6.5 Solve Proportions and their Applications

Learning objectives

Learning objectives

• 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 $\frac{1}{2}=\frac{4}{8}$ is a proportion because the two fractions are equal. The proportion $\frac{1}{2}=\frac{4}{8}$ is read $“1$ is to $2$ as $4$ is to $8\mathrm{".}$

If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion $\frac{\mathrm{20 students}}{\mathrm{1 teacher}}=\frac{\mathrm{60 students}}{\mathrm{3 teachers}}$ 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

Solution

1. Identify the quantities in each part. A proportion compares the same two kinds of quantities in two ratios, so each sentence supplies two ordered pairs.
2. Write the first proportion. The words 3 is to 7 as 15 is to 35 place 3 over 7 and 15 over 35, giving $\frac{3}{7}=\frac{15}{35}$.
3. Write the second proportion. Hits belong over at-bats on both sides, so the ratio of $5$ hits in $8$ at-bats to $30$ hits in $48$ at-bats is $\frac{5}{8}=\frac{30}{48}$.
4. Write the third proportion. Dollars belong over ounces on both sides, so the price comparison becomes $\frac{1.50}{6}=\frac{2.25}{9}$.

Takeaway: Example 6.40 translates each comparison into equal ratios with matching units: $\frac{3}{7}=\frac{15}{35}$, $\frac{5}{8}=\frac{30}{48}$, and $\frac{1.50}{6}=\frac{2.25}{9}$.

Look at the proportions $\frac{1}{2}=\frac{4}{8}$ and $\frac{2}{3}=\frac{6}{9}.$ 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 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.

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

Solution

1. State the cross-product test. An equation of two ratios is a proportion exactly when the cross products are equal.
2. Test the first equation. For $\frac{4}{9}=\frac{12}{28}$, the cross products are $4(28)=112$ and $9(12)=108$. They are not equal, so the equation is not a proportion.
3. Test the second equation. For $\frac{17.5}{37.5}=\frac{7}{15}$, the cross products are $17.5(15)=262.5$ and $37.5(7)=262.5$. They are equal, so the equation is a proportion.

Takeaway: Example 6.41 compares cross products and finds that $4/9=12/28$ is not a proportion, while $17.5/37.5=7/15$ is.

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

Solution

1. State the proportion. The equation is $\frac{x}{63}=\frac{4}{7}$, and the first figure displays that fraction equation.
2. Clear the denominator. Multiplying both sides by $63$ as the second figure shows gives $x=63\left(\frac{4}{7}\right)=9(4)=36$, matching the third and fourth figures.
3. Verify the solution. Substituting $x=36$ gives $\frac{36}{63}=\frac{4(9)}{7(9)}=\frac{4}{7}$, which the final figures confirm with a checkmark. Therefore, $x=36$.

Takeaway: To solve a proportion with a variable in the denominator, clear that denominator and check by substitution.

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

Solution

1. Recognize the variable denominator. The proportion is $\frac{144}{a}=\frac{9}{4}$, so cross multiplication will remove the denominators.
2. Form the cross products. Multiplying by opposite denominators gives $144(4)=a(9)$, which simplifies to $576=9a$.
3. Solve the linear equation. Dividing both sides by $9$ gives $a=64$.
4. Verify in the original proportion. Substituting $a=64$ gives $\frac{144}{64}=\frac{9}{4}$ because both sides simplify to $\frac{9}{4}$, confirming the solution.

Takeaway: Cross-multiplying $\frac{144}{a}=\frac{9}{4}$ gives $576=9a$, so Example 6.43 has the verified solution $a=64$.

Example 6.44

Solution

1. Cross-multiply the proportion. The first and second figures use $\frac{52}{91}=\frac{-4}{y}$ to form $52y=91(-4)=-364$.
2. Solve for $y$. Dividing both sides by $52$ as the fourth figure shows gives $y=\frac{-364}{52}=-7$, matching the fifth figure.
3. Verify the solution. Substitution produces $\frac{52}{91}=\frac{-4}{-7}=\frac{4}{7}$ because both numerators and denominators are divided by $13$. The final figures confirm $4/7=4/7$, so $y=-7$.

Takeaway: Dividing both the numerator and denominator of a negative fraction by the same negative factor leaves its value unchanged.

