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0.2 Proportions

Karen Perilloux; Prakash Ghimire; and Lauren Johnson

Learning Objectives

By the end of this section, you will be able to:

  • Use the definition of proportion
  • Solve proportions
  • Solve applications using proportions
  • Write percent equations as proportions
  • Translate and solve percent proportions

Use the Definition of Proportion

In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a proportion.

Proportion

A proportion is an equation of the form [latex]\frac{a}{b} = \frac{c}{d}[/latex], where [latex]b neq 0[/latex] and [latex]d neq 0[/latex]. The proportion states two ratios or rates are equal. The proportion is read “a is to b, as c is to d”.

The equation [latex]\frac{1}{2} = \frac{4}{8}[/latex] is a proportion because the two fractions are equal. The proportion [latex]\frac{1}{2} = \frac{4}{8}[/latex] is read “1 is to 2 as 4 is to 8″

If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion [latex]\frac{20 text{ students}}{1 text{ teacher}} = \frac{60 text{ students}}{3 text{ teachers}}[/latex], we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

Example

Write each sentence as a proportion:

  1. 3 is to 7 as 15 is to 35
  2. 5 hits in 8 at-bats is the same as 30 hits in 48 at-bats.
  3. $1.50 for 6 ounces is equivalent to @2.25 for 9 ounces.
Show Solution
1. 3 is to 7 as 15 is to 35. Write as a proportion. 37=1535
2. 5 hits in 8 at-bats is the same as 30 hits in 48 at-bats.

Write each fraction to compare hits to at-bats.

 hitsat-bats=hitsat-bats

Write as a proportion.

 58=3048
3. $1.50 for 6 ounces is equivalent to $2.25 for 9 ounces.

Write each fraction to compare dollars to ounces.

 $ounces=$ounces

Write as a proportion.

 1.506=2.259

Try It

Write each sentence as a proportion:

  1. 5 is to 9 as 20 is to 36.
  2. 7 hits in 11 at-bats is the same as 28 hits in 44 at-bats.
  3. $2.50 for 8 ounces is equivalent to $3.75 for 12 ounces.
    Show Solution

    1. [latex]\frac{5}{9} = \frac{20}{36}[/latex]

    2. [latex]\frac{7}{11} = \frac{28}{44}[/latex]

    3. [latex]\frac{2.50}{8} = \frac{3.75}{12}[/latex]

Try It

Write each sentence as a proportion:

    1. 6 is to 7 as 36 is to 42.
    2. 8 adults for 36 children is the same as 12 adults for 54 children.
    3. $3.75 for 6 ounces is equivalent to $2.50 for 4 ounces.
Show Solution

1. [latex]\frac{6}{8} = \frac{36}{42}[/latex]

2. [latex]\frac{8}{36} = \frac{12}{54}[/latex]

3. [latex]\frac{3.75}{6} = \frac{2.50}{4}[/latex]

Look at the proportions [latex]\frac{1}{2} = \frac{4}{8}[/latex] and [latex]\frac{2}{3} = \frac{6}{9}[/latex].

From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?

To determine if a proportion is true, we find the cross products of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross product because of the cross formed. If, and only if, the given proportion is true, that is, the two sides are equal, then the cross products of a proportion will be equal.

The figure shows cross multiplication of two proportions. There is the proportion 1 is to 2 as 4 is to 8. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 8 · 1 = 8 and 2 · 4 = 8. There is the proportion 2 is to 3 as 6 is to 9. Arrows are shown diagonally across the equal sign to show cross products. The equations formed by cross multiplying are 9 · 2 = 18 and 3 · 6 = 18.

Cross Products of a Proportion

For any proportion of the form [latex]\frac{a}{b} = \frac{c}{d}[/latex], where [latex]b \neq 0, d \neq 0[/latex], its cross products are equal.

The figure shows cross multiplication a proportion. There is the proportion a is to b as c is to d. Arrows are shown diagonally across the equal sign to show cross products. The equation formed by cross multiplying is a times d equals b times c.

 

Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are both equal, we have a proportion.

