Solve Proportions and their Applications

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.

The equation $\frac{1}{2} = \frac{4}{8}$ is a proportion because the two fractions are equal. The proportion $\frac{1}{2} = \frac{4}{8}$ 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 $\frac{20 students}{1 teacher} = \frac{60 students}{3 teachers}$ we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

Write each sentence as a proportion:

ⓐ $3$ is to $7$ as $15$ is to $35 .$
ⓑ $5$ hits in $8$ at bats is the same as $30$ hits in $48$ at-bats.
ⓒ $$1.50$ for $6$ ounces is equivalent to $$2.25$ for $9$ ounces.

Solution

Example illustrating how to convert a verbal proportion statement into its equivalent mathematical expression.
ⓐ
	3 is to 7 as 15 is to 35.
Write as a proportion.	$\frac{3}{7} = \frac{15}{35}$

Comparison of two equivalent batting ratios (hits to at-bats) demonstrated through fractions and proportions.
ⓑ
	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.	$\frac{hits}{at-bats} = \frac{hits}{at-bats}$
Write as a proportion.	$\frac{5}{8} = \frac{30}{48}$

Illustrates the steps to set up and verify equivalent ratios comparing dollars to ounces, demonstrating a proportion calculation.
ⓒ
	$1.50 for 6 ounces is equivalent to $2.25 for 9 ounces.
Write each fraction to compare dollars to ounces.	$\frac{$}{ounces} = \frac{$}{ounces}$
Write as a proportion.	$\frac{1.50}{6} = \frac{2.25}{9}$

Look at the proportions $\frac{1}{2} = \frac{4}{8}$ and $\frac{2}{3} = \frac{6}{9} .$ 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 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.

Determine whether each equation is a proportion:

ⓐ $\frac{4}{9} = \frac{12}{28}$
ⓑ $\frac{17.5}{37.5} = \frac{7}{15}$

Solution

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

ⓐ
	$A mathematical equation showing the fractions 4/9 equals 12/28. Both fractions are written in black against a plain white background.$
Find the cross products.	$28 \cdot 4 = 112 9 \cdot 12 = 108$ $The image illustrates an inequality between two fractions, 4/9 and 12/28, using cross-multiplication with blue arrows to show that 4 multiplied by 28 is not equal to 9 multiplied by 12.$

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

ⓑ
	$A mathematical equation displays the fraction 17.5/37.5 simplified to 7/15, demonstrating equivalence between the two ratios.$
Find the cross products.	$15 \cdot 17.5 = 262.5 37.5 \cdot 7 = 262.5$

Since the cross products are equal, $15 \cdot 17.5 = 37.5 \cdot 7,$ the equation is 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.

Solve: $\frac{x}{63} = \frac{4}{7} .$

Solution

		$A mathematical equation is displayed, showing a fraction on the left side, x over 63, which is equal to the fraction 4 over 7 on the right side.$
To isolate $x$ , multiply both sides by the LCD, 63.		$A mathematical equation shows 63 multiplied by the fraction x over 63, which equals 63 multiplied by the fraction 4 over 7. The number 63 is highlighted in red on both sides of the equation.$
Simplify.
Divide the common factors.
Check: To check our answer, we substitute into the original proportion.

	$A mathematical equation displays the fraction 36/63, a question mark, and the fraction 4/7, inviting a comparison or to determine if the fractions are equivalent. The number 36 is highlighted in red.$
Show common factors.	$A mathematical expression asks if the fraction (4 times 9) divided by (7 times 9) is equal to 4/7. This illustrates the principle of simplifying fractions by canceling common factors.$
Simplify.

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.

Solve: $\frac{144}{a} = \frac{9}{4} .$

Solution

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


Find the cross products and set them equal.
Simplify.
Divide both sides by 9.		$A mathematical equation shows a fraction 576 over 9 equal to a fraction 9a over 9.$
Simplify.
Check your answer.

