6.5 Solve Proportions and their Applications

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.

The equation 12=48 is a proportion because the two fractions are equal. The proportion 1/2=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 20 students1 teacher=60 students3 teachers we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

Look at the proportions 1/2=4/8 and 2/3=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 products because of the cross formed. The cross products of a proportion are 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 the equal, we have 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.

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 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.

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.

Translate and Solve Percent Proportions

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

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

Section 6.5 Exercises

Practice Makes Perfect

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.
  2. 7 is to 9 as 35 is to 45.
  3. 12 is to 5 as 96 is to 40.
  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.
  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.
  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.
  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.
  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
  14. 512=45108
  15. 116=2116
  16. 94=3934
  17. 1218=4.997.56
  18. 916=2.163.89
  19. 13.58.5=31.0519.55
  20. 10.18.4=3.032.52
    Solve Proportions

In the following exercises, solve each proportion.

  1. x56=78
  2. n91=813
  3. 4963=z9
  4. 5672=y9
  5. 5a=65117
  6. 4b=64144
  7. 98154=−7p
  8. 72156=−6q
  9. a−8=−4248
  10. b−7=−3042
  11. 2.63.9=c3
  12. 2.73.6=d4
  13. 2.7j=0.90.2
  14. 2.8k=2.11.5
  15. 121=m8
  16. 133=9n
    Solve Applications Using Proportions

In the following exercises, solve the proportion problem.

  1. 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?
  2. 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?
  3. 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?
  4. 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?
  5. A new energy drink advertises 106 calories for 8 ounces. How many calories are in 12 ounces of the drink?
  6. 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?
  7. Karen eats 12 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?
  8. An oatmeal cookie recipe calls for 12 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?
  9. 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?
  10. 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?
  11. Steve changed $600 into 480 Euros. How many Euros did he receive per US dollar?
  12. Martha changed $350 US into 385 Australian dollars. How many Australian dollars did she receive per US dollar?
  13. At the laundromat, Lucy changed $12.00 into quarters. How many quarters did she get?
  14. When she arrived at a casino, Gerty changed $20 into nickels. How many nickels did she get?
  15. 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?
  16. 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?
  17. 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?
  18. 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?
  19. 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?
  20. 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.

  1. What number is 35% of 250?
  2. What number is 75% of 920?
  3. What number is 110% of 47?
  4. What number is 150% of 64?
  5. 45 is 30% of what number?
  6. 25 is 80% of what number?
  7. 90 is 150% of what number?
  8. 77 is 110% of what number?
  9. What percent of 85 is 17?
  10. What percent of 92 is 46?
  11. What percent of 260 is 340?
  12. What percent of 180 is 220?
    Translate and Solve Percent Proportions

In the following exercises, translate and solve using proportions.

  1. What number is 65% of 180?
  2. What number is 55% of 300?
  3. 18% of 92 is what number?
  4. 22% of 74 is what number?
  5. 175% of 26 is what number?
  6. 250% of 61 is what number?
  7. What is 300% of 488?
  8. What is 500% of 315?
  9. 17% of what number is $7.65?
  10. 19% of what number is $6.46?
  11. $13.53 is 8.25% of what number?
  12. $18.12 is 7.55% of what number?
  13. What percent of 56 is 14?
  14. What percent of 80 is 28?
  15. What percent of 96 is 12?
  16. What percent of 120 is 27?

Everyday Math

327

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 33 ounces of concentrate with 55 ounces of water. If he puts 1212 ounces of concentrate in a bucket, how many ounces of water should he add? How many ounces of the solution will he have altogether?328. 

Mixing a concentrate Travis is going to wash his car. The directions on the bottle of car wash concentrate say to mix 22 ounces of concentrate with 1515 ounces of water. If Travis puts 66 ounces of concentrate in a bucket, how much water must he mix with the concentrate?

Writing Exercises

329

To solve “what number is 45%45% of 350”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.330. 

To solve “what percent of 125125 is 25”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.

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