6.6: Solve Proportions and their Applications

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    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

    Be Prepared 6.11

    Before you get started, take this readiness quiz.

    Simplify:
    If you missed this problem, review Example 4.44.

    Be Prepared 6.12

    Solve:
    If you missed this problem, review Example 4.99.

    Be Prepared 6.13

    Write as a rate: Sale rode his bike miles in hours.
    If you missed this problem, review Example 5.63.

    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 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 6.40

    Write each sentence as a proportion:

    1. ⓐ is to as is to
    2. ⓑ hits in at bats is the same as hits in at-bats.
    3. ⓒ for ounces is equivalent to for ounces.
    Answer

    ⓐ
    3 is to 7 as 15 is to 35.
    Write as a proportion.
    ⓑ
    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.
    ⓒ
    $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 6.79

    Write each sentence as a proportion:

    1. ⓐ is to as is to
    2. ⓑ hits in at-bats is the same as hits in at-bats.
    3. ⓒ for ounces is equivalent to for ounces.

    Try It 6.80

    Write each sentence as a proportion:

    1. ⓐ is to as is to
    2. ⓑ adults for children is the same as adults for children.
    3. ⓒ for ounces is equivalent to for ounces.

    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 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 where its cross products are equal.

    No Alt Text

    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 6.41

    Determine whether each equation is a proportion:

    1. ⓐ
    2. ⓑ
    Answer

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

    ⓐ
    .
    Find the cross products.
    .

    Since the cross products are not equal, the equation is not a proportion.

    ⓑ
    .
    Find the cross products.
    .

    Since the cross products are equal, the equation is a proportion.

    Try It 6.81

    Determine whether each equation is a proportion:

    1. ⓐ
    2. ⓑ

    Try It 6.82

    Determine whether each equation is a proportion:

    1. ⓐ
    2. ⓑ

    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 6.42

    Solve:

    Answer

      .
    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.
    .
    . .
    Show common factors. .
    Simplify. .

    Try It 6.83

    Solve the proportion:

    Try It 6.84

    Solve the proportion:

    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 6.43

    Solve:

    Answer

    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. .
    Simplify. .
    Check your answer.
    .
    . .
    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 6.85

    Solve the proportion:

    Try It 6.86

    Solve the proportion:

    Example 6.44

    Solve:

    Answer

    Find the cross products and set them equal. .
      .
    Simplify. .
    Divide both sides by 52. .
    Simplify. .
    Check:
    .
    . .
    Show common factors. .
    Simplify. .

    Try It 6.87

    Solve the proportion:

    Try It 6.88

    Solve the proportion:

    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 6.45

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

    Answer

    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 6.89

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

    Try It 6.90

    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?

    Example 6.46

    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?

    Answer

    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 3.5. .
    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.

    Try It 6.91

    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?

    Try It 6.92

    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?

    Example 6.47

    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?

    Answer

    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 4075.5 pesos for his spring break trip.

    Try It 6.93

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

    Try It 6.94

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

    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 6.48

    Translate to a proportion. What number is of

    Answer

    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 .

    Try It 6.95

    Translate to a proportion: What number is of

    Try It 6.96

    Translate to a proportion: What number is of

    Example 6.49

    Translate to a proportion. is of what number?

    Answer

    Identify the parts of the percent proportion. .
    Restate as a proportion. .
    Set up the proportion. Let .

    Try It 6.97

    Translate to a proportion: is of what number?

    Try It 6.98

    Translate to a proportion: is of what number?

    Example 6.50

    Translate to a proportion. What percent of is

    Answer

    Identify the parts of the percent proportion. .
    Restate as a proportion. .
    Set up the proportion. Let .

    Try It 6.99

    Translate to a proportion: What percent of is

    Try It 6.100

    Translate to a proportion: What percent of is

    Translate and Solve Percent Proportions

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

    Example 6.51

    Translate and solve using proportions: What number is of

    Answer

    Identify the parts of the percent proportion. .
    Restate as a proportion. .
    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 6.101

    Translate and solve using proportions: What number is of

    Try It 6.102

    Translate and solve using proportions: What number is of

    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 6.52

    Translate and solve using proportions: of is what number?

    Answer

    Identify the parts of the percent proportion. .
    Restate as a proportion. .
    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 6.103

    Translate and solve using proportions: of is what number?

    Try It 6.104

    Translate and solve using proportions: of is what number?

    Percents with decimals and money are also used in proportions.

    Example 6.53

    Translate and solve: of what number is

    Answer

    Identify the parts of the percent proportion. .
    Restate as a proportion. .
    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 6.105

    Translate and solve using proportions: of what number is

    Try It 6.106

    Translate and solve using proportions: of what number is

    Example 6.54

    Translate and solve using proportions: What percent of is

    Answer

    Identify the parts of the percent proportion. .
    Restate as a proportion. .
    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 6.107

    Translate and solve using proportions: What percent of is

    Try It 6.108

    Translate and solve using proportions: What percent of is

    Section 6.5 Exercises

    Practice Makes Perfect

    Use the Definition of Proportion

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

    243.

    is to as is to

    244.

    is to as is to

    245.

    is to as is to

    246.

    is to as is to

    247.

    wins in games is the same as wins in games.

    248.

    wins in games is the same as wins in games.

    249.

    campers to counselor is the same as campers to counselors.

    250.

    campers to counselor is the same as campers to counselors.

    251.

    for ounces is the same as for ounces.

    252.

    for ounces is the same as for ounces.

    253.

    for pounds is the same as for pounds.

    254.

    for pounds is the same as for pounds.

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

    255.

    256.

    257.

    258.

    259.

    260.

    261.

    262.

    Solve Proportions

    In the following exercises, solve each proportion.

    263.

    264.

    265.

    266.

    267.

    268.

    269.

    270.

    271.

    272.

    273.

    274.

    275.

    276.

    277.

    278.

    Solve Applications Using Proportions

    In the following exercises, solve the proportion problem.

    279.

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

    280.

    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?

    281.

    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?

    282.

    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?

    283.

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

    284.

    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?

    285.

    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?

    286.

    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?

    287.

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

    288.

    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?

    289.

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

    290.

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

    291.

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

    292.

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

    293.

    Jesse’s car gets miles per gallon of gas. If Las Vegas 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?

    294.

    Danny wants to drive to Phoenix to see his grandfather. Phoenix 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 Phoenix? If gas is per gallon, what is the total cost for the gas to drive to see his grandfather?

    295.

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

    296.

    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?

    297.

    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?

    298.

    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.

    299.

    What number is of

    300.

    What number is of

    301.

    What number is of

    302.

    What number is of

    303.

    is of what number?

    304.

    is of what number?

    305.

    is of what number?

    306.

    is of what number?

    307.

    What percent of is

    308.

    What percent of is

    309.

    What percent of is

    310.

    What percent of is

    Translate and Solve Percent Proportions

    In the following exercises, translate and solve using proportions.

    311.

    What number is of

    312.

    What number is of

    313.

    of is what number?

    314.

    of is what number?

    315.

    of is what number?

    316.

    of is what number?

    317.

    What is of

    318.

    What is of

    319.

    of what number is

    320.

    of what number is

    321.

    is of what number?

    322.

    is of what number?

    323.

    What percent of is

    324.

    What percent of is

    325.

    What percent of is

    326.

    What percent of is

    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 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?

    328.

    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

    329.

    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.

    330.

    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.

    Self Check

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

    .

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

    This page titled 6.6: Solve Proportions and their Applications was last modified on Sun, 28 May 2023 07:42:29 GMT and is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax.