3.3 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
When two ratios or rates are equal, the equation relating them is called a proportion.
Proportion
A proportion is an equation of the form , where .
The proportion states two ratios or rates are equal. The proportion is read is to , as is to
The equation is a proportion because the two fractions are equal. The proportion is read is to as is to
If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.
EXAMPLE 1
Write each sentence as a proportion:
- is to as is to .
- hits in at bats is the same as hits in at-bats.
- for ounces is equivalent to for ounces.
| a) | 3 is to 7 as 15 is to 35. |
| Write as a proportion. |
| b) | 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. | |
| Write as a proportion. |
| c) | $1.50 for 6 ounces is equivalent to $2.25 for 9 ounces. |
| Write each fraction to compare dollars to ounces. | |
| Write as a proportion. |
TRY IT 1.1
Write each sentence as a proportion:
- is to as is to .
- hits in at-bats is the same as hits in at-bats.
- for ounces is equivalent to for ounces.
Answer
TRY IT 1.2
Write each sentence as a proportion:
-
- is to as is to .
- adults for children is the same as adults for children.
- for ounces is equivalent to for ounces.
Answer
Look at the proportions and . 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.
Cross Products of a Proportion
For any proportion of the form , where , its cross products are equal.
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.
EXAMPLE 2
Determine whether each equation is a proportion:
To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.
| a) | |
| Find the cross products. |
Since the cross products are not equal, , the equation is not a proportion.
| b) | |
| Find the cross products. |
Since the cross products are equal, , the equation is a proportion.
TRY IT 2.1
Determine whether each equation is a proportion:
Answer
- no
- yes
TRY IT 2.2
Determine whether each equation is a proportion:
Answer
- no
- no
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 3
Solve: .
| Given fraction: | |
| To isolate , multiply both sides by the LCD, 63. | |
| Simplify. | |
| Divide the common factors. | |
| Check: To check our answer, we substitute into the original proportion. | |
| Substitute | |
| Show common factors. | |
| Simplify. |
TRY IT 3.1
Solve the proportion: .
Answer
77
TRY IT 3.2
Solve the proportion: .
Answer
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 4
Solve: .
Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.
| Given fraction | |
| Find the cross products and set them equal. | |
| Simplify. | |
| Divide both sides by 9. | |
| Simplify. | |
| Check your answer: | |
| Substitute | |
| Show common factors. | |
| Simplify. |
Another method to solve this would be to multiply both sides by the LCD, . Try it and verify that you get the same solution.
TRY IT 4.1
Solve the proportion: .
Answer
65
TRY IT 4.2
Solve the proportion: .
Answer
24
EXAMPLE 5
Solve: .
| Find the cross products and set them equal. | |
| Simplify. | |
| Divide both sides by 52. | |
| Simplify. | |
| Check: | |
| Substitute | |
| Show common factors. | |
| Simplify. |
TRY IT 5.1
Solve the proportion: .
Answer
−7
TRY IT 5.2
Solve the proportion: .
Answer
−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 6
When pediatricians prescribe acetaminophen to children, they prescribe millilitres (ml) of acetaminophen for every pounds of the child’s weight. If Zoe weighs pounds, how many millilitres of acetaminophen will her doctor prescribe?
| Identify what you are asked to find. | How many ml of acetaminophen the doctor will prescribe |
| Choose a variable to represent it. | Let 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. | |
| Multiply both sides by 80. | |
| Multiply and show common factors. | |
| 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.
TRY IT 6.1
Pediatricians prescribe millilitre s (ml) of acetaminophen for every pounds of a child’s weight. How many millilitre s of acetaminophen will the doctor prescribe for Emilia, who weighs pounds?
Answer
12 ml
TRY IT 6.2
For every kilogram (kg) of a child’s weight, pediatricians prescribe milligrams (mg) of a fever reducer. If Isabella weighs kg, how many milligrams of the fever reducer will the pediatrician prescribe?
Answer
180 mg
EXAMPLE 7
One brand of microwave popcorn has calories per serving. A whole bag of this popcorn has servings. How many calories are in a whole bag of this 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 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. | |
| Multiply both sides by . | |
| Simplify. | |
| 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 7.1
Marissa loves the Caramel Macchiato at the coffee shop. The oz. medium size has calories. How many calories will she get if she drinks the large oz. size?
Answer
300
TRY IT 7.2
Yaneli loves Starburst candies, but wants to keep her snacks to calories. If the candies have calories for pieces, how many pieces can she have in her snack?
Answer
5
EXAMPLE 8
Josiah went to Mexico for spring break and changed dollars into Mexican pesos. At that time, the exchange rate had U.S. is equal to Mexican pesos. How many Mexican pesos did he get for his trip?
| Identify what you are asked to find. | How many Mexican pesos did Josiah get? |
| Choose a variable to represent it. | Let 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 4075.50 pesos for his spring break trip. |
TRY IT 8.1
Yurianna is going to Europe and wants to change dollars into Euros. At the current exchange rate, Canadian dollar is equal to Euro. How many Euros will she have for her trip?
Answer
520 Euros
TRY IT 8.2
Corey and Nicole are traveling to Japan and need to exchange CAD into Japanese yen. If each dollar is yen, how many yen will they get?
Answer
54,162 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, and we can simplify . Since the equation shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:
Percent Proportion
The amount is to the base as the percent is to .
If we restate the problem in the words of a proportion, it may be easier to set up the proportion:
We could also say:
First we will practice translating into a percent proportion. Later, we’ll solve the proportion.
EXAMPLE 9
Translate to a proportion. What number is of
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. | What number out 90 is the same as 75 out of 100? |
| Set up the proportion. Let . |
TRY IT 9.1
Translate to a proportion: What number is of
Answer
TRY IT 9.2
Translate to a proportion: What number is of
Answer
EXAMPLE 10
Translate to a proportion. is of what number?
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | 19 out what number is the same as 25 out of 100? |
| Set up the proportion. Let . |
TRY IT 10.1
Translate to a proportion: is of what number?
Answer
TRY IT 10.2
Translate to a proportion: is of what number?
Answer
EXAMPLE 11
Translate to a proportion. What percent of is
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | 9 out 27 is the same as what number out of 100? |
| Set up the proportion. Let . |
TRY IT 11.1
Translate to a proportion: What percent of is
Answer
TRY IT 11.2
Translate to a proportion: What percent of is
Answer
Translate and Solve Percent Proportions
Now that we have written percent equations as proportions, we are ready to solve the equations.
EXAMPLE 12
Translate and solve using proportions: What number is of
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | What number out of 80 is the same as 45 out of 100? |
| Set up the proportion. Let number. | |
| 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. |
TRY IT 12.1
Translate and solve using proportions: What number is of
Answer
26
TRY IT 12.2
Translate and solve using proportions: What number is of
Answer
34
In the next example, the percent is more than , which is more than one whole. So the unknown number will be more than the base.
EXAMPLE 13
Translate and solve using proportions: of is what number?
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | What number out of 25 is the same as 125 out of 100? |
| Set up the proportion. Let 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. |
TRY IT 13.1
Translate and solve using proportions: of is what number?
Answer
80
TRY IT 13.2
Translate and solve using proportions: of is what number?
Answer
147
Percents with decimals and money are also used in proportions.
EXAMPLE 14
Translate and solve: of what number is
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | out of what number is the same as out of |
| Set up the proportion. Let 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. |
TRY IT 14.1
Translate and solve using proportions: of what number is
Answer
38
TRY IT 14.2
Translate and solve using proportions: of what number is
Answer
64
EXAMPLE 15
Translate and solve using proportions: What percent of is
| Identify the parts of the percent proportion. | |
| Restate as a proportion. | 9 out of 72 is the same as what number out of 100? |
| Set up the proportion. Let number. | |
| Find the cross products and set them equal. | |
| Simplify. | |
| Divide both sides by 72. | |
| Simplify. | |
| Check if the answer is reasonable. | Yes. 9 is of 72 and is 12.5%. |
| Write a complete sentence that answers the question. | 12.5% of 72 is 9. |
TRY IT 15.1
Translate and solve using proportions: What percent of is
Answer
37.5%
TRY IT 15.2
Translate and solve using proportions: What percent of is
Answer
25%
Key Concepts
- Proportion
- A proportion is an equation of the form , where , .The proportion states two ratios or rates are equal. The proportion is read “ is to , as is to ”.
- Cross Products of a Proportion
- For any proportion of the form , where , its cross products are equal: .
- Percent Proportion
- The amount is to the base as the percent is to 100.
Glossary
- proportion
- A proportion is an equation of the form , where , .The proportion states two ratios or rates are equal. The proportion is read “ is to , as is to ”.
Practice Makes Perfect
Use the Definition of Proportion
In the following exercises, write each sentence as a proportion.
| 1. is to as is to . | 2. is to as is to . |
| 3. is to as is to . | 4. is to as is to . |
| 5. wins in games is the same as wins in games. | 6. wins in games is the same as wins in games. |
| 7. campers to counsellor is the same as campers to counsellors. | 8. campers to counselor is the same as campers to counselors. |
| 9. for ounces is the same as for ounces. | 10. for ounces is the same as for ounces. |
| 11. for pounds is the same as for pounds. | 12. for pounds is the same as for pounds. |
In the following exercises, determine whether each equation is a proportion.
| 13. | 14. |
| 15. | 16. |
| 17. | 18. |
| 19. | 20. |
Solve Proportions
In the following exercises, solve each proportion.
| 21. | 22. |
| 23. | 24. |
| 25. | 26. |
| 27. | 28. |
| 29. | 30. |
| 31. | 32. |
| 33. | 34. |
| 35. | 36. |
Solve Applications Using Proportions
In the following exercises, solve the proportion problem.
