5.5: Ratios and Proportions

Figure 5.15 This bar graph shows popular social media app usage. (Source)

Learning Objectives

After completing this section, you should be able to:

1. Construct ratios to express comparison of two quantities.
2. Use and apply proportional relationships to solve problems.
3. Determine and apply a constant of proportionality.
4. Use proportions to solve scaling problems.

Ratios and proportions are used in a wide variety of situations to make comparisons. For example, using the information from Figure 5.15, we can see that the number of Facebook users compared to the number of Twitter users is 2,006 M to 328 M. Note that the "M" stands for million, so 2,006 million is actually 2,006,000,000 and 328 million is 328,000,000. Similarly, the number of Qzone users compared to the number of Pinterest users is in a ratio of 632 million to 175 million. These types of comparisons are ratios.

Constructing Ratios to Express Comparison of Two Quantities

Note there are three different ways to write a ratio, which is a comparison of two numbers that can be written as: /**/(\$ 1 =0.82\,{€})/**/, how many dollars should you receive? Round to the nearest cent if necessary.

Example 5.31

Solving a Proportion Involving Weights on Different Planets

A person who weighs 170 pounds on Earth would weigh 64 pounds on Mars. How much would a typical racehorse (1,000 pounds) weigh on Mars? Round your answer to the nearest tenth.

Answer

Step 1: Set up the two ratios into a proportion. Notice the Earth weights are both in the numerator and the Mars weights are both in the denominator.

$$\frac{170}{64}=\frac{\mathrm{1,000}}{x}$$

Step 2: Cross multiply, and then divide to solve.

$$\begin{array}{ccc}\hfill 170x& \hfill =\hfill & \mathrm{1,000}\left(64\right)\hfill \\ \hfill 170x& \hfill =\hfill & \mathrm{64,000}\hfill \\ \hfill \frac{170x}{170}& \hfill =\hfill & \frac{\mathrm{64,000}}{170}\hfill \\ \hfill x& \hfill =\hfill & 376.5\hfill \end{array}$$

So the 1,000-pound horse would weigh about 376.5 pounds on Mars.

Your Turn 5.31

1.

A person who weighs 200 pounds on Earth would weigh only 33 pounds on the moon. A 2021 Toyota Prius weighs 3,040 pounds on Earth; how much would it weigh on the moon? Round to the nearest tenth if necessary.

Example 5.32

Solving a Proportion Involving Baking

A cookie recipe needs /**/1{\text{ inch}} =/**/ how many miles). Then use that scale to determine the approximate lengths of the other borders of the state of Wyoming.

Example 5.38

Solving a Scaling Problem Involving Model Cars

Die-cast NASCAR model cars are said to be built on a scale of 1:24 when compared to the actual car. If a model car is 9 inches long, how long is a real NASCAR automobile? Write your answer in feet.

Answer

The scale tells us that 1 inch of the model car is equal to 24 inches (2 feet) on the real automobile. So set up the two ratios into a proportion. Notice that the model lengths are both in the numerator and the NASCAR automobile lengths are both in the denominator.

$$\begin{array}{ccc}\hfill \frac{1}{24}& \hfill =\hfill & \frac{9}{x}\hfill \\ \hfill 24(9)& \hfill =\hfill \\ x\hfill \\ \hfill 216\\ \hfill =\hfill & x\hfill \end{array}$$

This amount (216) is in inches. To convert to feet, divide by 12, because there are 12 inches in a foot (this conversion from inches to feet is really another proportion!). The final answer is:

$$\frac{216}{12}=18$$

The NASCAR automobile is 18 feet long.

Your Turn 5.38

1.

A toy Jeep is built on a /**/1:16/**/ scale. The website for the toy Jeep says the toy is /**/11.5/**/ inches long. Based on this, how long is the real Jeep?

Check Your Understanding

32.

If /**/a{:}b = c{:}d/**/, then /**/b{:}a = d{:}c/**/ for all non-zero whole numbers /**/a/**/, /**/b/**/, /**/c/**/, and /**/d/**/.

1. True
2. False

33.

If the ratio of wolves to rabbits in a national park is /**/3{:}5/**/, then the ratio of rabbits to (wolves and rabbits) is /**/5{:}8/**/.

1. True
2. False

34.

All fractions are ratios but not all ratios are fractions.

1. True
2. False

35.

In the following equation,/**/\frac{x}

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= \frac2090 + 4/**/, cross multiplication can be used as the first step towards solving for /**/x/**/.

1. True
2. False

36.

All fractions are ratios but not all ratios are fractions.

1. True
2. False

37.

There are 16 math majors and 12 non-math majors in Ms. Kraft’s class. What is not a correct way to express the ratio of math majors to non-math majors?

/**/16:12/**/	/**/12:16/**/	/**/4:3/**/

38.

There are 16 math majors and 12 non-math majors in Ms. Kraft’s class. What shows the ratio of math majors to all the students in Ms. Kraft’s class?

/**/16:12/**/

/**/12:16/**/

/**/16:28/**/

/**/28:12/**/

None of these

39.

One U.S. dollar is worth /**/0.72/**/ British pounds. Damon is traveling to Great Britain and wishes to exchange $450 U.S. dollars for British pounds. How many British pounds should Damon get in return?

625

6,250

3,456

345.6

None of these

40.