Example

Determine whether each equation is a proportion:

a)[latex]\frac{4}{9} = \frac{12}{28}[/latex]

b)[latex]\frac{17.5}{37.5} = \frac{7}{15}[/latex]

Show Solution

To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.

a) The figure shows the proportion 4 is to 9 equals 12 is to 28.
Find the cross products. 28⋅4=1129⋅12=10828⋅4=1129⋅12=108The figure shows cross multiplication a proportion. There is the proportion 4 is to 9 as 12 is to 28. Arrows are shown diagonally across the equal sign to show cross products. The equation formed by cross multiplying is 4 times 28 equals 9 times 12.

Since the cross products are not equal, [latex]28 \cdot 4 \neq 9 \cdot 12[/latex], the equation is not a proportion.

b)
The figure shows the proportion such that 17.5 is to 37.5 equals 7 is to 15.
Find the cross products. 15⋅17.5=262.537.5⋅7=262.515⋅17.5=262.537.5⋅7=262.5


The figure shows cross multiplication a proportion. There is the proportion 17.5 is to 37.5 as 7 is to 15. Arrows are shown diagonally across the equal sign to show cross products. The equation formed by cross multiplying is 17.5 times 15 equals 37.5 times 7.

Since the cross products are equal, [latex]15 \cdot 17.5 = 37.5 \cdot 7[/latex], the equation is a proportion.

Try It

Determine whether each equation is a proportion:

a) [latex]\frac{7}{9} = \frac{45}{72}[/latex]

b) [latex]\frac{24.5}{45.5} = \frac{7}{13}[/latex]

Show Solution

a) not a proportion

b) yes a proportion

Try It

Determine whether each equation is a proportion:

a) [latex]\frac{8}{9} = \frac{56}{73}[/latex]

b) [latex]\frac{28.5}{52.5} = \frac{8}{15}[/latex]

Show Solution

a) not a proportion

b) not a proportion

Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.

Example

Solve: [latex]\frac{x}{63} = \frac{4}{7}[/latex]

Show Solution
The figure shows the proportion x is to 63 equals 4 is to 7.
To isolate [latex]x[/latex], multiply both sides by the LCD, 63. The figure shows the proportion 63 times x is to 63 equals 63 times 4 is to 7.
Simplify. The figure shows an equation where x is equal to 9 times 7 times 4 divided by 7, with the 7 canceled from the numerator and denominator.
Divide the common factors. The figure shows an equation where x equals 36.
Check: To check our answer, we substitute into the original proportion.
The figure shows the proportion x is to 63 equals 4 is to 7.
. The figure shows the proportion 36 is to 63 as 4 is to 7, with a question mark symbol between the two fractions
Show common factors. The figure shows the proportion 4 times 9 is to 7 times 9 as 4 is to 7, with a question mark symbol between the two fractions
Simplify. The figure shows the proportion 4 is to 7 as 4 is to 7, with a check mark symbol

Try It

Solve the proportion:

n84=1112.

Show Solution

n=77

 

Try It

Solve the proportion: [latex]\frac{y}{96} = \frac{13}{12}[/latex]

 

Show Solution

y=104

 

 

When the variable is in a denominator, we’ll use the fact that the cross products of a proportion are equal to solve the proportions.

We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

Example

Solve:

144a=94.144a=94.
Show Solution

Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.

The figure shows cross multiplication a proportion. There is the proportion 144 is to a as 9 is to 4. Arrows are shown diagonally across the equal sign to show cross products. The equation formed by cross multiplying is 144 times a equals 9 times 4.
Find the cross products and set them equal. This figure shows an equation where 4 times 144 equals a times 9
Simplify. The figure shows an equation where 576 equals 9 a.
Divide both sides by 9. The figure shows the proportion 576 is to 9 equals 9 a is to 9.
Simplify. The figure shows an equation where 64 equals a.
Check your answer.
The figure shows the proportion 144 is to a equals 9 a is to 4.
. The figure shows the proportion 144 is to 64 as 9 is to 4, with a question mark
Show common factors.. The figure shows the proportion 9 times 16 is to 4 times 16 as 9 is to 4, with a question mark
Simplify. The figure shows the proportion 9 is to 4 as 9 is to 4, with a check mark symbol

Another method to solve this would be to multiply both sides by the LCD, [latex]4a[/latex].

4a.Try it and verify that you get the same solution.

 

 

 

Try It

Solve the proportion:

91b=75.

