 4: Cancel 0.45 on the left side:[/B][/SIZE]
2r = 0.8
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2r/2 = 0.8/2
r = 0.4
Substitute r = 0.4 into equation (2) above:
y = r + 0.45
y = 0.4 + 0.45
r = [B]0.85
[URL='https://www.mathcelebrity.com/1unk.php?num=r%2Br%2B0.45%3D1.25&pl=Solve']Source[/URL][/B]
Start with q. Multiply by p. Add 3. Divide AStart 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 t , add 6, divide by 2, then subtract 5.Start with t , add 6, divide by 2, then subtract 5.
Start with t:
t
Add 6:
t + 6
Divide by 2:
(t + 6)/2 [I]Add parentheses because we're dividing the [U]quantity[/U] by 2
[/I]
Then subtract 5:
[B](t + 6)/2 - 5[/B]
Stephanie spent $6.95 on these 12 chocolates. What was the cost of each chocolate? Give your answerStephanie spent $6.95 on these 12 chocolates. What was the cost of each chocolate? Give your answer to the nearest 5 cents
$6.95/12 chocolates
Divide top and bottom by 12 to get the cost per one chocolate:
$6.95/12 = 0.58 cents per chocolate
The problem asks us to round to the nearest [I]5 cents[/I].
5 * 11 = 55
5 * 12 = 60
Since 58 cents is closer to 60, we have [B]60 cents[/B] as our answer
subtract half of a number from 10subtract half of a number from 10
The phrase [I]a number[/I] means an arbitrary variable, let's call it x:
x
half of a number means we divide x by 2:
x/2
subtract half of a number from 10
[B]10 - x/2[/B]
Subtract the quotient of m and 7 from 4Subtract the quotient of m and 7 from 4
The quotient of m and 7 means we add divide m by 7
m/7
Subtract this quotient from 4
[B]4 - m/7[/B]
sum of 3 consecutive odd integers equals 1 hundred 17sum of 3 consecutive odd integers equals 1 hundred 17
The sum of 3 consecutive odd numbers equals 117. What are the 3 odd numbers?
1) Set up an equation where our [I]odd numbers[/I] are n, n + 2, n + 4
2) We increment by 2 for each number since we have [I]odd numbers[/I].
3) We set this sum of consecutive [I]odd numbers[/I] equal to 117
n + (n + 2) + (n + 4) = 117
[SIZE=5][B]Simplify this equation by grouping variables and constants together:[/B][/SIZE]
(n + n + n) + 2 + 4 = 117
3n + 6 = 117
[SIZE=5][B]Subtract 6 from each side to isolate 3n:[/B][/SIZE]
3n + 6 - 6 = 117 - 6
[SIZE=5][B]Cancel the 6 on the left side and we get:[/B][/SIZE]
3n + [S]6[/S] - [S]6[/S] = 117 - 6
3n = 111
[SIZE=5][B]Divide each side of the equation by 3 to isolate n:[/B][/SIZE]
3n/3 = 111/3
[SIZE=5][B]Cancel the 3 on the left side:[/B][/SIZE]
[S]3[/S]n/[S]3 [/S]= 111/3
n = 37
Call this n1, so we find our other 2 numbers
n2 = n1 + 2
n2 = 37 + 2
n2 = 39
n3 = n2 + 2
n3 = 39 + 2
n3 = 41
[SIZE=5][B]List out the 3 consecutive odd numbers[/B][/SIZE]
([B]37, 39, 41[/B])
37 ? 1st number, or the Smallest, Minimum, Least Value
39 ? 2nd number
41 ? 3rd or the Largest, Maximum, Highest Value
sum of the cube of x and half of ysum of the cube of x and half of y
The cube of x means we raise x to the 3rd power:
x^3
half of y means we divide y by 2:
y/2
sum of the cube of x and half of y means we add y/2 to x^3
[B]x^3 + y/2[/B]
sum of x plus y divided by 2sum of x plus y divided by 2
sum of x plus y:
x + y
sum of x plus y divided by 2
[B](x + y)/2[/B]
Suppose Briley has 10 coins in quarters and dimes and has a total of 1.45. How many of each coin doeSuppose 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]
Suppose x is a natural number. When you divide x by 7 you get a quotient of q and a remainder of 6.Suppose x is a natural number. When you divide x by 7 you get a quotient of q and a remainder of 6. When you divide x by 11 you get the same quotient but a remainder of 2. Find x.
[U]Use the quotient remainder theorem[/U]
A = B * Q + R where 0 ? R < B where R is the remainder when you divide A by B
Plugging in our numbers for Equation 1 we have:
[LIST]
[*]A = x
[*]B = 7
[*]Q = q
[*]R = 6
[*]x = 7 * q + 6
[/LIST]
Plugging in our numbers for Equation 2 we have:
[LIST]
[*]A = x
[*]B = 11
[*]Q = q
[*]R = 2
[*]x = 11 * q + 2
[/LIST]
Set both x values equal to each other:
7q + 6 = 11q + 2
Using our [URL='http://www.mathcelebrity.com/1unk.php?num=7q%2B6%3D11q%2B2&pl=Solve']equation calculator[/URL], we get:
q = 1
Plug q = 1 into the first quotient remainder theorem equation, and we get:
x = 7(1) + 6
x = 7 + 6
[B]x = 13[/B]
Plug q = 1 into the second quotient remainder theorem equation, and we get:
x = 11(1) + 2
x = 11 + 2
[B]x = 13[/B]
T = mg - mf for fT = 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 wt 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]
tammy earns $18000 salary with 4% comission on sales. How much should she sell to earn $55,000 totaltammy earns $18000 salary with 4% comission on sales. How much should she sell to earn $55,000 total
We have a commission equation below:
Sales * Commission percent = Salary
We're given 4% commission percent and 55,000 salary. With 4% as 0.04, we have:
Sales * 0.04 = 55,000
Divide each side of the equation by 0.04, and we get:
Sales = [B]1,375,000[/B]
Taylor is playing a game using a die and a spinner. The spinner is divided into 4 equal parts with cTaylor is playing a game using a die and a spinner. The spinner is divided into 4 equal parts with colors green, red, yellow, and purple. Taylor rolls the die and spins the spinner. What is the probability the die shows a 2 and the spinner lands on purple?
Probability of rolling a 2 on the die is 1/6
Probability of getting a purple on the spinner is 1/4
Since each event is independent, our joint probability is:
P(2 on the die and Purple on the spinner) = P(2 on the die) x P(Purple on the Spinner)
P(2 on the die and Purple on the spinner) = 1/6 x 1/4
P(2 on the die and Purple on the spinner) = [B]1/24[/B]
The average height of a family of 6 is 6 feet. After the demise of the mother, the average height reThe 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 171 and x?The average of 171 and x?
The phrase [I]average[/I] means add up all the items in the number set, divided by the count of items in the number set.
Our number set in this case is {171, x} which has 2 elements. Therefore, our average is:
[B](171 + x)/2[/B]
The average of a number and double the number is 25.5Let 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 ninetythe 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 average of two numbers x and ythe average of two numbers x and y
Average is the sum divided by the count:
Sum:
x + y
We have 2 numbers, so we divide (x + y) by 2
[B](x + y)/2[/B]
The base of a triangle with a height of 7 units is represented by the formula b=2/7A. The base of thThe 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 club uses the function S(t) = -4,500t + 54,000 to determine the salvage S(t) of a fertilizer bleThe club uses the function S(t) = -4,500t + 54,000 to determine the salvage S(t) of a fertilizer blender t years after its purchase. How long will it take the blender to depreciate completely?
Complete depreciation means the salvage value is 0.
So S(t) = 0. We need to find t to make S(t) = 0
-4,500t + 54,000 = 0
Subtract 54,000 from each side
-4,500t = -54,000
Divide each side by -4,500
[B]t = 12[/B]
The cost of renting a rototiller is $19.50 for the first hour and $7.95 for each additional hour. HoThe cost of renting a rototiller is $19.50 for the first hour and $7.95 for each additional hour. How long can a person have the rototiller if the cost must be less than $95?
Setup the inequality:
$19.50 + $7.95x < $95
Subtract 19.50 from both sides:
7.95x < 75.50
Divide each side of the inequality by 7.95 to isolate x
x < 9.5
The next lowest integer is 9. So we take 9 + the first hour of renting to get [B]10 total hours[/B].