	$A mathematical equation shows two fractions, 144/64 and 9/4, with a question mark and an equals sign between them, asking if they are equivalent. The denominator 64 is in red.$
Show common factors..	$A mathematical equation questions if (9 multiplied by 16) divided by (4 multiplied by 16) is equal to 9 divided by 4, illustrating the concept of simplifying fractions by cancelling common factors.$
Simplify.

Another method to solve this would be to multiply both sides by the LCD, $4 a .$ Try it and verify that you get the same solution.

Solve: $\frac{52}{91} = \frac{−4}{y} .$

Solution

Find the cross products and set them equal.

Simplify.
Divide both sides by 52.
Simplify.
Check:
	$A mathematical equation displays the fraction 52 over 91, which is set equal to the fraction -4 over y.$
	$A mathematical expression displaying two fractions, 52/91 and -4/-7, separated by an equals sign with a question mark above it, indicating a query about their equivalence.$
Show common factors.
Simplify.

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.

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?

Solution

This table outlines the sequential steps for calculating acetaminophen dosage using proportions, from problem identification to final solution.
Identify what you are asked to find.	How many ml of acetaminophen the doctor will prescribe
Choose a variable to represent it.	Let $a =$ 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.
Substitute given values—be careful of the units.	$A mathematical equation showing the fraction 5/25 equal to the fraction a/80.$
Multiply both sides by 80.	$A mathematical equation shows '80 multiplied by the fraction 5 over 25' on the left side, which is set equal to '80 multiplied by the fraction a over 80' on the right side.$
Multiply and show common factors.	$A mathematical equation illustrating fraction simplification: (16 x 5 x 5) / (5 x 5) = 80a / 80.$
Simplify.
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.

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?

Solution

Outlines systematic steps to solve a word problem using proportions, illustrated by calculating total calories in a bag of microwave popcorn.
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 $c =$ 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.
Substitute given values.	$A mathematical equation shows a fraction 120 over 1 equal to the fraction c over 3.5. This represents a proportion where 120 divided by 1 is equal to c divided by 3.5.$
Multiply both sides by 3.5.	$An algebraic equation: (3.5)(120/1) = (3.5)(c/3.5), demonstrating the multiplication of a decimal by a fraction on both sides.$
Multiply.
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.

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?

Solution

Step-by-step process for solving a currency conversion problem using proportions, detailing each stage from problem identification to final answer.
Identify what you are asked to find.	How many Mexican pesos did Josiah get?
Choose a variable to represent it.	Let $p =$ 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.
Substitute given values.
The variable is in the denominator, so find the cross products and set them equal.
Simplify.
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.

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, $60% = \frac{60}{100}$ and we can simplify $\frac{60}{100} = \frac{3}{5} .$ Since the equation $\frac{60}{100} = \frac{3}{5}$ shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:

\frac{amount}{base} = \frac{percent}{100}

\frac{3}{5} = \frac{60}{100}

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.

We could also say:

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.

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

Solution

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

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n = number$ .	$\frac{n}{90} = \frac{75}{100}$

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

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n = number$ .	$\frac{19}{n} = \frac{25}{100}$

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

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $p = percent$ .	$\frac{9}{27} = \frac{p}{100}$

Translate and Solve Percent Proportions

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

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

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n =$ number.	$A mathematical equation displays the fraction n over 80 set equal to the fraction 45 over 100, which is commonly used to represent a proportion or percentage problem.$
Find the cross products and set them equal.
Simplify.
Divide both sides by 100.
Simplify.
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.

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.

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

Solution

Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n =$ number.
Find the cross products and set them equal.
Simplify.
Divide both sides by 100.
Simplify.
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.

Percents with decimals and money are also used in proportions.

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

Solution

Step-by-step guide demonstrating how to solve a percentage problem using proportions, illustrating the procedure and mathematical solution for 6.5% of $24.
Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n =$ number.
Find the cross products and set them equal.
Simplify.
Divide both sides by 6.5 to isolate the variable.
Simplify.
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.