Show Solution

b=65

 

Try It

Solve the proportion:

39c=138.

Show Solution

c=24

 

Example

Solve:

5291=−4y.5291=−4y.
Show Solution

 

Find the cross products and set them equal. The figure shows cross multiplication a proportion. There is the proportion 52 is to 91 as -4 is to y. Arrows are shown diagonally across the equal sign to show cross products.
  [latex]y \cdot 52   = 91 ( -4)[/latex]
Simplify.  [latex] 52 y   = -364[/latex]
Divide both sides by 52. [latex]\frac{52y}{52}   = \frac{-364}{52}[/latex]
Simplify. [latex] y   = - 7[/latex]
Check:
    [latex]\frac{52}{91}   = \frac{-4}{y}[/latex]
Substitute [latex] y   = - 7[/latex] The figure shows the proportion 52 is to 91 as -4 is to -7, with a question mark
Show common factors. The figure shows the proportion 13 times 4 is to 13 times 4 as -4 is to -7, with a question mark
Simplify. The figure shows the proportion 4 is to 7 as 4 is to 7, with a check mark symbol

 

Try It

Solve the proportion:

8498=−6x.

 

 

Show Solution

x=-7

 

Try It

Solve the proportion:

−7y=105135.

Show Solution

y=-9

 

Solve Applications Using Proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion, we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

Example

When pediatricians prescribe acetaminophen to children, they prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of the child’s weight. If Zoe weighs 80 pounds, how many milliliters of acetaminophen will her doctor prescribe?

Show Solution

 

Identify what you are asked to find. How many ml of acetaminophen the doctor will prescribe
Choose a variable to represent it. Let

a=a=ml of acetaminophen.
Write a sentence that gives the information to find it. If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds?
Translate into a proportion. [latex]\frac{ml}{pounds}   = \frac{ml}{pounds}[/latex]
Substitute given values—be careful of the units. [latex]\frac{5}{25}   = \frac{a}{80}[/latex]
Multiply both sides by 80. [latex]80 \cdot \frac{5}{25}   =80\cdot  \frac{a}{80}[/latex]
Multiply and show common factors.  [latex]\frac{16 \cdot 5 \cdot 5}{5\cdot 5}   = \frac  {80a}{80}[/latex]
Simplify. [latex]16  =  a[/latex]
Check if the answer is reasonable.
Yes. Since 80 is about 3 times 25, the medicine should be about 3 times 5.
Write a complete sentence. The pediatrician would prescribe 16 ml of acetaminophen to Zoe.

You could also solve this proportion by setting the cross products equal.

 

Try It

Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs 60 pounds?

Show Solution

The doctor will prescribe 12 milliliters of acetaminophen for Emilia.

 

 

Try It

For every 1 kilogram (kg) of a child’s weight, pediatricians prescribe 15 milligrams (mg) of a fever reducer. If Isabella weighs 12 kg, how many milligrams of the fever reducer will the pediatrician prescribe?

Show Solution

180 mg

 

 

 

Example

One brand of microwave popcorn has 120 calories per serving. A whole bag of this popcorn has 3.5 servings. How many calories are in a whole bag of this microwave popcorn?

Show Solution

 

Identify what you are asked to find. How many calories are in a whole bag of microwave popcorn?
Choose a variable to represent it. Let

c=c=number of calories.
Write a sentence that gives the information to find it. If there are 120 calories per serving, how many calories are in a whole bag with 3.5 servings?
Translate into a proportion.  [latex]\frac{calories}{serving}   = \frac{calories}{serving}[/latex]
Substitute given values. [latex]\frac{120}{1}   = \frac{c}{3.5}[/latex]
Multiply both sides by 3.5. [latex]3.5 \cdot (\frac{120}{1})   =3.5\cdot  (\frac{c}{3.5})[/latex]
Multiply. [latex]420= c[/latex]
Check if the answer is reasonable.
Yes. Since 3.5 is between 3 and 4, the total calories should be between 360 (3⋅120) and 480 (4⋅120).
Write a complete sentence. The whole bag of microwave popcorn has 420 calories.

 

Try It

Marissa loves the Caramel Macchiato at the coffee shop. The 16 oz. medium size has 240 calories. How many calories will she get if she drinks the large 20 oz. size?