Check our work:
$7.95 * 9.5 + $19.50
$71.55 + $19.50 = $91.05
The cube of the difference of 5 times the square of y and 7 divided by the square of 2 times yThe cube of the difference of 5 times the square of y and 7 divided by the square of 2 times y
Take this in algebraic expression in parts:
[U]Term 1[/U]
[LIST]
[*]The square of y means we raise y to the 2nd power: y^2
[*]5 times the square of y: 5y^2
[/LIST]
[U]Term 2[/U]
[LIST]
[*]2 times y: 2y
[*]The square of 2 times y: (2y)^2 = 4y^2
[*]7 divide by the square of 2 times y: 7/4y^2
[/LIST]
[U]The difference of these terms is written as Term 1 minus Term 2:[/U]
[LIST]
[*]5y^2/4y^2
[/LIST]
[U]The cube of the difference means we raise the difference to the power of 3:[/U]
[B](5y^2/4y^2)^3[/B]
The difference between the quotient of x and y, and twice zThe 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 the squares of two consecutive numbers is 141. Find the numbersThe difference between the squares of two consecutive numbers is 141. Find the numbers
Take two consecutive numbers:
n- 1 and n
Given a difference (d) between the squares of two consecutive numbers, the shortcut for this is:
2n - 1 = d
Proof of this:
n^2- (n - 1)^2 = d
n^2 - (n^2 - 2n + 1) = d
n^2 - n^2 + 2n - 1 = d
2n - 1 = d
Given d = 141, we have
2n - 1 = 141
Add 1 to each side:
2n = 142
Divide each side by 2:
2n/2 = 142/2
n = [B]71[/B]
Therefore, n - 1 = [B]70
Our two consecutive numbers are (70, 71)[/B]
Check your work
70^2 = 4900
71^2 = 5041
Difference = 5041 - 4900
Difference = 141
[MEDIA=youtube]vZJtZyYWIFQ[/MEDIA]
The difference of 2 positive numbers is 54. The quotient obtained on dividing the 1 by the other isThe 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 4 and the quotient of 18 and a numberthe difference of 4 and the quotient of 18 and a number
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
The quotient of 18 and a number means we divide 18 by the variable x.
18/x
The difference of 4 and the quotient above means we subtract 18/x from 4:
[B]4 - 18/x[/B]
The difference of five and five y is the same as eight and two yThe difference of five and five y
5 - 5y
eight and two y
8 + 2y
The phrase [I]is the same as[/I] means equal to. Set 5 - 5y equal to 8 + 2y for our final algebraic expression
[B]5 - 5y = 8 + 2y[/B]
[B][/B]
If the problem asks you to solve for y:
Add 5y to each side:
5 = 8 + 7y
Subtract 8 from each side:
7y = -3
Divide each side by 7:
[B]y= -3/7[/B]
The doctor told Anna that tomorrow she will have to check her blood pressure every 15 minutes from 1The doctor told Anna that tomorrow she will have to check her blood pressure every 15 minutes from 10:00 AM to 4:00 PM. How many times does she have to take her blood pressure?
10:00 A.M. to 4:00 P.M. is 6 hours.
Each hour is 60 minutes
60 minutes divided by 15 minutes equals 4 blood pressure checks per hour.
4 blood pressure checks per hour * 6 hours = [B]24 blood pressure checks[/B]
The enrollment at a gymnastics academy increased 120% from 2016 to 2017. The enrollment in 2017 wasThe enrollment at a gymnastics academy increased 120% from 2016 to 2017. The enrollment in 2017 was 210. What is 2016's enrollment?
We take 2017's enrollment of 210 and divide by 1.2 since 120% is 1.2 as a multiplier:
2016 enrollment = 2017 enrollment / 1.2
2016 enrollment =210/1.2
2016 enrollment = [B]175[/B]
The first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered tThe first significant digit in any number must be 1, 2, 3, 4, 5, 6, 7, 8, or 9. It was discovered that first digits do not occur with equal frequency. Probabilities of occurrence to the first digit in a number are shown in the accompanying table. The probability distribution is now known as Benford's Law. For example, the following distribution represents the first digits in 231 allegedly fraudulent checks written to a bogus company by an employee attempting to embezzle funds from his employer.
Digit, Probability
1, 0.301
2, 0.176
3, 0.125
4, 0.097
5, 0.079
6, 0.067
7, 0.058
8, 0.051
9, 0.046
[B][U]Fradulent Checks[/U][/B]
Digit, Frequency
1, 36
2, 32
3, 45
4, 20
5, 24
6, 36
7, 15
8, 16
9, 7
Complete parts (a) and (b).
(a) Using the level of significance α = 0.05, test whether the first digits in the allegedly fraudulent checks obey Benford's Law. Do the first digits obey the Benford's Law?
Yes or No
Based on the results of part (a), could one think that the employe is guilty of embezzlement?
Yes or No
Show frequency percentages
Digit Fraud Probability Benford Probability
1 0.156 0.301
2 0.139 0.176
3 0.195 0.125
4 0.087 0.097
5 0.104 0.079
6 0.156 0.067
7 0.065 0.058
8 0.069 0.051
9 0.03 0.046
Take the difference between the 2 values, divide it by the Benford's Value. Sum up the squares to get the Test Stat of 2.725281277
Critical Value Excel: =CHIINV(0.95,8) = 2.733
Since test stat is less than critical value, we cannot reject, so [B]YES[/B], it does obey Benford's Law and [B]NO[/B], there is not enough evidence to suggest the employee is guilty of embezzlement.
The fraction has a value of 3/5. The sum of the numerator and the denominator was 40. What was the fThe 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) = x^3 - 48x has a local minimum at x = and a local maximum at x = ?The function f(x) = x^3 - 48x has a local minimum at x = and a local maximum at x = ?
f'(x) = 3x^2 - 48
Set this equal to 0:
3x^2 - 48 = 0
Add 48 to each side:
3x^2 = 48
Divide each side by 3:
x^2 = 16
Therefore, x = -4, 4
Test f(4)
f(4) = 4^3 - 48(4)
f(4) = 64 - 192
f(4) = [B]-128 <-- Local minimum[/B]
Test f(-4)
f(-4) = -4^3 - 48(-4)
f(-4) = -64 + 192
f(-4) = [B]128 <-- Local maximum[/B]
The function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the baThe function P(x) = -30x^2 + 360x + 785 models the profit, P(x), earned by a theatre owner on the basis of a ticket price, x. Both the profit and the ticket price are in dollars. What is the maximum profit, and how much should the tickets cost?
Take the [URL='http://www.mathcelebrity.com/dfii.php?term1=-30x%5E2+%2B+360x+%2B+785&fpt=0&ptarget1=0&ptarget2=0&itarget=0%2C1&starget=0%2C1&nsimp=8&pl=1st+Derivative']derivative of the profit function[/URL]:
P'(x) = -60x + 360
We find the maximum when we set the profit derivative equal to 0
-60x + 360 = 0
Subtract 360 from both sides:
-60x = -360
Divide each side by -60
[B]x = 6 <-- This is the ticket price to maximize profit[/B]
Substitute x = 6 into the profit equation:
P(6) = -30(6)^2 + 360(6) + 785
P(6) = -1080 + 2160 + 785
[B]P(6) = 1865[/B]
The height of an object t seconds after it is dropped from a height of 300 meters is s(t)=-4.9t^2 +3The height of an object t seconds after it is dropped from a height of 300 meters is s(t)=-4.9t^2 +300. Find the average velocity of the object during the first 3 seconds? (b) Use the Mean value Theorem to verify that at some time during the first 3 seconds of the fall the instantaneous velocity equals the average velocity. Find that time.
Average Velocity:
[ f(3) - f(0) ] / ( 3 - 0 )
Calculate f(3):
f(3) = -4.9(3^2) + 300
f(3) = -4.9(9) + 300
f(3) = -44.1 + 300
f(3) = 255.9
Calculate f(0):
f(0) = -4.9(0^2) + 300
f(0) = 0 + 300
f(0) = 300
So we have average velocity:
Average velocity = (255.9 - 300)/(3 - 0)
Average velocity = -44.1/3
Average velocity = -[B]14.7
[/B]
Velocity is the first derivative of position
s(t)=-4.9t^2 +300
s'(t) = -9.8t
So we set velocity equal to average velocity:
-9.8t = -14.7
Divide each side by -9.8 to solve for t, we get [B]t = 1.5[/B]
The IQ scores are normally distributed with a mean of 100 and a standard deviation of 15. a) What iThe IQ scores are normally distributed with a mean of 100 and a standard deviation of 15.
a) What is the probability that a randomly person has an IQ between 85 and 115?
b) Find the 90th percentile of the IQ distribution
c) If a random sample of 100 people is selected, what is the standard deviation of the sample mean?
a) [B]68%[/B] from the [URL='http://www.mathcelebrity.com/probnormdist.php?xone=50&mean=100&stdev=15&n=1&pl=Empirical+Rule']empirical rule calculator[/URL]
b) P(z) = 0.90. so z = 1.28152 using Excel NORMSINV(0.9)
(X - 100)/10 = 1.21852
X = [B]113[/B] rounded up
c) Sample standard deviation is the population standard deviation divided by the square root of the sample size
15/sqrt(100) = 15/10 =[B] 1.5[/B]
The length of a rectangle is three times its width.If the perimeter is 80 feet, what are the dimensiThe length of a rectangle is three times its width.If the perimeter is 80 feet, what are the dimensions?