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

Solution

A step-by-step guide demonstrating how to solve for a percentage in a proportion problem.
Identify the parts of the percent proportion.
Restate as a proportion.
Set up the proportion. Let $n =$ number.	$A mathematical equation shows the fraction 9/72 equal to the fraction n/100, which can be used to solve for the variable 'n' in a proportion or percentage problem.$
Find the cross products and set them equal.
Simplify.
Divide both sides by 72.	$A mathematical equation is displayed, showing '72n over 72 equals 900 over 72' with fractions represented by horizontal lines.$
Simplify.
Check if the answer is reasonable.
Yes. 9 is $\frac{1}{8}$ of 72 and $\frac{1}{8}$ is 12.5%.
Write a complete sentence that answers the question.	12.5% of 72 is 9.

Key Concepts

Proportion
- A proportion is an equation of the form $\frac{a}{b} = \frac{c}{d}$ , where $b \neq 0$ , $d \neq 0$ .The proportion states two ratios or rates are equal. The proportion is read “ $a$ is to $b$ , as $c$ is to $d$ ”.
Cross Products of a Proportion
- For any proportion of the form $\frac{a}{b} = \frac{c}{d}$ , where $b \neq 0$ , its cross products are equal: $a \cdot d = b \cdot c$ .
Percent Proportion
- The amount is to the base as the percent is to 100. $\frac{amount}{base} = \frac{percent}{100}$

Section Exercises

Practice Makes Perfect

Use the Definition of Proportion

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

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

Solution

$\frac{4}{15} = \frac{36}{135}$

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

$12$ is to $5$ as $96$ is to $40 .$

Solution

$\frac{12}{5} = \frac{96}{40}$

$15$ is to $8$ as $75$ is to $40 .$

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

Solution

$\frac{5}{7} = \frac{115}{161}$

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

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

Solution

$\frac{8}{1} = \frac{48}{6}$

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

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

Solution

$\frac{9.36}{18} = \frac{2.60}{5}$

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

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

Solution

$\frac{18.04}{11} = \frac{4.92}{3}$

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

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

$\frac{7}{15} = \frac{56}{120}$

Solution

yes

$\frac{5}{12} = \frac{45}{108}$

$\frac{11}{6} = \frac{21}{16}$

Solution

$\frac{9}{4} = \frac{39}{34}$

$\frac{12}{18} = \frac{4.99}{7.56}$

Solution

$\frac{9}{16} = \frac{2.16}{3.89}$

$\frac{13.5}{8.5} = \frac{31.05}{19.55}$

Solution

yes

$\frac{10.1}{8.4} = \frac{3.03}{2.52}$

Solve Proportions

In the following exercises, solve each proportion.

$\frac{x}{56} = \frac{7}{8}$

Solution

x = 49

$\frac{n}{91} = \frac{8}{13}$

$\frac{49}{63} = \frac{z}{9}$

Solution

z = 7

$\frac{56}{72} = \frac{y}{9}$

$\frac{5}{a} = \frac{65}{117}$

Solution

a = 9

$\frac{4}{b} = \frac{64}{144}$

$\frac{98}{154} = \frac{−7}{p}$

Solution

p = −11

$\frac{72}{156} = \frac{−6}{q}$

$\frac{a}{−8} = \frac{−42}{48}$

Solution

a = 7

$\frac{b}{−7} = \frac{−30}{42}$

$\frac{2.6}{3.9} = \frac{c}{3}$

Solution

c = 2

$\frac{2.7}{3.6} = \frac{d}{4}$

$\frac{2.7}{j} = \frac{0.9}{0.2}$

Solution

j = 0.6

$\frac{2.8}{k} = \frac{2.1}{1.5}$

$\frac{\frac{1}{2}}{1} = \frac{m}{8}$

Solution

m = 4

$\frac{\frac{1}{3}}{3} = \frac{9}{n}$

Solve Applications Using Proportions

In the following exercises, solve the proportion problem.