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

Solution

1. Form the proportion. The first figure uses the matching units of milliliters per pound, so $\frac{5\text{ ml}}{25\text{ lb}}=\frac{a\text{ ml}}{80\text{ lb}}$, or numerically $\frac{5}{25}=\frac{a}{80}$ as the second figure shows.
2. Clear the denominator. Multiplying both sides by $80$ gives $80\left(\frac{5}{25}\right)=a$, which the third and fourth figures simplify to $16=a$.
3. State the dosage. Therefore, Zoe's doctor will prescribe $16$ milliliters of acetaminophen, as stated in the last figure.

Takeaway: In an application proportion, place corresponding quantities in matching positions so the units cancel consistently.

Example 6.46

Solution

1. Form the proportion. The first figure matches calories with servings, giving $\frac{120\text{ calories}}{1\text{ serving}}=\frac{c\text{ calories}}{3.5\text{ servings}}$. Numerically, the second figure shows $\frac{120}{1}=\frac{c}{3.5}$.
2. Clear the denominator. Multiplying both sides by $3.5$ as the third figure shows gives $c=3.5(120)$.
3. Evaluate the calories. The last figure records $420=c$. A whole bag contains $420$ calories.

Takeaway: Keep identical units in corresponding positions so a proportion remains meaningful when the arithmetic is performed.

Example 6.47

Solution

1. Form the exchange proportion. The first figure matches dollars to pesos, giving $\frac{1\text{ dollar}}{12.54\text{ pesos}}=\frac{325\text{ dollars}}{p\text{ pesos}}$, or numerically $\frac{1}{12.54}=\frac{325}{p}$ as the second figure shows.
2. Cross-multiply. The third figure forms $p(1)=12.54(325)$.
3. Evaluate the amount. Multiplying gives $p=4,075.5$, as the last figure displays. Josiah received $4,075.5$ Mexican pesos.

Takeaway: In an exchange-rate proportion, keep the two currencies in corresponding numerator and denominator positions.

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, $\mathrm{60\%}=\frac{60}{100}$ and we can simplify $\frac{60}{100}=\frac{3}{5}.$ Since the equation $\frac{60}{100}=\frac{3}{5}$ shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:

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

Solution

1. Identify the percent parts. The first figure labels the unknown number as the amount, $75\%$ as the percent, and $90$ as the base; the word of identifies the base.
2. Write the proportion. The second figure asks what number out of $90$ is the same as $75$ out of $100$. If $n$ is the unknown number, this gives $\frac{n}{90}=\frac{75}{100}$.
3. State the translated proportion. Thus, the requested translation is $\frac{n}{90}=\frac{75}{100}$, with the amount over the base equal to the percent over 100.

Takeaway: For a percent translation, put the amount over the base and the percent over 100 in a proportion.

Example 6.49

Solution

1. Identify the percent parts. The first figure labels $19$ as the amount, $25\%$ as the percent, and the unknown number as the base.
2. Write the proportion. The second figure expresses the relationship as $19$ out of the unknown number equals $25$ out of $100$. If $n$ is the unknown base, the proportion is $\frac{19}{n}=\frac{25}{100}$.
3. State the translated proportion. Therefore, the required translation is $\frac{19}{n}=\frac{25}{100}$.

Takeaway: Even when the base is unknown, the amount and base occupy corresponding positions in the percent proportion.

Example 6.50

Solution

1. Identify the percent parts. The first figure labels $9$ as the amount, $27$ as the base, and the unknown percent as the percent part.
2. Write the proportion. The second figure asks what number out of $27$ is the same as what number out of $100$. If $n$ is the unknown number out of 100, the proportion is $\frac{9}{27}=\frac{n}{100}$.
3. State the translated proportion. Thus, the requested translation is $\frac{9}{27}=\frac{n}{100}$, where $n$ represents the percent out of 100.

Takeaway: An unknown percent can be represented as the numerator of a fraction with denominator 100.

Translate and Solve Percent Proportions

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

Example 6.51

Solution

1. Form the percent proportion. The first two figures identify $n$ as the amount and show that what number out of $80$ equals $45$ out of $100$. This gives $\frac{n}{80}=\frac{45}{100}$, as the third figure displays.
2. Cross-multiply. The fourth and fifth figures give $100n=80(45)=3,600$.
3. Solve for the amount. Dividing both sides by $100$ as the sixth figure shows gives $n=36$, matching the last figure. Therefore, $45\%$ of $80$ is $36$.