We're given 2 equations:
[LIST=1]
[*]l = 3w
[*]P = 80 = 2l + 2w = 80
[/LIST]
Substitute (1) into (2) for l:
2(3w) + 2w = 80
6w + 2w = 80
8w = 80
Divide each side by 8:
8w/8 = 80/8
w = [B]10
[/B]
Substitute w = 10 into (1)
l = 3(10)
l = [B]30[/B]
The length of a rectangular building is 6 feet less than 3 times the width. The perimeter is 120 feeThe 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 train car is 50.6 feet. This is 5.8 feet less than 6 times the width. What is the wiThe length of a train car is 50.6 feet. This is 5.8 feet less than 6 times the width. What is the width?
5.8 feet less than 6 times the width is an algebraic expression:
6w - 5.8
We set this equal to the length of 50.6
6w - 5.8 = 50.6
Add 5.8 to each side:
6w - 5.8 + 5.8 = 50.6 + 5.8
Cancel the 5.8 on the left side:
6w = 56.4
Divide each side by 6:
6w/6 = 56.4/6
[URL='http://www.mathcelebrity.com/1unk.php?num=6w-5.8%3D50.6&pl=Solve']Typing this problem into the search engine[/URL], we get [B]w = 9.4[/B].
[MEDIA=youtube]gfM-d_Ej728[/MEDIA]
The length of the flag is 2 cm less than 7 times the width. The perimeter is 60cm. Find the length aThe length of the flag is 2 cm less than 7 times the width. The perimeter is 60cm. Find the length and width.
A flag is a rectangle shape. So we have the following equations
Since P = 2l + 2w, we have 2l + 2w = 60
l = 7w - 2
Substitute Equation 1 into Equation 2:
2(7w -2) + 2w = 60
14w - 4 + 2w = 60
16w - 4 = 60
Add 4 to each side
16w = 64
Divide each side by 16 to isolate w
w = 4
Which means l = 7(4) - 2 = 28 - 2 = 26
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 Palafoxes make $3,840 a month. They spend $1,600 for rent. What fraction of their income goes toThe Palafoxes make $3,840 a month. They spend $1,600 for rent. What fraction of their income goes to rent?
Rent Payment Fraction = Rent Payment / Total Income
Rent Payment Fraction = 1600 / 3840
Our greatest common factor of 1600 and 3840 is 320.
So if we divide 1600 and 3840 by 320, we get:
Rent Payment Fraction = [B]5/12
[MEDIA=youtube]DsXk6AKT18M[/MEDIA][/B]
The parent company contributed $5 million for the 50 million votes cast. What did they pay each voteWe want to see votes per dollar. So we divide 50 million votes by $5 million dollars.
50,000,0000
------------
5,000,000
We have 10 votes for every dollar spent.
Or, ten cents per vote.
The perimeter of a rectangle is 400 meters. The length is 15 meters less than 4 times the width. FinThe perimeter of a rectangle is 400 meters. The length is 15 meters less than 4 times the width. Find the length and the width of the rectangle.
l = 4w - 15
Perimeter = 2l + 2w
Substitute, we get:
400 = 2(4w - 15) + 2w
400 = 8w - 30 + 2w
10w - 30 = 400
Add 30 to each side
10w = 370
Divide each side by 10 to isolate w
w = 37
Plug that back into our original equation to find l
l = 4(37) - 15
l = 148 - 15
l = 133
So we have (l, w) = (37, 133)
The phone company charges Rachel 12 cents per minute for her long distance calls. A discount companyThe phone company charges Rachel 12 cents per minute for her long distance calls. A discount company called Rachel and offered her long distance service for 1/2 cent per minute, but will charge a $46 monthly fee. How many minutes per month must Rachel talk on the phone to make the discount a better deal?
Minutes Rachel talks = m
Current plan cost = 0.12m
New plan cost = 0.005m + 46
Set new plan equal to current plan:
0.005m + 46 = 0.12m
Solve for [I]m[/I] in the equation 0.005m + 46 = 0.12m
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 0.005m and 0.12m. To do that, we subtract 0.12m from both sides
0.005m + 46 - 0.12m = 0.12m - 0.12m
[SIZE=5][B]Step 2: Cancel 0.12m on the right side:[/B][/SIZE]
-0.115m + 46 = 0
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 46 and 0. To do that, we subtract 46 from both sides
-0.115m + 46 - 46 = 0 - 46
[SIZE=5][B]Step 4: Cancel 46 on the left side:[/B][/SIZE]
-0.115m = -46
[SIZE=5][B]Step 5: Divide each side of the equation by -0.115[/B][/SIZE]
-0.115m/-0.115 = -46/-0.115
m = [B]400
She must talk over 400 minutes for the new plan to be a better deal
[URL='https://www.mathcelebrity.com/1unk.php?num=0.005m%2B46%3D0.12m&pl=Solve']Source[/URL][/B]
The price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she paThe price of a cheap backpack is $15 less than an expensive backpack. When Emily bought both, she paid $75. What is the cost of the cheap backpack?
backpack cost = b
Cheap backpack = b - 15
The total of both items equals 75:
b + b - 15 = 75
Solve for [I]b[/I] in the equation b + b - 15 = 75
[SIZE=5][B]Step 1: Group the b terms on the left hand side:[/B][/SIZE]
(1 + 1)b = 2b
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
2b - 15 = + 75
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants -15 and 75. To do that, we add 15 to both sides
2b - 15 + 15 = 75 + 15
[SIZE=5][B]Step 4: Cancel 15 on the left side:[/B][/SIZE]
2b = 90
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2b/2 = 90/2
b = 45
Cheap backpack = 45 - 15 = [B]30
[URL='https://www.mathcelebrity.com/1unk.php?num=b%2Bb-15%3D75&pl=Solve']Source[/URL][/B]
the product of a number and 15 is not less than 15the product of a number and 15 is not less than 15
The phrase [I]a number[/I] means an arbitrary variable. Let's call it x.
x
the product of a number and 15 means we multiply x by 15
15x
The phrase [I]not less than[/I] means greater than or equal to. We set 15x greater than prequel to 15
[B]15x >= 15 <-- This is our algebraic expression
[/B]
[U]If the problem asks you to solve for x:[/U]
Divide each side by 15:
15x/15 >= 15/15
[B]x >= 1[/B]
The quantity x minus y divided by 4The quantity x minus y divided by 4
The quantity x minus y
x - y
The quantity x minus y divided by 4
[B](x - y)/4[/B]
The quotient of 2 and the sum of a number and 1The quotient of 2 and the sum of a number and 1.
The phrase [I]a number[/I] represents an arbitrary variable, let's call it x.
The sum of a number and 1 is written as:
x + 1
The word [I]quotient[/I] means a fraction. So we divide 2 by x + 1
2
--------
( x + 1)
the quotient of 3 and u is equal to 52 divided by uthe quotient of 3 and u is equal to 52 divided by u
Take this algebraic expression in 3 parts:
[LIST=1]
[*]The quotient of 3 and u means we divide 3 by u: 3/u
[*]52 divided by u means we divide 52 by u: 52/u
[*]The phrase [I]is equal to[/I] means an equation, so we set (1) equal to (2)
[/LIST]
[B]3/u = 52/u[/B]
the quotient of 4 more than a number and 7 is 10the quotient of 4 more than a number and 7 is 10
Take this algebraic expression in pieces:
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
4 more than a number means we add 4 to x:
x + 4
The quotient of 4 more than a number and 7 means we divide x + 4 by 7
(x + 4)/7
The word [I]is[/I] means an equation, so we set (x + 4)/7 equal to 10
[B](x + 4)/7 = 10[/B]
the quotient of a number and twice another numberthe 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 m and the sum of n and p.the quotient of m and the sum of n and p.