Show Solution

300 calories

 

 

Try It

Yaneli loves Starburst candies, but wants to keep her snacks to 100 calories. If the candies have 160 calories for 8 pieces, how many pieces can she have in her snack?

Show Solution

5 pieces of Starburst candies.

 

 

Example

Josiah went to Mexico for spring break and changed $325 dollars into Mexican pesos. At that time, the exchange rate had $1 U.S. is equal to 12.54 Mexican pesos. How many Mexican pesos did he get for his trip?

Show Solution

 

Identify what you are asked to find. How many Mexican pesos did Josiah get?
Choose a variable to represent it. Let

p=p=number of pesos.
Write a sentence that gives the information to find it. If $1 U.S. is equal to 12.54 Mexican pesos, then $325 is how many pesos?
Translate into a proportion.  [latex]\frac{\ $}{pesos}   = \frac{\ $}{pesos}[/latex]
Substitute given values. [latex]\frac{1}{12.54}   = \frac{325}{p}[/latex]
The variable is in the denominator, so find the cross products and set them equal. [latex]p \cdot 1   = 12.54(325)[/latex]
Simplify. [latex]p   = 4,075.5[/latex]
Check if the answer is reasonable.
Yes, $100 would be $1,254 pesos. $325 is a little more than 3 times this amount.
Write a complete sentence. Josiah has 4075.5 pesos for his spring break trip.

 

 

Try It

Yurianna is going to Europe and wants to change $800 dollars into Euros. At the current exchange rate, $1 US is equal to 0.738 Euro. How many Euros will she have for her trip?

Show Solution

590.4 Euros

 

 

Try It

Corey and Nicole are traveling to Japan and need to exchange $600 into Japanese yen. If each dollar is 94.1 yen, how many yen will they get?

Show Solution

56460 Yen

 

 

Write Percent Equations As Proportions

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.

For example,

60%=6010060%=60100and we can simplify

60100=35.60100=35.Since the equation

60100=3560100=35shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:

amountbase=percent100amountbase=percent100
35=6010035=60100

Percent Proportion

The amount is to the base as the percent is to

100.100.

amountbase=percent100amountbase=percent100

If we restate the problem in the words of a proportion, it may be easier to set up the proportion:

The amount is to the base as the percent is to one hundred.The amount is to the base as the percent is to one hundred.

We could also say:

The amount out of the base is the same as the percent out of one hundred.The amount out of the base is the same as the percent out of one hundred.

First we will practice translating into a percent proportion. Later, we’ll solve the proportion.

Example

Translate to a proportion. What number is 75% of 90? 

Show Solution

If you look for the word “of”, it may help you identify the base.

Identify the parts of the percent proportion. The image shows the percent equation in words: 'What number is 75% of 90?' Braces label the parts: 'What number' is the amount, '75%' is the percent, and '90' is the base.
Restate as a proportion.  What number out of [latex]90[/latex] is the same as [latex]75[/latex] out of [latex]100[/latex]?
Set up the proportion. Let

n=numbern=number 

 n90=75100n90=75100

 

 

Try It

Translate to a proportion: What number is 60% of 105?

Show Solution

[latex]\frac{x}{105} = \frac{60}{100}[/latex]

 

 

105?

Try It

Translate to a proportion: What number is 40% of 85?

Show Solution

[latex]\frac{x}{85} = \frac{40}{100}[/latex]

 

 

85?

Example

Translate to a proportion. 19 is 25% of what number?

 

Show Solution

 

Identify the parts of the percent proportion. The image shows the percent equation in words: '19 is 25% of what number?' Braces label the parts: '19' is the amount, '25%' is the percent, and 'what number' is the base.
Restate as a proportion.  [latex]19[/latex] out of what number is the same as [latex]25[/latex] out of [latex]100[/latex]?
Set up the proportion. Let

n=numbern=number 

 19n=2510019n=25100

 

Try It 

Translate to a proportion: 36 is 25% of what number?

Show Solution

[latex]\frac{36}{x} = \frac{25}{100}[/latex]

 

 

Try It

Translate to a proportion: 27 is 36% of what number?