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?

Solution

9 ml

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?

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?

Solution

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

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?

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

Solution

159 cal

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?

Karen eats $\frac{1}{2}$ 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?

Solution

$\frac{3}{4} cup$

An oatmeal cookie recipe calls for $\frac{1}{2}$ 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?

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?

Solution

$252.50

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?

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

Solution

0.8 Euros

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

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

Solution

48 quarters

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

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?

Solution

19 gallons, $58.71

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?

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?

Solution

12.8 hours

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?

Phil wants to fertilize his lawn. Each bag of fertilizer covers about $4,000$ square feet of lawn. Phil’s lawn is approximately $13,500$ square feet. How many bags of fertilizer will he have to buy?

Solution

4 bags

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.

What number is $35%$ of $250 ?$

Solution

$\frac{n}{250} = \frac{35}{100}$

What number is $75%$ of $920 ?$

What number is $110%$ of $47 ?$

Solution

$\frac{n}{47} = \frac{110}{100}$

What number is $150%$ of $64 ?$

$45$ is $30%$ of what number?

Solution

$\frac{45}{n} = \frac{30}{100}$

$25$ is $80%$ of what number?

$90$ is $150%$ of what number?

Solution

$\frac{90}{n} = \frac{150}{100}$

$77$ is $110%$ of what number?

What percent of $85$ is $17 ?$

Solution

$\frac{17}{85} = \frac{p}{100}$

What percent of $92$ is $46 ?$

What percent of $260$ is $340 ?$

Solution

$\frac{340}{260} = \frac{p}{100}$

What percent of $180$ is $220 ?$

Translate and Solve Percent Proportions

In the following exercises, translate and solve using proportions.

What number is $65%$ of $180 ?$

Solution

$\frac{n}{180} = \frac{65}{100}$ ; $117$

What number is $55%$ of $300 ?$

$18%$ of $92$ is what number?

Solution

$\frac{n}{92} = \frac{18}{100}$ ; $16.56$

$22%$ of $74$ is what number?

$175%$ of $26$ is what number?

Solution

$\frac{n}{26} = \frac{175}{100}$ ; $45.5$

$250%$ of $61$ is what number?

What is $300%$ of $488 ?$

Solution

$\frac{n}{488} = \frac{300}{100}$ ; $1464$

What is $500%$ of $315 ?$

$17%$ of what number is $$7.65 ?$

Solution

$\frac{7.65}{n} = \frac{17}{100}$ ; $45$

$19%$ of what number is $$6.46 ?$

$$13.53$ is $8.25%$ of what number?

Solution

$\frac{13.53}{n} = \frac{8.25}{100}$ ; $164$

$$18.12$ is $7.55%$ of what number?

What percent of $56$ is $14 ?$

Solution

$\frac{14}{56} = \frac{p}{100}$ ; $25 %$

What percent of $80$ is $28 ?$

What percent of $96$ is $12 ?$

Solution

$\frac{12}{96} = \frac{p}{100}$ ; $12.5 %$

What percent of $120$ is $27 ?$

Everyday Math

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?

Solution

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

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

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.

Solution

Answers will vary.

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.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

A self-assessment checklist for students to rate their understanding of proportions and percentages, with options: Confidently, With some help, or No-I don't get it!

ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?

Chapter Review Exercises

Understand Percent

In the following exercises, write each percent as a ratio.

$32%$ admission rate for the university

Solution

$\frac{32}{100}$

$53.3%$ rate of college students with student loans

In the following exercises, write as a ratio and then as a percent.

$13$ out of $100$ architects are women.

Solution

$\frac{13}{100}, 13%$

$9$ out of every $100$ nurses are men.

In the following exercises, convert each percent to a fraction.