Takeaway: Solving a percent proportion by cross multiplication gives the same result as the equivalent decimal percent equation.

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.

Example 6.52

Solution

1. Form the percent proportion. The first two figures identify $n$ as the amount and the question as what number out of $25$ equals $125$ out of $100$. Thus, $\frac{n}{25}=\frac{125}{100}$, as the third figure shows.
2. Cross-multiply. The next figures give $100n=25(125)=3,125$.
3. Solve for the amount. Dividing both sides by $100$ gives $n=31.25$, matching the last figure. Therefore, $125\%$ of $25$ is $31.25$.

Takeaway: A percent proportion handles a rate greater than 100 percent just as readily as a rate below 100 percent.

Percents with decimals and money are also used in proportions.

Example 6.53

Solution

1. Form the percent proportion. The first two figures identify $1.56$ as the amount, the unknown base as the denominator, and $6.5\%$ as the percent. With $n$ as the base, the third figure gives $\frac{1.56}{n}=\frac{6.5}{100}$.
2. Cross-multiply. The fourth and fifth figures produce $100(1.56)=6.5n$, so $156=6.5n$.
3. Solve for the base. Dividing both sides by $6.5$ as the sixth figure shows gives $n=24$, matching the last figure. Thus, $6.5\%$ of $24$ is $1.56$.

Takeaway: With a decimal percent, use the exact percent value over 100 in the proportion and keep it during cross multiplication.

Example 6.54

Solution

1. Form the percent proportion. The first two figures identify $9$ as the amount and $72$ as the base, so if $n$ is the number out of 100, the third figure gives $\frac{9}{72}=\frac{n}{100}$.
2. Cross-multiply. The next figures show $72n=100(9)=900$.
3. Solve for the percent number. Dividing both sides by $72$ gives $n=\frac{900}{72}=12.5$, as the last figure displays. Therefore, $9$ is $12.5\%$ of $72$.

Takeaway: The number obtained over 100 in a percent proportion is the percentage itself, provided it is interpreted accordingly.

Section 6.5 Exercises

Section 6.5 Exercises

Use the Definition of Proportion

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

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

Solve Proportions

In the following exercises, solve each proportion.

Solve Applications Using Proportions

In the following exercises, solve the proportion problem.

Write Percent Equations as Proportions

In the following exercises, translate to a proportion.

Translate and Solve Percent Proportions

In the following exercises, translate and solve using proportions.

Everyday Math

Writing Exercises

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?

1. Write each sentence as a proportion:

1. ⓐ $\,5$ is to $9$ as $20$ is to $36.$

2. ⓑ $\,7$ hits in $11$ at-bats is the same as $28$ hits in $44$ at-bats.

3. ⓒ $\,\text{\textdollar 2.50}$ for $8$ ounces is equivalent to $\text{\textdollar 3.75}$ for $12$ ounces.

2. Write each sentence as a proportion:

1. ⓐ $\,6$ is to $7$ as $36$ is to $42.$

2. ⓑ $\,8$ adults for $36$ children is the same as $12$ adults for $54$ children.

3. ⓒ $\,\text{\textdollar 3.75}$ for $6$ ounces is equivalent to $\text{\textdollar 2.50}$ for $4$ ounces.

3. Determine whether each equation is a proportion:

1. ⓐ $\,\frac{7}{9}=\frac{54}{72}$

2. ⓑ $\frac{24.5}{45.5}=\frac{7}{13}$

4. Determine whether each equation is a proportion:

1. ⓐ $\,\frac{8}{9}=\frac{56}{73}\,$

2. ⓑ $\,\frac{28.5}{52.5}=\frac{8}{15}$

5. Solve the proportion: $\frac{84}{98}=\frac{-6}{x}.$

1. Solve the proportion: $\frac{-7}{y}=\frac{105}{135}.$

2. 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 Emilia, who weighs $60$ pounds?

3. For every $1$ kilogram (kg) of a child’s weight, pediatricians prescribe $15$ milligrams (mg) of a fever reducer. If Isabella weighs $12$ kg, how many milligrams of the fever reducer will the pediatrician prescribe?