Show Solution

[latex]\frac{27}{x} = \frac{36}{100}[/latex]

 

 

Example

Translate to a proportion. What percent of 27 is 9?

Show Solution

 

Identify the parts of the percent proportion. The image shows the percent equation in words: what percent of 27 is 9?' Braces label the parts: 'what percent' is the percent, '27' is the base, and '9' is the amount.
Restate as a proportion. [latex]9[/latex] out of [latex]27[/latex] is the same as what number of [latex]100[/latex]?
Set up the proportion. Let

p=percentp=percent 

 927=p100927=p100

 

Try It

Translate to a proportion: What percent of 52 is 39?

Show Solution

[latex]\frac{x}{100} = \frac{39}{52}[/latex]

 

 

Try It

Translate to a proportion: What percent of 92 is 23?

Show Solution

[latex]\frac{x}{100} = \frac{23}{92}[/latex]

 

 

Translate and Solve Percent Proportions

Now that we have written percent equations as proportions, we are ready to solve the equations.

Example

Translate and solve using proportions: What number is 45% of 80?

Show Solution

 

Identify the parts of the percent proportion. The image shows the percent equation in words: 'What number is 45% of 80?' Braces label the parts: 'What number' is the amount, '45%' is the percent, and '80' is the base.
Restate as a proportion.  What number out of [latex]80[/latex] is the same as [latex]45[/latex] out of [latex]100[/latex]?
Set up the proportion. Let

n=n=number.

 [latex]\frac{n}{80}   = \frac{45}{100}[/latex]
Find the cross products and set them equal. [latex]100 \cdot n   = 80 \cdot 45[/latex]
Simplify.  [latex]100n   = 3,600[/latex]
Divide both sides by 100. [latex]\frac{100n}{100}   = \frac{3,600}{100}[/latex]
Simplify.   [latex]n   = 36[/latex]
Check if the answer is reasonable.
Yes. 45 is a little less than half of 100 and 36 is a little less than half 80.
Write a complete sentence that answers the question. 36 is 45% of 80.

 

 

Try It

Translate and solve using proportions: What number is 65% of 40?

Show Solution

[latex]26[/latex]

 

 

40?

Try It

Translate and solve using proportions: What number is 85% of 40?

Show Solution

[latex]34[/latex]

 

 

40?

In the next example, the percent is more than 100, which is more than one whole. So the unknown number will be more than the base.

Example

Translate and solve using proportions: 125% of 25 is what number? 

Show Solution

 

Identify the parts of the percent proportion. The image shows the percent equation in words: '125% is 25 of what number?' Braces label the parts: '125%' is the percent, '25' is the base, and 'what number' is the amount.
Restate as a proportion. What number out of [latex]25[/latex] is the same as [latex]125[/latex] out of [latex]100[/latex]?
Set up the proportion. Let

n=n=number.

 [latex]\frac{n}{25}   = \frac{125}{100}[/latex]
Find the cross products and set them equal.  [latex]100 \cdot n   = 25 \cdot 125[/latex]
Simplify.  [latex]100 n   = 3,125[/latex]
Divide both sides by 100. [latex]\frac{100n}{100}   = \frac{3,125}{100}[/latex]
Simplify. [latex]n   = 31.25[/latex]
Check if the answer is reasonable.
Yes. 125 is more than 100 and 31.25 is more than 25.
Write a complete sentence that answers the question. 125% of 25 is 31.25.

 

 

Try It

Translate and solve using proportions: 125% of 64 is what number?

Show Solution

[latex]80[/latex]

 

Try It

Translate and solve using proportions: 175% of 84 is what number?

Show Solution

[latex]147[/latex]

 

 

 

Percents with decimals and money are also used in proportions.

Example

Translate and solve: 6.5% of what number is $1.56? 

Show Solution

 

Identify the parts of the percent proportion. The image shows the percent equation in words: '6.5% of what number is $1.56?' Braces label the parts: '6.5%' is the percent, 'what number' is the base, and '$1.56' is the amount.
Restate as a proportion. [latex]\ $1.56[/latex] out of what number is the same as [latex]6.5[/latex] out of [latex]100[/latex]?
Set up the proportion. Let

n=n=number.