The sum of n and p means we add p to n:
n + p
The quotient means a fraction, so we divide m by (n + p)
[B]m/(n + p)[/B]
the quotient of m squared and a squaredthe quotient of m squared and a squared
[U]m squared means we raise m to the power of 2:[/U]
m^2
[U]a squared means we raise a to the power of 2:[/U]
a^2
[U]The [I]quotient[/I] means we divide m^2 by a^2:[/U]
[B]m^2/a^2[/B]
the ratio of 50 and a number added to the quotient of a number and 10the ratio of 50 and a number added to the quotient of a number and 10
Take this algebraic expression in parts.
The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
x
The ratio of 50 and x means we divide by 50 by x
50/x
The quotient of a number and 10 means we have a fraction:
x/10
The phrase [I]added to[/I] means we add 50/x to x/10
[B]50/x + x/10[/B]
The ratio of the measures of the 3 angles of a triangle is 1:2:3. What is the measure of the largestThe ratio of the measures of the 3 angles of a triangle is 1:2:3. What is the measure of the largest angle in degrees?
Let the smallest angle be x.
Then we have 3 angles based on the ratio: x, 2x, 3x
We know the sum of the angles of a triangle equals 180. So we have:
x + 2x + 3x = 180
6x = 180
Divide each side by 6:
6x/6 = 180/6
x = 30
The largest angle is 3(30) = [B]90
[MEDIA=youtube]l8Lc6YtK9dg[/MEDIA][/B]
The sales price of a new compact disc player is $210 at a local discount store. At the store where tThe sales price of a new compact disc player is $210 at a local discount store. At the store where the sale is going on, each new cd is on sale for $11. If Kyle purchases a player and some cds for $243 how many cds did he purchase?
If Kyle bought the player, he has 243 - 210 = 33 left over.
Each cd is 11, so set up an equation to see how many CDs he bought:
11x = 33
Divide each side by 11
[B]x = 3[/B]
The sales tax on a computer was $33.60. If the sales tax rate is 7%, how much did the computer costThe sales tax on a computer was $33.60. If the sales tax rate is 7%, how much did the computer cost without tax?
Let the cost of the computer be c. We have:
0.07c = 33.60
Solve for [I]c[/I] in the equation 0.07c = 33.60
[SIZE=5][B]Step 1: Divide each side of the equation by 0.07[/B][/SIZE]
0.07c/0.07 = 33.60/0.07
c = $[B]480[/B]
[URL='https://www.mathcelebrity.com/1unk.php?num=0.07c%3D33.60&pl=Solve']Source[/URL]
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 beThe 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 science club charges 4.50 per car at their car wash. Write and solve and inequality to find howThe science club charges 4.50 per car at their car wash. Write and solve and inequality to find how many cars they have to wash to earn at least 300
Let x be the number of cars they wash. Set up our inequality. Note, at least 300 means 300 or greater, so we use greater than or equal to.
[U]Inequality:[/U]
[B]4.50x >= 300
[/B]
[U]So solve for x, divide each side by 4[/U]
[B]x >= 66.67[/B]
the set of natural numbers less than 7 that are divisible by 3the set of natural numbers less than 7 that are divisible by 3
Natural Numbers less than 7
{1, 2, 3, 4, 5, 6}
Only 2 of them are divisible by 3. Divisible means the number is divided evenly, with no remainder:
[B]{3, 6}[/B]
The sum of 2 numbers is 70. The difference of these numbers is 24. Write and solve a system of equatThe sum of 2 numbers is 70. The difference of these numbers is 24. Write and solve a system of equations to determine the numbers.
Let the two numbers be x and y. We have the following equations:
[LIST=1]
[*]x + y = 70
[*]x - y = 24
[/LIST]
Add (1) to (2):
2x = 94
Divide each side by 2
[B]x = 47[/B]
Plug this into (1)
47 + y = 70
Subtract 47 from each side, we have:
[B]y = 23[/B]
the sum of 3 numbers divided by its productthe 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 3h and k divided by 2The sum of 3h and k divided by 2
The sum of 3h and k
3h + k
Divided by 2:
[B](3h + k)/2[/B]
the sum of 4 and x split into 5 equal partsthe sum of 4 and x split into 5 equal parts
The sum of x and 4 means we add 4 to x:
x + 4
Whenever you see the phrase [I]split into[/I], think of divide or divided by:
[B](x + 4)/5[/B]
The sum of 80 and 40 is divided by 5The sum of 80 and 40 is divided by 5
The sum of 80 and 40:
80 + 40
Divided by 5:
[B](80 + 40)/5[/B]
The sum of a and b divided by their productThe 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 dthe 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 5 all divided by 2 is 17The sum of a number and 5 all divided by 2 is 17
The phrase [I]a number[/I] means an arbitrary variable, let's call it x
x
The sum of a number and 5:
x + 5
All divided by 2:
(x + 5)/2
The word [I]is[/I] means equal to, so we set (x + 5)/2 equal to 17:
[B](x + 5)/2 = 17[/B]
The sum of a number and 5 divided by 8The sum of a number and 5 divided by 8.
Let's take this algebraic expression in parts.
Part 1: The phrase [I]a number[/I] means an arbitrary variable, let's call it x.
Part 2: The sum of a number and 5 means we add 5 to the number x
x + 5
Part 3: Next, we divide this expression by 8
[B](x + 5)/8[/B]
the sum of a number and itself is 8A number means an arbitrary variable, let's call it x.
The sum of a number and itself means adding the number to itself
x + x
Simplified, we have 2x
The word is means equal to, so we have an algebraic expression of:
[B]2x= 8
[/B]
IF you need to solve this equation, divide each side by 2
[B]x = 4[/B]
the sum of a number divided by 8 and 3 equals 6"A Number" means an arbitrary variable, let's call it x.
x divide d by 8 is written as a quotient
x/8
The sum of x/8 and 3 means we add:
x/8 + 3
Finally, equals means we have an equation, so we set our expression above equal to 6
x/8 + 3 = 6
The sum of m and 3 divided by the difference of m minus 3The sum of m and 3 divided by the difference of m minus 3.
Sum of m and 3:
m + 3
Difference of m minus 3
m - 3
Take a quotient of these expressions:
[B]m + 3
-------
m - 3[/B]
the sum of n and twice n is 12Twice 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 digits of a certain two-digit number is 16. Reversing its digits increases the numberThe sum of the digits of a certain two-digit number is 16. Reversing its digits increases the number by 18. What is the number?
Let x and (16-x) represent the original ten and units digits respectively
Reversing its digits increases the number by 18
Set up the relational equation
[10x + (16-x)] + 18 = 10(16 - x) + x
Solving for x
9x + 34 = 160 - 9x
Combine like terms
18x = 126
Divide each side of the equation by 18
18x/18 = 126/18
x = 7
So 16 - x = 16 - 7 = 9
The first number is 79, the other number is 97. and 97 - 79 = 18 so we match up.
The number in our answer is [B]79[/B]
The sum of the squares of two consecutive positive integers is 61. Find these two numbers.The sum of the squares of two consecutive positive integers is 61. Find these two numbers.
Let the 2 consecutive integers be x and x + 1. We have:
x^2 + (x + 1)^2 = 61
Simplify:
x^2 + x^2 + 2x + 1 = 61
2x^2 + 2x + 1 = 61
Subtract 61 from each side:
2x^2 + 2x - 60 = 0
Divide each side by 2
x^2 + x - 30
Using our [URL='http://www.mathcelebrity.com/quadratic.php?num=x%5E2%2Bx-30&pl=Solve+Quadratic+Equation&hintnum=+0']quadratic equation calculator[/URL], we get:
x = 5 and x = -6
The question asks for [I]positive integers[/I], so we use [B]x = 5. [/B]This means the other number is [B]6[/B].
The sum of three consecutive integers is 42Let the 3 integers be x, y, and z.
y = x + 1
z = y + 1, or x + 2.