$48%$

Solution

$\frac{12}{25}$

$175%$

$64.1%$

Solution

$\frac{641}{1000}$

$8 \frac{1}{4} %$

In the following exercises, convert each percent to a decimal.

$6%$

Solution

0.06

$23%$

$128%$

Solution

1.28

4.9%

In the following exercises, convert each percent to ⓐ a simplified fraction and ⓑ a decimal.

In $2012, 13.5%$ of the United States population was age $65$ or over. (Source: www.census.gov)

Solution

ⓐ $\frac{27}{200}$
ⓑ $0.135$

In $2012, 6.5%$ of the United States population was under $5$ years old. (Source: www.census.gov)

When a die is tossed, the probability it will land with an even number of dots on the top side is $50% .$

Solution

ⓐ $\frac{1}{2}$
ⓑ $0.5$

A couple plans to have three children. The probability they will all be girls is $12.5% .$

In the following exercises, convert each decimal to a percent.

$0.04$

Solution

$0.15$

$2.82$

Solution

282%

$3$

$0.003$

Solution

0.3%

$1.395$

In the following exercises, convert each fraction to a percent.

$\frac{3}{4}$

Solution

75%

$\frac{11}{5}$

$3 \frac{5}{8}$

Solution

362.5%

$\frac{2}{9}$

According to the Centers for Disease Control, $\frac{2}{5}$ of adults do not take a vitamin or supplement.

Solution

40%

According to the Centers for Disease Control, among adults who do take a vitamin or supplement, $\frac{3}{4}$ take a multivitamin.

In the following exercises, translate and solve.

What number is $46%$ of $350 ?$

Solution

161

$120%$ of $55$ is what number?

$84$ is $35%$ of what number?

Solution

240

$15$ is $8%$ of what number?

$200%$ of what number is $50 ?$

Solution

$7.9%$ of what number is $$4.74 ?$

What percent of $120$ is $81.6 ?$

Solution

68%

What percent of $340$ is $595 ?$

Solve General Applications of Percents

In the following exercises, solve.

When Aurelio and his family ate dinner at a restaurant, the bill was $$83.50 .$ Aurelio wants to leave $20%$ of the total bill as a tip. How much should the tip be?

Solution

$16.70

One granola bar has $2$ grams of fiber, which is $8%$ of the recommended daily amount. What is the total recommended daily amount of fiber?

The nutrition label on a package of granola bars says that each granola bar has $190$ calories, and $54$ calories are from fat. What percent of the total calories is from fat?

Solution

28.4%

Elsa gets paid $$4,600$ per month. Her car payment is $$253 .$ What percent of her monthly pay goes to her car payment?

In the following exercises, solve.

Jorge got a raise in his hourly pay, from $$19.00$ to $$19.76 .$ Find the percent increase.

Solution

Last year Bernard bought a new car for $$30,000 .$ This year the car is worth $$24,000 .$ Find the percent decrease.

Solve Sales Tax, Commission, and Discount Applications

In the following exercises, find ⓐ the sales tax ⓑ the total cost.

The cost of a lawn mower was $$750 .$ The sales tax rate is $6%$ of the purchase price.

Solution

ⓐ $45
ⓑ $795

The cost of a water heater is $$577 .$ The sales tax rate is $8.75%$ of the purchase price.

In the following exercises, find the sales tax rate.

Andy bought a piano for $$4,600 .$ The sales tax on the purchase was $$333.50 .$

Solution

7.25%

Nahomi bought a purse for $$200 .$ The sales tax on the purchase was $$16.75 .$

In the following exercises, find the commission.

Ginny is a realtor. She receives $3%$ commission when she sells a house. How much commission will she receive for selling a house for $$380,000 ?$

Solution

$11,400

Jackson receives $16.5%$ commission when he sells a dinette set. How much commission will he receive for selling a dinette set for $$895 ?$

In the following exercises, find the rate of commission.