 [latex]\frac{1.56}{n}   = \frac{6.5}{100}[/latex]
Find the cross products and set them equal. [latex]100 ( 1.56)   = n \cdot 6.5[/latex]
Simplify. [latex]156  = 6.5 n[/latex]
Divide both sides by 6.5 to isolate the variable. [latex]\frac{156}{6.5}   = \frac{6.5n}{6.5}[/latex]
Simplify. [latex]24  = n[/latex] 
Check if the answer is reasonable.
Yes. 6.5% is a small amount and $1.56 is much less than $24.
Write a complete sentence that answers the question. 6.5% of $24 is $1.56.

 

 

Try It

Translate and solve using proportions: 8.5% of what number is $3.23?

Show Solution

[latex]$3.23[/latex]

 

Try It

Translate and solve using proportions: 7.25% of what number is $4.64?

Show Solution

[latex]$4.64[/latex]

 

Example

Translate and solve using proportions: What percent of 72 is 9? 

Show Solution

 

Identify the parts of the percent proportion. The image shows the percent equation in words: What percent of 72 is 9?' Braces label the parts: 'what percent' is the percent, '72' is the base, and '9' is the amount.
Restate as a proportion. [latex]\ 9[/latex] out of [latex]72[/latex]  is the same as what number out of [latex]100[/latex]?
Set up the proportion. Let

n=n=number.

 [latex]\frac{9}{72}   = \frac{n}{100}[/latex]
Find the cross products and set them equal. [latex]72 \cdot n    = 100 \cdot 9[/latex]
Simplify.  [latex]72  n    = 900[/latex]
Divide both sides by 72.  [latex]\frac{72n}{72}   = \frac{900}{72}[/latex]
Simplify.  [latex]n    = 12.5[/latex]
Check if the answer is reasonable.
Yes. 9 is [latex]\frac{1}{8}[/latex] of 72 and [latex]\frac{1}{8}[/latex] is 12.5%.
Write a complete sentence that answers the question. 12.5% of 72 is 9.

 

 

Try It

Translate and solve using proportions: What percent of 72 is 27?

Show Solution

37.5%

 

 

Try It

Translate and solve using proportions: What percent of 92 is 23?

Show Solution

25%

 

 

 

Key Concepts

Proportion: A proportion is an equation of the form \[\frac{a}{b} = \frac{c}{d}\]

where [latex]b \neq 0[/latex] and [latex]d \neq 0[/latex]. The proportion states that two ratios or rates are equal. The proportion is read “[latex]a[/latex]  is to  [latex]b[/latex], as  [latex]c [/latex] is to  [latex]d[/latex] ”.
 

Cross Products of a Proportion: For any proportion of the form [latex]\frac{a}{b} = \frac{c}{d}[/latex] where  [latex]b \neq 0[/latex] , its cross products are equal: [latex]a \cdot d = b \cdot c[/latex]

Percent Proportion: The amount is to the base as the percent is to 100:
[latex]\frac{\text{amount}}{\text{base}} = \frac{\text{percent}}{100}[/latex]

 

Section Exercises

Use the Definition of Proportion

In the following exercises, write each sentence as a proportion.

1. 4 is to 15 as 36 is to 135.

Show Solution

[latex]\frac{4}{15} = \frac{36}{135}[/latex]

 

2. 7 is to 9 as 35 is to 45.

 

3. 12 is to 5 as 96 is to 40.
Show Solution
125=9640
4. 15 is to 8 as 75 is to 40. 

5. 5 wins in 7 games is the same as 115 wins in 161 games.

Show Solution
57=115161

 

6. 4 wins in 9 games is the same as 36 wins in 81 games.

7. 8 campers to 1 counselor is the same as 48 campers to 6 counselors.

Show Solution
81=486

 

8. 6 campers to 1 counselor is the same as 48 campers to 8 counselors.

9. $9.36 for 18 ounces is the same as $2.60 for 5 ounces.

Show Solution
9.3618=2.605

 

 

10. 3.92 for 8 ounces is the same as $1.47 for 3 ounces. 

 

11. $18.04 for 11 pounds is the same as $4.92 for 3 pounds. 

Show Solution
18.0411=4.923

 

12. 12.42 for 27 pounds is the same as $5.52 for 12 pounds. 

In the following exercises, determine whether each equation is a proportion.

13.