Set up our equation:
x + (x + 1) + (x + 2) = 42
Group our variables and constants:
(x + x + x) + (1 + 2) = 42
3x + 3 = 42
Subtract 3 from each side:
3x = 39
Divide each side of the equation by 3:
[B]x = 13
So y = x + 1 = 14
z = x + 2 = 15
(x,y,z) = (13,14,15)[/B]
The sum of twice an integer and 3 times the next consecutive integer is 48The sum of twice an integer and 3 times the next consecutive integer is 48
Let the first integer be n
This means the next consecutive integer is n + 1
Twice an integer means we multiply n by 2:
2n
3 times the next consecutive integer means we multiply (n + 1) by 3
3(n + 1)
The sum of these is:
2n + 3(n + 1)
The word [I]is[/I] means equal to, so we set 2n + 3(n + 1) equal to 48:
2n + 3(n + 1) = 48
Solve for [I]n[/I] in the equation 2n + 3(n + 1) = 48
We first need to simplify the expression removing parentheses
Simplify 3(n + 1): Distribute the 3 to each term in (n+1)
3 * n = (3 * 1)n = 3n
3 * 1 = (3 * 1) = 3
Our Total expanded term is 3n + 3
Our updated term to work with is 2n + 3n + 3 = 48
We first need to simplify the expression removing parentheses
Our updated term to work with is 2n + 3n + 3 = 48
[SIZE=5][B]Step 1: Group the n terms on the left hand side:[/B][/SIZE]
(2 + 3)n = 5n
[SIZE=5][B]Step 2: Form modified equation[/B][/SIZE]
5n + 3 = + 48
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 3 and 48. To do that, we subtract 3 from both sides
5n + 3 - 3 = 48 - 3
[SIZE=5][B]Step 4: Cancel 3 on the left side:[/B][/SIZE]
5n = 45
[SIZE=5][B]Step 5: Divide each side of the equation by 5[/B][/SIZE]
5n/5 = 45/5
Cancel the 5's on the left side and we get:
n = [B]9[/B]
The sum of two y and the quantity of three plus y plus twice the quantity two y minus two equals fifThe sum of two y and the quantity of three plus y plus twice the quantity two y minus two equals fifteen
The sum of two y and the quantity of three plus y
2y + (3 + y)
twice the quantity two y minus two
2(2y - 2)
The sum of two y and the quantity of three plus y plus twice the quantity two y minus two
2y + (3 + y) + 2(2y - 2)
Equals 15 to get our algebraic expression
[B]2y + (3 + y) + 2(2y - 2) = 15[/B]
[B][/B]
If the problem asks you to solve for yL
2y + 3 + y + 4y - 4 = 15
Group like terms:
7y - 1 = 15
Add 1 each side:
7y = 16
Divide each side by 7:
y = [B]16/7[/B]
the sum of w and v divided by their differencethe sum of w and v divided by their difference
the sum of w and v:
w + v
their difference:
w - v
the sum of w and v divided by their difference
[B](w + v)/(w - v)[/B]
the sum of x and 3 is divided by 2the sum of x and 3 is divided by 2
The sum of x and 3
x + 3
Divide this expression by 2
(x + 3)/2
the sum of X and 3 is divided by 2the sum of X and 3 is divided by 2
The sum of X and 3
X + 3
Is divided by 2
[B](X + 3)/2[/B]
the sum of x and 96 equals half of xthe sum of x and 96 equals half of x
half of x means we divide x by 2:
x/2
The sum of x and 96:
x + 96
The phrase equals means we set x + 96 equal to x/2:
[B]x + 96 = x/2[/B]
The top part of the tree is 3 times as long as the trunk which equation can tulia use to find t theThe top part of the tree is 3 times as long as the trunk which equation can tulia use to find t the length of the trunk
Let p be the top part of the tree.
We have p = 3t.
Divide by 3, we get t = [B]p/3[/B]
the total of a and 352 equals a divided by 195the total of a and 352 equals a divided by 195
Take this algebraic expression in 3 parts:
[LIST=1]
[*]The total of a and 352 means we add 352 to a: a + 352
[*]a divided by 195: a/195
[*]The phrase [I]equals[/I] means we set (1) equal to (2) to get our final algebraic expression:
[/LIST]
[B]a + 352 = a/195[/B]
The volleyball team and the wrestling team at Clarksville High School are having a joint car wash tThe volleyball team and the wrestling team at Clarksville High School are having a joint car wash today, and they are splitting the revenues. The volleyball team gets $4 per car. In addition, they have already brought in $81 from past fundraisers. The wrestling team has raised $85 in the past, and they are making $2 per car today. After washing a certain number of cars together, each team will have raised the same amount in total. What will that total be? How many cars will that take?
Set up the earnings equation for the volleyball team where w is the number of cars washed:
E = Price per cars washed * w + past fundraisers
E = 4w + 81
Set up the earnings equation for the wrestling team where w is the number of cars washed:
E = Price per cars washed * w + past fundraisers
E = 2w + 85
If the raised the same amount in total, set both earnings equations equal to each other:
4w + 81 = 2w + 85
Solve for [I]w[/I] in the equation 4w + 81 = 2w + 85
[SIZE=5][B]Step 1: Group variables:[/B][/SIZE]
We need to group our variables 4w and 2w. To do that, we subtract 2w from both sides
4w + 81 - 2w = 2w + 85 - 2w
[SIZE=5][B]Step 2: Cancel 2w on the right side:[/B][/SIZE]
2w + 81 = 85
[SIZE=5][B]Step 3: Group constants:[/B][/SIZE]
We need to group our constants 81 and 85. To do that, we subtract 81 from both sides
2w + 81 - 81 = 85 - 81
[SIZE=5][B]Step 4: Cancel 81 on the left side:[/B][/SIZE]
2w = 4
[SIZE=5][B]Step 5: Divide each side of the equation by 2[/B][/SIZE]
2w/2 = 4/2
w = [B]2 <-- How many cars it will take
[/B]
To get the total earnings, we take either the volleyball or wrestling team's Earnings equation and plug in w = 2:
E = 4(2) + 81
E = 8 + 81
E = [B]89 <-- Total Earnings[/B]
There are 100 teachers in a school of 3300 students find the ratio of number of teachers to the numbThere are 100 teachers in a school of 3300 students find the ratio of number of teachers to the number of students.
The Ratio is 100/3300.
Divide top and bottom by 100:
1/330 or [B]1:33
[/B]
You can also this into the search engine: [URL='https://www.mathcelebrity.com/ratio.php?simpratio=100%3A3300&rs=+7%3A5&rtot=+36&ab=+7%3A3&bc=+2%3A5&pl=Simplify+Ratio']Ratio of 100 to 3300[/URL].
there are 120 calories in 3/4 cup serving of cereal. How many Calories are there in 6 cups of cereal120/3/4 = x/6
Cross multiply:
0.75x = 720
Divide each side of the equation by 0.75
[B]x = 960[/B]
There are 13 animals in the barn. some are chickens and some are pigs. there are 40 legs in all. HowThere are 13 animals in the barn. some are chickens and some are pigs. there are 40 legs in all. How many of each animal are there?
Chickens have 2 legs, pigs have 4 legs. Let c be the number of chickens and p be the number of pigs. Set up our givens:
(1) c + p = 13
(2) 2c + 4p = 40
[U]Rearrange (1) to solve for c by subtracting p from both sides:[/U]
(3) c = 13 - p
[U]Substitute (3) into (2)[/U]
2(13 - p) + 4p = 40
26 - 2p + 4p = 40
[U]Combine p terms[/U]
2p + 26 = 40
[U]Subtract 26 from each side:[/U]
2p = 14
[U]Divide each side by 2[/U]
[B]p = 7[/B]
[U]Substitute p = 7 into (3)[/U]
c = 13 - 7
[B]c = 6[/B]
For a shortcut, you could use our [URL='http://www.mathcelebrity.com/simultaneous-equations.php?term1=c+%2B+p+%3D+13&term2=2c+%2B+4p+%3D+40&pl=Cramers+Method']simultaneous equations calculator[/URL]
There are 2 consecutive integers. Twice the first increased by the second yields 16. What are the nuThere are 2 consecutive integers. Twice the first increased by the second yields 16. What are the numbers?
Let x be the first integer. y = x + 1 is the next integer. We have the following givens:
[LIST=1]
[*]2x + y = 16
[*]y = x + 1
[/LIST]
Substitute (2) into (1)
2x + (x + 1) = 16
Combine x terms
3x + 1 = 16
Subtract 1 from each side
3x = 15
Divide each side by 3
[B]x = 5[/B]
So the other integer is 5 + 1 = [B]6[/B]
There are 2 piles of papers on a desk. Each pile has the same number of papers. There are 12 papersThere are 2 piles of papers on a desk. Each pile has the same number of papers. There are 12 papers in all on the desk. How many papers are in each pile?
12 papers on the desk / 2 piles of papers
Divide top and bottom by 2
[B]6 papers per pile.[/B]
There are 32 female performers in a dance recital. The ratio of men to women is 3:8. How many men arThere 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 72 boys and 90 girls on the Math team For the next math competition Mr Johnson would likeThere are 72 boys and 90 girls on the Math team For the next math competition Mr Johnson would like to arrange all of the students in equal rows with only girls or boys in each row with only girls or boys in each row. What is the greatest number of students that can be put in each row?