Ruben received $$675$ commission when he sold a $$4,500$ painting at the art gallery where he works. What was the rate of commission?

Solution

15%

Tori received $$80.75$ for selling a $$950$ membership at her gym. What was her rate of commission?

In the following exercises, find the sale price.

Aya bought a pair of shoes that was on sale for $$30$ off. The original price of the shoes was $$75 .$

Solution

$45

Takwanna saw a cookware set she liked on sale for $$145$ off. The original price of the cookware was $$312 .$

In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.

Nga bought a microwave for her office. The microwave was discounted $30%$ from an original price of $$84.90 .$

Solution

ⓐ $25.47
ⓑ $59.43

Jarrett bought a tie that was discounted $65%$ from an original price of $$45 .$

In the following exercises, find ⓐ the amount of discount ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)

Hilda bought a bedspread on sale for $$37 .$ The original price of the bedspread was $$50 .$

Solution

ⓐ $13
ⓑ 26%

Tyler bought a phone on sale for $$49.99 .$ The original price of the phone was $$79.99 .$

In the following exercises, find

ⓐ the amount of the mark-up
ⓑ the list price

Manny paid $$0.80$ a pound for apples. He added $60%$ mark-up before selling them at his produce stand. What price did he charge for the apples?

Solution

ⓐ $0.48
ⓑ $1.28

It cost Noelle $$17.40$ for the materials she used to make a purse. She added a $325%$ mark-up before selling it at her friend’s store. What price did she ask for the purse?

Solve Simple Interest Applications

In the following exercises, solve the simple interest problem.

Find the simple interest earned after $4$ years on $$2,250$ invested at an interest rate of $5% .$

Solution

$450

Find the simple interest earned after $7$ years on $$12,000$ invested at an interest rate of $8.5% .$

Find the principal invested if $$660$ interest was earned in $5$ years at an interest rate of $3% .$

Solution

$4400

Find the interest rate if $$2,898$ interest was earned from a principal of $$23,000$ invested for $3$ years.

Kazuo deposited $$10,000$ in a bank account with interest rate $4.5% .$ How much interest was earned in $2$ years?

Solution

$900

Brent invested $$23,000$ in a friend’s business. In $5$ years the friend paid him the $$23,000$ plus $$9,200$ interest. What was the rate of interest?

Fresia lent her son $$5,000$ for college expenses. Three years later he repaid her the $$5,000$ plus $$375$ interest. What was the rate of interest?

Solution

2.5%

In $6$ years, a bond that paid $5.5%$ earned $$594$ interest. What was the principal of the bond?

Solve Proportions and their Applications

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

$3$ is to $8$ as $12$ is to $32 .$

Solution

$\frac{3}{8} = \frac{12}{32}$

$95$ miles to $3$ gallons is the same as $475$ miles to $15$ gallons.

$1$ teacher to $18$ students is the same as $23$ teachers to $414$ students.

Solution

$\frac{1}{18} = \frac{23}{414}$

$$7.35$ for $15$ ounces is the same as $$2.94$ for $6$ ounces.

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

$\frac{5}{13} = \frac{30}{78}$

Solution

yes

$\frac{16}{7} = \frac{48}{23}$

$\frac{12}{18} = \frac{6.99}{10.99}$

Solution

$\frac{11.6}{9.2} = \frac{37.12}{29.44}$

In the following exercises, solve each proportion.

$\frac{x}{36} = \frac{5}{9}$

Solution

$\frac{7}{a} = \frac{−6}{84}$

$\frac{1.2}{1.8} = \frac{d}{6}$

Solution

$\frac{\frac{1}{2}}{2} = \frac{m}{20}$

In the following exercises, solve the proportion problem.

The children’s dosage of acetaminophen is $5$ milliliters (ml) for every $25$ pounds of a child’s weight. How many milliliters of acetaminophen will be prescribed for a $60$ pound child?