715=56120
Show Solution

yes

 

14.

512=45108512=45108

 

15.

116=2116116=2116
Show Solution

no

 

16.

94=393494=3934

 

17.

1218=4.997.561218=4.997.56
Show Solution

no

18.

916=2.163.89916=2.163.89

 

19.

13.58.5=31.0519.5513.58.5=31.0519.55

 

Show Solution

yes

 

20.

10.18.4=3.032.5210.18.4=3.032.52

 

Solve Proportions

In the following exercises, solve each proportion.

21.

x56=78x56=78
Show Solution

x = 49

 

 

22.

n91=813n91=813

 

23.

4963=z94963=z9
Show Solution

z = 7

 

24.

5672=y95672=y9

 

25.

5a=65117
Show Solution

a = 9

 

26.

4b=641444b=64144

 

27.

98154=−7p
Show Solution

p = −11

 

28.

72156=−6q72156=−6q

 

29.

a−8=−4248a−8=−4248
Show Solution

a = 7

 

30.

b−7=−3042b−7=−3042

 

31.

2.63.9=c32.63.9=c3
Show Solution

c = 2

 

32.

2.73.6=d42.73.6=d4

 

33.

2.7j=0.90.22.7j=0.90.2
Show Solution

j = 0.6

 

34.

2.8k=2.11.52.8k=2.11.5

 

35.

121=m8121=m8
Show Solution

m = 4

 

36.

133=9n133=9n

 

Solve Applications Using Proportions

In the following exercises, solve the proportion problem.

 

37. Pediatricians prescribe 5  milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 45 pounds?
Show Solution

9 ml

 

38. Brianna, who weighs 6 kg, just received her shots and needs a pain killer. The pain killer is prescribed for children at 15 milligrams (mg) for every 1 kilogram (kg) of the child’s weight. How many milligrams will the doctor prescribe?

 

39. At the gym, Carol takes her pulse for 10 sec and counts 19 beats. How many beats per minute is this? Has Carol met her target heart rate of 140 beats per minute?

Show Solution

114 beats/minute. Carol has not met her target heart rate.

 

40. Kevin wants to keep his heart rate at 160 beats per minute while training. During his workout he counts 27 beats in 10 seconds. How many beats per minute is this? Has Kevin met his target heart rate?

 

41. A new energy drink advertises 106 calories for 8 ounces. How many calories are in 12 ounces of the drink?

Show Solution

159 cal

 

42.

One 12 ounce can of soda has 150 calories. If Josiah drinks the big 32 ounce size from the local mini-mart, how many calories does he get?

 

43.

Karen eats [latex]\frac{1}{2}[/latex] cup of oatmeal that counts for 2 points on her weight loss program. Her husband, Joe, can have 3 points of oatmeal for breakfast. How much oatmeal can he have?

Show Solution
34cup

 

44.

An oatmeal cookie recipe calls for [latex]\frac{1}{2}[/latex] cup of butter to make 4 dozen cookies. Hilda needs to make 10 dozen cookies for the bake sale. How many cups of butter will she need?

 

45.

Janice is traveling to Canada and will change $250 US dollars into Canadian dollars. At the current exchange rate,  $1 US is equal to $1.01 Canadian. How many Canadian dollars will she get for her trip?

Show Solution

$252.50

 

46.

Todd is traveling to Mexico and needs to exchange $450 into Mexican pesos. If each dollar is worth 12.29 pesos, how many pesos will he get for his trip?

 

47.

Steve changed $600 into 480 Euros. How many Euros did he receive per US dollar?

Show Solution

0.8 Euros

 

48.

Martha changed $350 US into 385 Australian dollars. How many Australian dollars did she receive per US dollar?

 

49.

At the laundromat, Lucy changed $12.00 into quarters. How many quarters did she get?

Show Solution

48 quarters

 

50.

When she arrived at a casino, Gerty changed $20 into nickels. How many nickels did she get?

51.

Jesse’s car gets 30 miles per gallon of gas. If Las Vegas is 285 miles away, how many gallons of gas are needed to get there and then home? If gas is $3.09 per gallon, what is the total cost of the gas for the trip?

Show Solution

19 gallons, $58.71

 

52.