To find the maximum number (n) of boys or girls in each row, we want the GCF (Greatest Common Factor) of 72 and 90.
[URL='https://www.mathcelebrity.com/gcflcm.php?num1=72&num2=90&num3=&pl=GCF+and+LCM']Using our GCF calculator for GCF(72,90)[/URL], we get 18.
[LIST]
[*]72 boys divided by 18 = [B]4 rows of boys[/B]
[*]90 girls divided by 18 = [B]5 rows of girls[/B]
[/LIST]
There are 8 lions, 4 tigers, 5 cheetahs, 6 giraffes, 7 hippos, and 78 monkeys at the City Zoo. If eaThere are 8 lions, 4 tigers, 5 cheetahs, 6 giraffes, 7 hippos, and 78 monkeys at the City Zoo. If each of the 4 zookeepers feeds the same number of animals, how many animals does each zookeeper feed?
Calculate Total Animals:
8 + 4 + 5 + 6 + 7 + 78 = 108
Now divide 108 animals equally into 4 zookeepers
108/4 = [B]27 animals each zookeeper will feed[/B]
There are 812 students in a school. There are 36 more girls than boys. How many girls are there?[SIZE=6]There are 812 students in a school. There are 36 more girls than boys. How many girls are there?
Let b be boys
Let g be girls
We're given two equations:[/SIZE]
[LIST=1]
[*][SIZE=6]b + g = 812[/SIZE]
[*][SIZE=6]g = b + 36[/SIZE]
[/LIST]
[SIZE=6]Rearrange equation 2 to subtract b from each side:
[/SIZE]
[LIST=1]
[SIZE=6]
[LIST][*]b + g = 812[/LIST]
[LIST][*]-b + g = 36[/LIST][/SIZE]
[/LIST]
[SIZE=6]Add equation (1) to equation (2):
b - b + 2g = 812 + 36
The b's cancel:
2g = 848
Divide each side by 2:
2g/2 = 848/2
g = [B]424[/B]
[B][/B]
To find b, we put g= 424 into equation 1:
b + 424 = 812
b = 812 - 424
b = [B]388[/B]
[MEDIA=youtube]JO1b7qVwWoI[/MEDIA]
[/SIZE]
There are two numbers. The sum of 4 times the first number and 3 times the second number is 24. TheThere are two numbers. The sum of 4 times the first number and 3 times the second number is 24. The difference between 2 times the first number and 3 times the second number is 24. Find the two numbers.
Let the first number be x and the second number be y. We have 2 equations:
[LIST=1]
[*]4x + 3y = 24
[*]2x - 3y = 24
[/LIST]
Without doing anything else, we can add the 2 equations together to eliminate the y term:
(4x + 2x) + (3y - 3y) = (24 + 24)
6x = 48
Divide each side by 6:
[B]x = 8
[/B]
Substitute this into equation (1)
4(8) + 3y = 24
32 + 3y = 24
[URL='https://www.mathcelebrity.com/1unk.php?num=32%2B3y%3D24&pl=Solve']Type 32 + 3y = 24 into our search engine[/URL] and we get [B]y = 2.6667[/B].
There were 150 students at a dance. There were 16 more boys than girls. How many boys were there?Set up two equations:
(1) b = g + 16
(2) b + g = 150
Substitute equation (1) into (2)
(g + 16) + g = 150
Combine like terms
2g + 16 = 150
Subtract 16 from each side
2g = 134
Divide each side by 2 to isolate g
g = 67
Substitute this into equation (1)
b = 67 + 16
[B]b = 83[/B]
Think of a number. Double the number. Subtract 6 from the result and divide the answer by 2. The quoThink of a number. Double the number. Subtract 6 from the result and divide the answer by 2. The quotient will be 20. What is the number
Let's call our number n.
Double the number means we multiply n by 2:
2n
Subtract 6 from the result means we subtract 6 from 2n:
2n - 6
Divide the answer by 2:
(2n - 6)/2
We can simplify this as n - 3
The quotient will be 20. This means the simplified term above is set equal to 20:
[B]n - 3 = 20 [/B] <-- This is our algebraic expression
If you want to take it a step further, and solve for n in the algebraic expression above, we [URL='https://www.mathcelebrity.com/1unk.php?num=n-3%3D20&pl=Solve']type this expression into our calculator[/URL], and get:
n = 23
Thirty is half of the sum of 4 and a numberThirty is half of the sum of 4 and a number.
The phrase [I]a number[/I] represents an arbitrary variable, let's call it x.
The sum of 4 and a number:
4 + x
Half of this sum means we divide by 2:
(4 + x)/2
Set this equal to 30:
[B](4 + x)/2 = 30[/B] <-- This is our algebraic expression
triple 5, raise the result to the 10th power, then divide p by what you havetriple 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 c divide the result by atriple 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 s add the result to q then divide what you have by rtriple 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]
Twice a number decreased by eight is zeroThe phrase [I]a number[/I] means an arbitrary variable. We can pick any letter a-z except for i and e.
Let's choose x.
Twice a number:
2x
decreased by eight
2x - 8
[I]is [/I]means equal to. Set 2x - 8 equal to zero for our algebraic expression:
[B]2x - 8 = 0
[/B]
If the problem asks you to solve for x, add 8 to each side:
2x = 8
Divide each side by 2:
x= [B]4[/B]
two mechanics worked on a car. the first mechanic worked for 10 hours, and the second mechanic worketwo mechanics worked on a car. the first mechanic worked for 10 hours, and the second mechanic worked for 5 hours. together they charged a total of 1225. what was the rate charged per hour by each mechanic if the sum of the two rates was 170 per hour?
Set up two equations:
(1) 10x + 5y = 1225
(2) x + y = 170
Rearrange (2)
x = 170 - y
Substitute that into (1)
10(170 - y) + 5y = 1225
1700 - 10y + 5y = 1225
1700 - 5y = 1225
Move 5y to the other side
5y + 1225 = 1700
Subtract 1225 from each side
5y =475
Divide each side by 5
[B]y = 95[/B]
Which means x = 170 - 95, [B]x = 75[/B]
Two numbers total 50 and have a difference of 28. Find the two numbers.Two numbers total 50 and have a difference of 28. Find the two numbers.
Using x and y as our two numbers, we write the following 2 equations:
[LIST=1]
[*]x + y = 50
[*]x - y = 28
[/LIST]
Add the 2 rows:
2x = 78
Divide each side by 2:
[B]x = 39[/B]
If x = 39, then from (1), we have
y = 50 - 39
[B]y = 11[/B]
Two numbers total 83 and have a difference of 17 find the two numbersLet the numbers be x and y. Set up our givens:
[LIST=1]
[*]x + y = 83
[*]x - y = 17
[/LIST]
[U]Add equation (1) to equation (2)[/U]
x + x + y - y = 83 + 17
[U]The y-terms cancel out:[/U]
2x = 100
[U]Divide each side by 2:[/U]
2x/2= 100/2
x = [B]50[/B]
[U]
Plug x = 50 into equation (1)[/U]
50 + y = 83
[U]Subtract 50 from each side:[/U]
50 - 50 + y = 83 - 50
[U]Cancel the 50 on the left side:[/U]
y = [B]33
[/B]
So our two numbers (x, y) = (33, 50)
[MEDIA=youtube]jajO043ChUM[/MEDIA]
Two years of local internet service costs 685, including the installation fee of 85. What is the monTwo years of local internet service costs 685, including the installation fee of 85. What is the monthly fee?
Subtract the installation fee of 85 from the total cost of 685 to get the service cost only:
685 - 85 = 600
Now, divide that by 24 months in 2 years to get a per month fee
600/24 = [B]25 per month[/B]
u=ak/b for aCross multiply:
ub = ak
Divide each side of the equation by k to isolate a:
a = ub/k
[MEDIA=youtube]A3NW3Y68iNY[/MEDIA]
U=ak/b, for aU=ak/b, for a
[U]Cross multiply:[/U]
Ub = ak
[U]Divide each side by k[/U]
[B]a = Ub/k[/B]
VectorsFree Vectors Calculator - Given 2 vectors A and B, this calculates:
* Length (magnitude) of A = ||A||
* Length (magnitude) of B = ||B||
* Sum of A and B = A + B (addition)
* Difference of A and B = A - B (subtraction)
* Dot Product of vectors A and B = A x B
A ÷ B (division)
* Distance between A and B = AB
* Angle between A and B = θ
* Unit Vector U of A.
* Determines the relationship between A and B to see if they are orthogonal (perpendicular), same direction, or parallel (includes parallel planes).