| 37. Pediatricians prescribe millilitre s (ml) of acetaminophen for every pounds of a child’s weight. How many millilitres of acetaminophen will the doctor prescribe for Jocelyn, who weighs pounds? | 38. Brianna, who weighs kg, just received her shots and needs a pain killer. The pain killer is prescribed for children at milligrams (mg) for every kilogram (kg) of the child’s weight. How many milligrams will the doctor prescribe? |
| 39. At the gym, Carol takes her pulse for sec and counts beats. How many beats per minute is this? Has Carol met her target heart rate of beats per minute? | 40. Kevin wants to keep his heart rate at beats per minute while training. During his workout he counts beats in seconds. How many beats per minute is this? Has Kevin met his target heart rate? |
| 41. A new energy drink advertises calories for ounces. How many calories are in ounces of the drink? | 42. One ounce can of soda has calories. If Josiah drinks the big ounce size from the local mini-mart, how many calories does he get? |
| 43. Karen eats cup of oatmeal that counts for points on her weight loss program. Her husband, Joe, can have points of oatmeal for breakfast. How much oatmeal can he have? | 44. An oatmeal cookie recipe calls for cup of butter to make dozen cookies. Hilda needs to make dozen cookies for the bake sale. How many cups of butter will she need? |
| 45. Janice is traveling to the US and will change Canadian dollars into US dollars. At the current exchange rate, Canadian is equal to US. How many Canadian dollars will she get for her trip? | 46. Todd is traveling to Mexico and needs to exchange into Mexican pesos. If each dollar is worth pesos, how many pesos will he get for his trip? |
| 47. Steve changed into Euros. How many Euros did he receive per Canadian dollar? | 48. Martha changed Canadian into Australian dollars. How many Australian dollars did she receive per US dollar? |
| 49. At the laundromat, Lucy changed into quarters. How many quarters did she get? | 50. When she arrived at a casino, Gerty changed into nickels. How many nickels did she get? |
| 51. Jesse’s car gets miles per gallon of gas. If Toronto is miles away, how many gallons of gas are needed to get there and then home? If gas is per gallon, what is the total cost of the gas for the trip? | 52. Danny wants to drive to Banff to see his grandfather. Banff is miles from Danny’s home and his car gets miles per gallon. How many gallons of gas will Danny need to get to and from Banff? If gas is 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 White Rock to go to Edmonton, miles away. After hours, he has gone miles. At that rate, how long will the whole drive take? | 54. Kelly leaves her home in Seattle to drive to Spokane, a distance of miles. After hours, she has gone miles. At that rate, how long will the whole drive take? |
| 55. Phil wants to fertilize his lawn. Each bag of fertilizer covers about square feet of lawn. Phil’s lawn is approximately square feet. How many bags of fertilizer will he have to buy? | 56. April wants to paint the exterior of her house. One gallon of paint covers about square feet, and the exterior of the house measures approximately 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 of | 58. What number is of |
| 59. What number is of | 60. What number is of |
| 61. is of what number? | 62. is of what number? |
| 63. is of what number? | 64. is of what number? |
| 64. is of what number? | 65. What percent of is |
| 66. What percent of is | 67. What percent of is |
| 68. What percent of is |
Translate and Solve Percent Proportions
In the following exercises, translate and solve using proportions.
| 69. What number is of | 70. What number is of |
| 71. of is what number? | 72. of is what number? |
| 73. of is what number? | 74. of is what number? |
| 75. What is of | 76. What is of |
| 77. of what number is | 78. of what number is |
| 79. is of what number? | 80. is of what number? |
| 81. What percent of is | 82. What percent of is |
| 83. What percent of is | 84. What percent of is |
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 ounces of concentrate with ounces of water. If he puts ounces of concentrate in a bucket, how many ounces of water should he add? How many ounces of the solution will he have altogether? | 86. Mixing a concentrate Travis is going to wash his car. The directions on the bottle of car wash concentrate say to mix ounces of concentrate with ounces of water. If Travis puts ounces of concentrate in a bucket, how much water must he mix with the concentrate? |
Writing Exercises
| 87. To solve “what number is of 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. | 88. To solve “what percent of is 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. |
Answers
| 1. | 3. | 5. |
| 7. | 9. | 11. |
| 13. yes | 15. no | 17. no |
| 19. yes | 21. 49 | 23. 47 |
| 25. 9 | 27. -11 | 29. 7 |
| 31. 2 | 33. 0.6 | 35. 4 |
| 37. 9 ml | 39. 114, no | 41. 159 cal |
| 43. | 45. $175.00 | 47. 0.65 |
| 49. 48 quarters | 51. 19, $58.71 | 53. 11.1 hours |
| 55. 4 bags | 57. | 59. |
| 61. | 63. | 65. |
| 67. | 69. 117 | 70. 165 |
| 71. 16.56 | 73. 45.5 | 75. 1464 |
| 77. $45 | 79. $164 | 81. 25% |
| 83. 12.5% | 85. 20, 32 | 87. Answers will vary. |
Attributions
This chapter has been adapted from “Solve Proportions and their Applications” in Prealgebra (OpenStax) by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.