Danny wants to drive to Phoenix to see his grandfather. Phoenix is 370 miles from Danny’s home and his car gets 18.5 miles per gallon. How many gallons of gas will Danny need to get to and from Phoenix? If gas is $3.19 per gallon, what is the total cost for the gas to drive to see his grandfather?

53.

Hugh leaves early one morning to drive from his home in Chicago to go to Mount Rushmore, 812 miles away. After 3 hours, he has gone 190 miles. At that rate, how long will the whole drive take?

Show Solution

12.8 hours

 

54.

Kelly leaves her home in Seattle to drive to Spokane, a distance of 280 miles. After 2 hours, she has gone 152 miles. At that rate, how long will the whole drive take?

55.

Phil wants to fertilize his lawn. Each bag of fertilizer covers about 4000 square feet of lawn. Phil’s lawn is approximately 13,500 square feet. How many bags of fertilizer will he have to buy?

Show Solution

4 bags

 

56.

April wants to paint the exterior of her house. One gallon of paint covers about 350 square feet, and the exterior of the house measures approximately 2000 square feet. How many gallons of paint will she have to buy?

Write Percent Equations as Proportions

In the following exercises, translate to a proportion.

57.

What number is 35% of 250?

Show Solution

 

n250=35100

 

58.

What number is 75% of 920?

 

59.

What number is 110% of 47?

Show Solution
n47=110100

60.

What number is 150% of 64?

 

61.

45 is 30% of what number?
Show Solution
45n=30100

 

62.

25 is 80% of what number?

63.

90 us 150% of what number?

Show Solution
90n=150100

 

64.

77 is 110% of what number?

 

65.

What percent of 85 is 17?

Show Solution
1785=p100

66.

What percent of 92 is 46?

67.

What percent of 260 is 340?

Show Solution
340260=p100

 

68.

What percent of 180 is 220?

Translate and Solve Percent Proportions

In the following exercises, translate and solve using proportions.

69.

What number is 65% of 180?

Show Solution
n180=65100

 

70.

What number is 55% of 300?

71.

18% of 92 is what number?

Show Solution
n92=18100

 

72.

22% if 74 is what number?

73.

26

175% of 26 is what number?

Show Solution
n26=175100

 

74.

250% of 61 is what number?

75.

What is 300% of 488?

Show Solution
n488=300100

 

76.

What  is 500% of 315?

315?

77.

17% of what number is $7.65?
Show Solution
7.65n=17100

 

78.

19% of what number is $6.46?

$6.46?

79.

$13.53 is 8.25% of what number?

Show Solution
13.53n=8.25100

 

80.

$18.12 is 7.55% of what number?

 

81.

What percent of 56 is 14?

Show Solution
1456=p100

 

82.

What percent of 80 is 28?

83.

What percent of 96 is 12?

Show Solution
1296=p100

 

84.

What percent of 120 is 27 ?

Everyday Math

85.

Mixing a concentrate Sam bought a large bottle of concentrated cleaning solution at the warehouse store. He must mix the concentrate with water to make a solution for washing his windows. The directions tell him to mix 3 ounces of concentrate with 5 ounces of water. If he puts 12 ounces of concentrate in a bucket, how many ounces of water should he add? How many ounces of the solution will he have altogether?

Show Solution

He must add 20 oz of water to obtain a final solution of 32 oz.

 

86.

Mixing a concentrate Travis is going to wash his car. The directions on the bottle of car wash concentrate say to mix 2 ounces of concentrate with 15 ounces of water. If Travis puts 6 ounces of concentrate in a bucket, how much water must he mix with the concentrate?

Writing Exercises

87.

To solve “what number is 45% of 350″ do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.

Show Solution

Answers will vary.

 

88.

To solve “what percent of 125 is 25″, do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.

 

 

Glossary

proportion
A proportion is an equation of the form \[\frac{a}{b} = \frac{c}{d}\] where [latex]b \neq 0[/latex] and [latex]d \neq 0[/latex]. The proportion states that two ratios or rates are equal. The proportion is read “[latex]a[/latex]  is to  [latex]b[/latex], as  [latex]c [/latex] is to  [latex]d[/latex] ”.

 

License

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0.2 Proportions Copyright © by Karen Perilloux; Prakash Ghimire; and Lauren Johnson is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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