* Cauchy-Schwarz Inequality
* The orthogonal projection of A on to B, projBA and and the vector component of A orthogonal to B → A - projBA
Also calculates the horizontal component and vertical component of a 2-D vector.
Video store movie rental plans. Plan A 25 membership fee plus 1.25 for movie. Plan B 40 for unlimiteVideo store movie rental plans. Plan A 25 membership fee plus 1.25 for movie. Plan B 40 for unlimited rentals. What number of movies rentals is plan B less than plan A?
Let x equal the number of movies rented and C the cost for rentals
Plan A: C = 1.25x + 25
Plan B: C = 40
Set up the inequality:
1.25x + 25 > 40
Subtract 25 from each side:
1.25x > 15
Divide each side of the inequality by 1.25
x > 12
So [B]13[/B] rentals or more make Plan B less than Plan A.
vw^2+y=x for wvw^2+y=x for w
This is an algebraic expression.
Subtract y from each side:
vw^2 + y - y = x - y
The y's cancel on the left side, so we're left with:
vw^2 = x - y
Divide each side by v
w^2 = (x - y)/v
Take the square root of each side:
w = [B]Sqrt((x - y)/v)[/B]
Water flows from tank A to tank B at the rate of 2 litres per minute.[QUOTE="Jahn, post: 78, member: 5"]Water flows from tank A to tank B at the rate of 2 litres per minute. Initially tank A has 200 litres in it and tank B has 100 Litres in it. Water drains from tank B at 0.5 litres per minute.
After how many minutes are there equal volumes of water in the 2 tanks?
Write an equation and solve it.[/QUOTE]
Tank A: V = 200 - 2x
Tank B: V = 100 - 0.5x
Where x is the number of minutes passed.
Set them equal to each other
200 - 2x = 100 - 0.5x
Subtract 100 from each side:
100 - 2x = -0.5x
Add 2x to each side:
1.5x = 100
Divide each side of the equation by x:
x = 66.66666667
Wendy is paid $7.50 per hour plus a bonus of $80 each week. Last week Wendy earned $312.50. How manyWendy is paid $7.50 per hour plus a bonus of $80 each week. Last week Wendy earned $312.50. How many hours did Wendy work last week?
Setup the earnings equation with h hours:
7.5h + 80 = 312.50
Solve for [I]h[/I] in the equation 7.5h + 80 = 312.50
[SIZE=5][B]Step 1: Group constants:[/B][/SIZE]
We need to group our constants 80 and 312.50. To do that, we subtract 80 from both sides
7.5h + 80 - 80 = 312.50 - 80
[SIZE=5][B]Step 2: Cancel 80 on the left side:[/B][/SIZE]
7.5h = 232.5
[SIZE=5][B]Step 3: Divide each side of the equation by 7.5[/B][/SIZE]
7.5h/7.5 = 232.5/7.5
h = [B]31
[URL='https://www.mathcelebrity.com/1unk.php?num=7.5h%2B80%3D312.50&pl=Solve']Source[/URL][/B]
What is the inverse of dividing by 3What 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 ratio 18b^2 to 45b written in simplest forWhat is the ratio 18b^2 to 45b written in simplest for
Using our [URL='https://www.mathcelebrity.com/monomial.php?num1=+%286xy%5E3%29%5E4&num2=+%283y%5E4%29%5E5%288x%5E6y%5E3%29&num3=18b%5E2%2F45b&pl=Divide']monomial calculator[/URL], we see that 18b^2/45b is
[B]2b/5[/B]
What is the value of x in the following equation: 2/3x + 1/6 = 1/3Answer Choices:
A. 6
B. 1/2
C. 1/3
D. 1/4
[U]Multiply through by 6:[/U]
2 * 6x/3 + 6/6 = 6/3
4x + 1 = 2
[U]Subtract 1 from each side:[/U]
4x + 1 - 1 = 2 - 1
4x = 1
[U]Divide each side by 4:[/U]
4x/4 = 1/4
x = [B]1/4[/B]
[MEDIA=youtube]jywMlPs3c2w[/MEDIA]
What number when multiplied by four exceeds itself by 42?What number when multiplied by four exceeds itself by 42?
Let the number be n. We have:
4n = n + 42
Subtract n from each side:
3n = 42
Divide each side by 3
[B]n = 14[/B]
When 3 consecutive positive integers are multiplied, the product is 16 times the sum of the 3 integeWhen 3 consecutive positive integers are multiplied, the product is 16 times the sum of the 3 integers. What is the difference of the product minus the sum?
Let the 3 consecutive positive integers be:
[LIST=1]
[*]x
[*]x + 1
[*]x + 2
[/LIST]
The product is:
x(x + 1)(x + 2)
The sum is:
x + x + 1 + x + 2 = 3x + 3
We're told the product is equivalent to:
x(x + 1)(x + 2) = 16(3x + 3)
x(x + 1)(x + 2) = 16 * 3(x + 1)
Divide each side by (x + 1)
x(x + 2) = 48
x^2 + 2x = 48
x^2 + 2x - 48 = 0
Now subtract the sum from the product:
x^2 + 2x - 48 - (3x + 3)
[B]x^2 - x - 51[/B]
When 4 times a number is increased by 40, the answer is the same as when 100 is decreased by the numWhen 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 Ms. Thelma turned on her oven, the temperature inside was 70 degrees F. The temperature began tWhen Ms. Thelma turned on her oven, the temperature inside was 70 degrees F. The temperature began to rise at a rate of 20 degrees per minute. How Long did it take for the oven to reach 350 degrees F?
Figure out how many degrees we have left:
350 - 70 = 280
Let m = minutes
20m = 280
Divide each side by m
[B]m = 14[/B]
When the circumference of a circle is increased from 10 pi inches to 15 pi inches, by how many incheWhen the circumference of a circle is increased from 10 pi inches to 15 pi inches, by how many inches is the radius increased?
C = 2pir
Smaller circle:
2pir = 10pi
Divide each side by 2pi:
r = 5
Larger circle:
2pir = 15pi
Divide each side by 2pi:
r = 7.5
Difference = 7.5 - 5 = [B]2.5 or 2&1/2
[MEDIA=youtube]HvMNNffcv78[/MEDIA][/B]
When the side of a square is doubled in length, its area increases by 432 square inches. What is theWhen the side of a square is doubled in length, its area increases by 432 square inches. What is the size of the original square?
Original square side length is s
Area = s^2
Double the side lengths to 2s
New area = (2s)^2 = 4s^2
Setup the difference relation:
4s^2 - s^2 = 432
3s^2 = 432
Divide each side by 3:
3s^2/3 = 432/3
s^2 = 144
s = [B]12[/B]
Which of the following equations represents a line that is parallel to the line with equation y = -3Which of the following equations represents a line that is parallel to the line with equation y = -3x + 4?
A) 6x + 2y = 15
B) 3x - y = 7
C) 2x - 3y = 6
D) x + 3y = 1
Parallel lines have the same slope, so we're looking for a line with a slope of 3, in the form y = mx + b. For this case, we want a line with a slope of -3, like our given line.
If we rearrange A) by subtracting 6x from each side, we get:
2y = -6x + 15
Divide each side by 2, we get:
y = -3x + 15/2
This line is in the form y = mx + b, where m = -3. So our answer is [B]A[/B].
Wilbie had candy to give to his 3 children. He first took 5 pieces and evenly divided the rest amongWilbie 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 an equation that relates the quantities. G varies jointly with t and q and inversely with theWrite 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]
wy - ma = ay/n for ywy - ma = ay/n for y
Subtract ay/n from each side:
wy - ma - ay/n = ay/n - ay/n
wy - ma - ay/n = 0
Now add ma to each side:
wy - ay/n = ma
Factor out y:
y(w - a/n) = ma
Divide each side by (w - a/n)
y = [B]ma/(w - a/n)[/B]
X divide by 6 subtract by 1X divide by 6 subtract by 1
x divide by 6
x/6
subtract by 1
[B]x/6 - 1[/B]
x/y + 9 = n for yx/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]
Xavier has $132 to buy a video game. Each game costs $12. Write an equation to find the number of gaXavier has $132 to buy a video game. Each game costs $12. Write an equation to find the number of games Xavier can purchase.
