# 6.6: Solve Proportions and their Applications (Part 2)

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)## 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% = \(\dfrac{60}{100}\) and we can simplify \(\dfrac{60}{100} = \dfrac{3}{5}\). Since the equation \(\dfrac{60}{100} = \dfrac{3}{5}\) shows a percent equal to an equivalent ratio, we call it a **percent proportion**. Using the vocabulary we used earlier:

\[\begin{split} \dfrac{ \text{amount}}{\text{base}} & = \dfrac{\text{percent}}{100} \\ \dfrac{3}{5} & = \dfrac{60}{100} \end{split}\]

The amount is to the base as the percent is to 100.

\[\dfrac{amount}{base} = \dfrac{percent}{100} \tag{6.5.37}\]

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. | $$What\; number\; out\; \textcolor{red}{of}\; 90\; is\; the\; same\; as\; 75\; out\; of\; 100?$$ |

Set up the proportion. Let n = number. | $$\dfrac{n}{90} = \dfrac{75}{100} \tag{6.5.38}$$ |

Translate to a proportion: What number is 60% of 105?

**Answer**-
\(\frac{n}{105} = \frac{60}{100}\)

Translate to a proportion: What number is 40% of 85?

**Answer**-
\(\frac{n}{85} = \frac{40}{100}\)

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

**Solution**

Identify the parts of the percent proportion. | |

Restate as a proportion. | $$19\; out\; \textcolor{red}{of}\; what\; number\; is\; the\; same\; as\; 25\; out\; of\; 100?$$ |

Set up the proportion. Let n = number. | $$\dfrac{19}{n} = \dfrac{25}{100} \tag{6.5.39}$$ |

Translate to a proportion: 36 is 25% of what number?

**Answer**-
\(\frac{36}{n} = \frac{25}{100}\)

Translate to a proportion: 27 is 36% of what number?

**Answer**-
\(\frac{27}{n} = \frac{36}{100}\)

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

**Solution**

Identify the parts of the percent proportion. | |

Restate as a proportion. | $$9\; out\; \textcolor{red}{of}\; 27\; is\; the\; same\; as\; what\; number\; out\; of\; 100$$ |

Set up the proportion. Let p = percent. | $$\dfrac{9}{27} = \dfrac{p}{100} \tag{6.5.40}$$ |

Translate to a proportion: What percent of 52 is 39?

**Answer**-
\(\frac{n}{100} = \frac{39}{52}\)

Translate to a proportion: What percent of 92 is 23?

**Answer**-
\(\frac{n}{100} = \frac{23}{92}\)

## 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. | $$What\; number\; out\; \textcolor{red}{of}\; 80\; is\; the\; same\; as\; 45\; out\; of\; 100?$$ |

Set up the proportion. Let n = number. | $$\dfrac{n}{80} = \dfrac{45}{100} \tag{6.5.41}$$ |

Find the cross products and set them equal. | $$100 \cdot n = 80 \cdot 45 \tag{6.5.42}$$ |

Simplify. | $$100n = 3,600 \tag{6.5.43}$$ |

Divide both sides by 100. | $$\dfrac{100n}{100} = \dfrac{3,600}{100} \tag{6.5.44}$$ |

Simplify. | $$n = 36 \tag{6.5.45}$$ |

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. |

Translate and solve using proportions: What number is 65% of 40?

**Answer**-
26

Translate and solve using proportions: What number is 85% of 40?

**Answer**-
34

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. | $$What\; number\; out\; \textcolor{red}{of}\; 25\; is\; the\; same\; as\; 125\; out\; of\; 100?$$ |

Set up the proportion. Let n = number. | $$\dfrac{n}{25} = \dfrac{125}{100} \tag{6.5.46}$$ |

Find the cross products and set them equal. | $$100 \cdot n = 25 \cdot 125 \tag{6.5.47}$$ |

Simplify. | $$100n = 3,125 \tag{6.5.48}$$ |

Divide both sides by 100. | $$\dfrac{100n}{100} = \dfrac{3,125}{100} \tag{6.5.49}$$ |

Simplify. | $$n = 31.25 \tag{6.5.50}$$ |

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. |

Translate and solve using proportions: 125% of 64 is what number?

**Answer**-
80

Translate and solve using proportions: 175% of 84 is what number?

**Answer**-
147

Percents with decimals and money are also used in proportions.

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

**Solution**

Identify the parts of the percent proportion. | |

Restate as a proportion. | $$ \$1.56\; out\; \textcolor{red}{of}\; what\; number\; is\; the\; same\; as\; 6.5\; out\; of\; 100?$$ |

Set up the proportion. Let n = number. | $$\dfrac{1.56}{n} = \dfrac{6.5}{100} \tag{6.5.51}$$ |

Find the cross products and set them equal. | $$100(1.56) = n \cdot 6.5 \tag{6.5.52}$$ |

Simplify. | $$156 = 6.5n \tag{6.5.53}$$ |

Divide both sides by 6.5 to isolate the variable. | $$\dfrac{156}{6.5} = \dfrac{6.5n}{6.5} \tag{6.5.54}$$ |

Simplify. | $$24 = n \tag{6.5.55}$$ |

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: 8.5% of what number is $3.23?