Solution

12 ml

After a workout, Dennis takes his pulse for $10$ sec and counts $21$ beats. How many beats per minute is this?

An $8$ ounce serving of ice cream has $272$ calories. If Lavonne eats $10$ ounces of ice cream, how many calories does she get?

Solution

340 calories

Alma is going to Europe and wants to exchange $$1,200$ into Euros. If each dollar is $0.75$ Euros, how many Euros will Alma get?

Zack wants to drive from Omaha to Denver, a distance of $494$ miles. If his car gets $38$ miles to the gallon, how many gallons of gas will Zack need to get to Denver?

Solution

13 gallons

Teresa is planning a party for $100$ people. Each gallon of punch will serve $18$ people. How many gallons of punch will she need?

In the following exercises, translate to a proportion.

What number is $62%$ of $395 ?$

Solution

$\frac{n}{395} = \frac{62}{100}$

$42$ is $70%$ of what number?

What percent of $1,000$ is $15 ?$

Solution

$\frac{15}{1000} = \frac{p}{100}$

What percent of $140$ is $210 ?$

In the following exercises, translate and solve using proportions.

What number is $85%$ of $900 ?$

Solution

$\frac{n}{900} = \frac{85}{100}, 765$

$6%$ of what number is $$24 ?$

$$3.51$ is $4.5%$ of what number?

Solution

$\frac{3.51}{n} = \frac{4.5}{100}, $78$

What percent of $3,100$ is $930 ?$

In the following exercises, convert each percent to ⓐ a decimal ⓑ a simplified fraction.

$24%$

Solution

$0.24, \frac{6}{25}$

$5%$

$350%$

Solution

$3.5, \frac{7}{2}$

In the following exercises, convert each fraction to a percent. (Round to $3$ decimal places if needed.)

$\frac{7}{8}$

$\frac{1}{3}$

Solution

$33. \bar{3} % o r 33 \frac{1}{3} %$

$\frac{11}{12}$

In the following exercises, solve the percent problem.

$65$ is what percent of $260 ?$

Solution

25%

What number is $27%$ of $3,000 ?$

$150%$ of what number is $60 ?$

Solution

Yuki’s monthly paycheck is $$3,825 .$ She pays $$918$ for rent. What percent of her paycheck goes to rent?

The total number of vehicles on one freeway dropped from $84,000$ to $74,000 .$ Find the percent decrease (round to the nearest tenth of a percent).

Solution

11.9%

Kyle bought a bicycle in Denver where the sales tax was $7.72%$ of the purchase price. The purchase price of the bicycle was $$600 .$ What was the total cost?

Mara received $$31.80$ commission when she sold a $$795$ suit. What was her rate of commission?

Solution

Kiyoshi bought a television set on sale for

$899 .

The original price was

$1,200 .

Find:

ⓐ the amount of discount
ⓑ the discount rate (round to the nearest tenth of a percent)

Oxana bought a dresser at a garage sale for $$20 .$ She refinished it, then added a $250%$ markup before advertising it for sale. What price did she ask for the dresser?

Solution

$70

Find the simple interest earned after $5$ years on $$3000$ invested at an interest rate of $4.2% .$

Brenda borrowed $$400$ from her brother. Two years later, she repaid the $$400$ plus $$50$ interest. What was the rate of interest?

Solution

6.25%

Write as a proportion: $4$ gallons to $144$ miles is the same as $10$ gallons to $360$ miles.

Solve for a: $\frac{12}{a} = \frac{−15}{65}$

Solution

−52

Vin read $10$ pages of a book in $12$ minutes. At that rate, how long will it take him to read $35$ pages?

proportion

A proportion is an equation of the form

\frac{a}{b} = \frac{c}{d}

, where

b \neq 0

d \neq 0

.The proportion states two ratios or rates are equal. The proportion is read “

a

is to

b

, as

c

is to

d

”.