The HO scale for model trains is the most common size of model trains. This scale is /**/1:87/**/. If a real locomotive is 73 feet long, how long should the model locomotive be (in inches)? Round your answer to the nearest inch.

41.

Albert’s Honda Civic gets 37 miles per gallon of gasoline. The gas tank on the Civic can hold /**/13.5/**/ gallons of gas. Albert is driving from Tucson, Arizona to Los Angeles, California, a distance of 485 miles. Albert thinks he can make it on one full tank of gasoline. Can he? Explain.

42.

The average price of a gallon of regular gasoline in the California on July 1, 2021 was /**/$4.28/**/ per gallon. Albert stops at a gas station in California and puts 9.5 gallons of gasoline into his Civic. How much did he pay for the gas?

Section 5.4 Exercises

For the following exercises, use this scenario: Kelly opened a bag of colored chocolate coated candies and counted the number of each color of candy. She found she had 9 green, 4 yellow, 13 black, 11 orange, 8 blue, and 7 red. What is the ratio of the following candy colors?

1.

Red candies to green candies

2.

Green candies to black candies

3.

Yellow candies to black candies

4.

Black candies to blue candies

5.

Orange candies to non-orange candies

6.

Yellow candies to non-yellow candies

7.

Red candies to all candies

8.

Pink candies to all candies

9.

Candies with the letter ‘r’ in their name to all candies

10.

Candies with the letter ‘r’ in their name to candies without the letter ‘r’ in their name

For the following exercises, solve each proportion for the unknown variable.

11.

/**/\frac{x}

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= \frac{4}{8}/**/

12.

/**/\frac27{x} = \frac3.514/**/

13.

/**/\frac16{7} = \frac{x}14/**/

14.

/**/\frac{1}0.8 = \frac{x}514/**/

15.

/**/\frac{203}

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= \frac{10}22/**/

16.

/**/\frac1{29.5} = \frac16

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/**/

17.

/**/\frac14

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= \frac{a}19/**/

18.

/**/\frac13

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= \frac{s}2961/**/

19.

/**/\frac{1}{k} = \frac69111/**/ (Round answer to the nearest hundredth.)

20.

/**/\frac{p}

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= \frac22{7}/**/ (Round answer to the nearest hundredth.)

21.

Pet Paradise has 20 cats and 16 dogs. Animal Acres has 15 cats. How many dogs must be at Animal Acres so that Pet Paradise and Animal Acres have the same ratio of cats to dogs?

22.

Pet Paradise has 20 cats and 16 dogs. Critter Corral has 28 dogs. How many cats must be at Critter Corral so that Pet Paradise and Critter Corral have the same ratio of cats to dogs?

23.

A high school has 960 students. The ratio of students to high school teachers is /**/16:1/**/. How many high school teachers are at the school?

24.

A high school has 960 students. The ratio of students to high school teachers is /**/16:1/**/. How many more teachers are needed to have a /**/12:1/**/ ratio at the high school of students to teachers?

25.

One U.S. dollar is worth $1.23 Canadian dollars. Bernice is traveling to Canada and wants to convert $550 U.S. to Canadian money. How much in Canadian money should she receive?

26.

One U.S. dollar is worth $1.23 Canadian dollars. Rene is traveling from Canada to the United States and wants to convert $550 of Canadian money to U.S. money. How much in U.S. money should he receive? Round your answer to the nearest cent.

27.

One U.S. dollar is worth $1.23 Canadian dollars. What is one Canadian dollar worth in U.S. funds? Round your answer to the nearest cent.

28.

A salad recipe needs one cup of crushed almonds. It will serve eight people. Rashida needs to make a salad to serve 20 people. How many cups of crushed almonds does she need?

29.

A salad recipe needs one cup of crushed almonds. It will serve eight people. Elmer has 4.75 cups of crushed almonds. If he uses all of the crushed almonds he has to make this salad, how many people will it serve?

30.

Jorge is 6 feet tall and casts a 7-foot shadow. At the same time, a nearby tree has a shadow of 56 feet. How tall is the tree?

31.

Tony can run 4 kilometers in 30 minutes. At that rate, how far could he run in 1 hour, 45 minutes?

32.

Kara’s parent owns a restaurant. When she came in one day, they asked her to figure out how much they were spending per ounce on steak they were buying from a vendor. They had their last four receipts, but unfortunately they spilled liquid on them and some parts were unreadable. Find out how much Kara’s parent is spending per ounce on steak; then use that information to fill in the unreadable parts of the receipts (labeled /**/a/**/, /**/b/**/, and /**/c/**/ below).

Receipt	1	2	3	4
Ounces	128	460	/**/b/**/	541
Cost	$163.84	/**/a/**/	$277.76	/**/c/**/

33.

The scale for a map reads “/**/{\text{1 inch = 250 miles}}/**/.” You measure the distance on the map from Fargo, North Dakota to Winnipeg, Manitoba and get /**/1.44/**/ inches. How far is it from Fargo to Winnipeg?

34.

Hot Wheels toy cars are said to be built on a scale of /**/1{:}64/**/ when compared to the actual car. If a real car is 18 feet long, how long should the Hot Wheels toy car be (in inches)?

35.

The Eiffel Tower in Paris, France, is 1,067 feet tall. The replica Eiffel Tower in Las Vegas, Nevada, is built on the scale of /**/1.976{:}1/**/. How tall is the replica Eiffel Tower in Las Vegas? Round your answer to the nearest foot.