**Answer**-
38

Translate and solve using proportions: 7.25% of what number is $4.64?

**Answer**-
64

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

**Solution**

Identify the parts of the percent proportion. | |

Restate as a proportion. | $$9\; out\; \textcolor{red}{of}\; 72\; is\; the\; same\; as\; what\; number\; out\; of\; 100?$$ |

Set up the proportion. Let n = number. | $$\dfrac{9}{72} = \dfrac{n}{100} \tag{6.5.56}$$ |

Find the cross products and set them equal. | $$72 \cdot n = 100 \cdot 9 \tag{6.5.57}$$ |

Simplify. | $$72n = 900 \tag{6.5.58}$$ |

Divide both sides by 72. | $$\dfrac{72n}{72} = \dfrac{900}{72} \tag{6.5.59}$$ |

Simplify. | $$n = 12.5 \tag{6.5.60}$$ |

Check if the answer is reasonable. | Yes. 9 is \(\dfrac{1}{8}\) of 72 and \(\dfrac{1}{8}\) is 12.5%. |

Write a complete sentence that answers the question. | 12.5% of 72 is 9. |

Translate and solve using proportions: What percent of 72 is 27?

**Answer**-
37.5%

Translate and solve using proportions: What percent of 92 is 23?

**Answer**-
25%

## 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.
- 7 is to 9 as 35 is to 45.
- 12 is to 5 as 96 is to 40.
- 15 is to 8 as 75 is to 40.
- 5 wins in 7 games is the same as 115 wins in 161 games.
- 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.
- 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.
- $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.
- $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.

- \(\dfrac{7}{15} = \dfrac{56}{120}\)
- \(\dfrac{5}{12} = \dfrac{45}{108}\)
- \(\dfrac{11}{6} = \dfrac{21}{16}\)
- \(\dfrac{9}{4} = \dfrac{39}{34}\)
- \(\dfrac{12}{18} = \dfrac{4.99}{7.56}\)
- \(\dfrac{9}{16} = \dfrac{2.16}{3.89}\)
- \(\dfrac{13.5}{8.5} = \dfrac{31.05}{19.55}\)
- \(\dfrac{10.1}{8.4} = \dfrac{3.03}{2.52}\)

### Solve Proportions

In the following exercises, solve each proportion.

- \(\dfrac{x}{56} = \dfrac{7}{8}\)
- \(\dfrac{n}{91} = \dfrac{8}{13}\)
- \(\dfrac{49}{63} = \dfrac{z}{9}\)
- \(\dfrac{56}{72} = \dfrac{y}{9}\)
- \(\dfrac{5}{a} = \dfrac{65}{117}\)
- \(\dfrac{4}{b} = \dfrac{64}{144}\)
- \(\dfrac{98}{154} = \dfrac{-7}{p}\)
- \(\dfrac{72}{156} = \dfrac{-6}{q}\)
- \(\dfrac{a}{-8} = \dfrac{-42}{48}\)
- \(\dfrac{b}{-7} = \dfrac{-30}{42}\)
- \(\dfrac{2.6}{3.9} = \dfrac{c}{3}\)
- \(\dfrac{2.7}{3.6} = \dfrac{d}{4}\)
- \(\dfrac{2.7}{j} = \dfrac{0.9}{0.2}\)
- \(\dfrac{2.8}{k} = \dfrac{2.1}{1.5}\)
- \(\dfrac{\dfrac{1}{2}}{1} = \dfrac{m}{8}\)
- \(\dfrac{\dfrac{1}{3}}{3} = \dfrac{9}{n}\)

### 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?
- 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?
- 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?
- 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 \(\dfrac{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?
- An oatmeal cookie recipe calls for \(\dfrac{1}{2}\) 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?
- 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?
- 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?
- 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?
- 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?
- 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?
- 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?
- What number is 75% of 920?
- What number is 110% of 47?
- What number is 150% of 64?
- 45 is 30% of what number?
- 25 is 80% of what number?
- 90 is 150% of what number?
- 77 is 110% of what number?
- What percent of 85 is 17?
- What percent of 92 is 46?
- What percent of 260 is 340?
- 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?
- What number is 55% of 300?
- 18% of 92 is what number?
- 22% of 74 is what number?
- 175% of 26 is what number?
- 250% of 61 is what number?
- What is 300% of 488?
- What is 500% of 315?
- 17% of what number is $7.65?
- 19% of what number is $6.46?
- $13.53 is 8.25% of what number?
- $18.12 is 7.55% of what number?
- What percent of 56 is 14?
- What percent of 80 is 28?
- What percent of 96 is 12?
- 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?**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.
- 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

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

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

## Contributors and Attributions

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."