Let g be the number of games, we have a cost function C(g)
C(g) = 12g
We want to find g such that C(g) = 132
12g = 132
Divide each side by 12
[B]g = 11[/B]
xy divided by 2xy divided by 2
[B]xy/2[/B]
Y add z then divide by xY add z then divide by x
y add z:
y + z
Then divide by x means we divide the sum (y + z) by x
[B](y + z)/x[/B]
y varies directly as x and inversely as iy 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]
Yolanda runs each lap in 7 minutes. She will run less than 35 minutes today. What are the possible nYolanda runs each lap in 7 minutes. She will run less than 35 minutes today. What are the possible numbers of laps she will run today?
7 minutes per lap must be less than 35 minutes. Let l be the number of laps
7l < 35
Divide each side by 7
[B]l < 5[/B]
You and your friend are playing a number-guessing game. You ask your friend to think of a positive nYou and your friend are playing a number-guessing game. You ask your friend to think of a positive number, square the number, multiply the result by 2, and then add three. If your friend's final answer is 53, what was the original number chosen?
Let n be our original number.
Square the number means we raise n to the power of 2:
n^2
Multiply the result by 2:
2n^2
And then add three:
2n^2 + 3
If the friend's final answer is 53, this means we set 2n^2 + 3 equal to 53:
2n^2 + 3 = 53
To solve for n, we subtract 3 from each side, to isolate the n term:
2n^2 + 3 - 3 = 53 - 3
Cancel the 3's on the left side, and we get:
2n^2 = 50
Divide each side of the equation by 2:
2n^2/2 = 50/2
Cancel the 2's, we get:
n^2 = 25
Take the square root of 25
n = +-sqrt(25)
n = +-5
We are told the number is positive, so we discard the negative square root and get:
n = [B]5[/B]
You and your friend are saving for a vacation. You start with the same amount and save for the sameYou and your friend are saving for a vacation. You start with the same amount and save for the same number of weeks. You save 75 per week, and your friend saves 50 per week. When vacation time comes, you have 950, and your friend has 800. How much did you start with, and for how many weeks did you save?
[U]Let w be the number of weeks. Set up two equations where s is the starting amount:[/U]
(1) s + 75w =950
(2) s + 50w = 800
[U]Rearrange (1) into (3) to solve for s by subtracting 75w[/U]
(3) s = 950 - 75w
[U]Rearrange (2) into (4) to solve for s by subtracting 50w[/U]
(4) s = 800 - 50w
[U]Set (3) and (4) equal to each other so solve for w[/U]
950 - 75w = 800 - 50w
[U]Add 75w to each side, and subtract 950 from each side:[/U]
25w = 150
[U]Divide each side by w[/U]
[B]w = 6[/B]
Now plug w = 6 into (3)
s = 950 - 75(6)
s = 950 - 450
[B]s = 500[/B]
You borrowed $25 from your friend. You paid him back in full after 6 months. He charged $2 for interYou borrowed $25 from your friend. You paid him back in full after 6 months. He charged $2 for interest. What was the annual simple interest rate that he charged you? Use the formula: I = Prt.
We have I = 2, P = 25, t = 0.5
2 = 25(r)0.5
Divide each side by 0.5
4 = 25r
Divide each side by 25
r = 4/25
[B]r = 0.16[/B]
As a percentage, this is [B]16%[/B]
You bought a magazine for $5 and four erasers. You spent a total of $25. How much did each eraser coYou bought a magazine for $5 and four erasers. You spent a total of $25. How much did each eraser cost?
Subtract the cost of the magazine from what you spent:
$25 - $5 = $20.
If you spent $20 on 4 erasers, we divide 20/4 = [B]$5 per eraser[/B]
You can pay a daily entrance fee of $3 or purchase a membership for the 12 week summer season for $8You can pay a daily entrance fee of $3 or purchase a membership for the 12 week summer season for $82 and pay only $1 per day to swim. How many days would you have to swim to make the membership worthwhile?
Set up cost equations:
Daily entrance fee:
3d where d is the number of days of membership
Membership fee
82 + 1d
Set them equal to each other
82 + 1d = 3d
Subtract d from each side:
2d = 82
Divide each side by 2
[B]d = 41[/B]
You deposit $2000 in an account that earns simple interest at an annual rate of 4%. How long must yoYou deposit $2000 in an account that earns simple interest at an annual rate of 4%. How long must you leave the money in the account to earn $500 in interest?
The simple interest formula for the accumulated balance is:
Prt = I
We are given P = 2,000, r = 0.04, and I = 500.
2000(0.04)t = 500
80t = 500
Divide each side by 80
t = [B]6.25 years
[MEDIA=youtube]Myz0FZgwZpk[/MEDIA][/B]
You had $22 to spend on 8 notebooks after buying them you had $6If you have $6 left over, then 8 notebooks cost $22 - $6 = $16.
8 notebooks = $16
Divide each side of the equation by 8
Each notebook is $2
You have $10.00 to spend on tacos. Each taco costs $0.50. Write and solve an inequality that explainYou have $10.00 to spend on tacos. Each taco costs $0.50. Write and solve an inequality that explains how many tacos you can buy.
Let's start with t as the number of tacos.
We know that cost = price * quantity, so we have the following inequality for our taco spend:
[B]0.5t <= 10
[/B]
Divide each side of the inequality by 0.5 to isolate t:
0.5t/0.5 <= 10/0.5
Cancel the 0.5 on the left side and we get:
t <= [B]20
[MEDIA=youtube]yy51EsGi1nM[/MEDIA][/B]
You have $250,000 in an IRA (Individual Retirement Account) at the time you retire. You have the opYou have $250,000 in an IRA (Individual Retirement Account) at the time you retire. You have the option of investing this money in two funds: Fund A pays 5.4% annually and Fund B pays 7.9% annually. How should you divide your money between fund Fund A and Fund B to produce an annual interest income of $14,750?
You should invest $______in Fund A and $___________in Fund B.
Equation is x(.079) + (250,000 - x).054 = 14,750
.025x + 13,500 = 14,750
.025x = 1,250
[B]x = 50,000 for Fund A[/B]
So at 5.4%, we have 250,000 - 50,000 = [B]200,000[/B] for the other fund B.
You have $37 to plant garden. If you spend $12.25 on seeds, how many packs of vegetable plants canYou have $37 to plant garden. If you spend $12.25 on seeds, how many packs of vegetable plants can you buy for 2.75 each?
[U]How much do we have to spend on plants?[/U]
$37 - 12.25 = $24.75
[U]Calculate how many vegetable plants we can buy. Set up an equation where x = vegetable plants[/U]
2.75x = 24.75
Divide each side by 2.75
[B]x = 9[/B]
You have a total of 42 math and science problems for homework. You have 10 more math problems than sYou have a total of 42 math and science problems for homework. You have 10 more math problems than science problems. How many problems do you have in each subject?
Let m be the math problems and s be the science problems. We have two equations:
(1) m + s = 42
(2) m = s + 10
Substitute (2) into (1)
(s + 10) + s = 42
Combine like terms
2s + 10 = 42
Subtract 10 from each side
2s = 32
Divide each side by 2
[B]s = 16[/B]
So that means m = 16 + 10 --> [B]m = 26
(m, s) = (26, 16)[/B]
your starting salary at a new company is 45000. Each year you receive a 2% raise. How long will it tyour starting salary at a new company is 45000. Each year you receive a 2% raise. How long will it take you to make $80000?
Let y be the number of years of compounding the 2% raise. Since 2% as a decimal is 0.02, we have the following equation for compounding the salary:
45000 * (1.02)^y = 80000
Divide each side by 45000:
(1.02)^y = 1.77777777778
To solve this equation for y, we [URL='https://www.mathcelebrity.com/natlog.php?num=1.02%5Ey%3D1.77777777778&pl=Calculate']type it in our search engine[/URL] and we get:
y = [B]29.05[/B]
[B]Or just over 29 years[/B]
z = (x + y)/mx; Solve for xz = (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 side by zm - 1
x = y/(zm - 1)
[MEDIA=youtube]ksxCS3YlCwY[/MEDIA]
z varies directly with x and inversely with yz 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/w=x+z/x+y for zz/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]
Zombies are doubling every 2 days. If two people are turned into zombies today, how long will it takZombies are doubling every 2 days. If two people are turned into zombies today, how long will it take for the population of about 600,000 to turn into zombies?
Let d = every 2 days. We set up the exponential equation
2 * 2^d = 600,000
Divide each side by 2:
2^d = 300000
To solve this equation for d, we [URL='https://www.mathcelebrity.com/natlog.php?num=2%5Ed%3D300000&pl=Calculate']type it in our math engine[/URL] and we get
d = 18.19 (2 day periods)
18.19 * days per period = 36.38 total days
Most problems like this ask you to round to full days, so we round up to [B]37 days[/B].
zy-dm=ky/t for yzy-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]