4. Yaneli loves Starburst candies, but wants to keep her snacks to $100$ calories. If the candies have $160$ calories for $8$ pieces, how many pieces can she have in her snack?

5. Yurianna is going to Europe and wants to change $\text{\textdollar 800}$ dollars into Euros. At the current exchange rate, $\text{\textdollar 1}$ US is equal to $0.738$ Euro. How many Euros will she have for her trip?

1. Corey and Nicole are traveling to Japan and need to exchange $\text{\textdollar 600}$ into Japanese yen. If each dollar is $94.1$ yen, how many yen will they get?

2. Translate to a proportion: What number is $\mathrm{60\%}$ of $105?$

3. Translate to a proportion: What number is $\mathrm{40\%}$ of $85?$

4. Translate to a proportion: $36$ is $\mathrm{25\%}$ of what number?

5. Translate to a proportion: $27$ is $\mathrm{36\%}$ of what number?

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

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

3. Translate and solve using proportions: What number is $\mathrm{65\%}$ of $40?$

4. Translate and solve using proportions: What number is $\mathrm{85\%}$ of $40?$

5. Translate and solve using proportions: $\mathrm{125\%}$ of $64$ is what number?

1. Translate and solve using proportions: $\mathrm{175\%}$ of $84$ is what number?

2. Translate and solve using proportions: $\mathrm{8.5\%}$ of what number is $\text{\textdollar 3.23}?$

3. Translate and solve using proportions: $\mathrm{7.25\%}$ of what number is $\text{\textdollar 4.64}?$

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

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

1. $4$ is to $15$ as $36$ is to $135.$

2. $12$ is to $5$ as $96$ is to $40.$

3. $5$ wins in $7$ games is the same as $115$ wins in $161$ games.

4. $8$ campers to $1$ counselor is the same as $48$ campers to $6$ counselors.

5. $\text{\textdollar 9.36}$ for $18$ ounces is the same as $\text{\textdollar 2.60}$ for $5$ ounces.

1. $\text{\textdollar 18.04}$ for $11$ pounds is the same as $\text{\textdollar 4.92}$ for $3$ pounds.

2. $\frac{7}{15}=\frac{56}{120}$

3. $\frac{11}{6}=\frac{21}{16}$

4. $\frac{12}{18}=\frac{4.99}{7.56}$

5. $\frac{13.5}{8.5}=\frac{31.05}{19.55}$

1. $\frac{x}{56}=\frac{7}{8}$

2. $\frac{49}{63}=\frac{z}{9}$

3. $\frac{5}{a}=\frac{65}{117}$

4. $\frac{98}{154}=\frac{-7}{p}$

5. $\frac{a}{-8}=\frac{-42}{48}$

1. $\frac{2.6}{3.9}=\frac{c}{3}$

2. $\frac{2.7}{j}=\frac{0.9}{0.2}$

3. $\frac{\frac{1}{2}}{1}=\frac{m}{8}$

4. 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?

5. 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?

1. A new energy drink advertises $106$ calories for $8$ ounces. How many calories are in $12$ ounces of the drink?

2. Karen eats $\frac{1}{2}$ 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?

3. Janice is traveling to Canada and will change $\text{\textdollar 250}$ US dollars into Canadian dollars. At the current exchange rate, $\text{\textdollar 1}$ US is equal to $\text{\textdollar 1.01}$ Canadian. How many Canadian dollars will she get for her trip?

4. Steve changed $\text{\textdollar 600}$ into $480$ Euros. How many Euros did he receive per US dollar?

5. At the laundromat, Lucy changed $\text{\textdollar 12.00}$ into quarters. How many quarters did she get?

1. $45$ is $\mathrm{30\%}$ of what number?

2. $90$ is $\mathrm{150\%}$ of what number?

3. What percent of $85$ is $17?$

4. What percent of $260$ is $340?$

5. What number is $\mathrm{65\%}$ of $180?$

1. $\mathrm{18\%}$ of $92$ is what number?

2. $\mathrm{175\%}$ of $26$ is what number?

3. What is $\mathrm{300\%}$ of $488?$

4. $\mathrm{17\%}$ of what number is $\text{\textdollar 7.65}?$

5. $\text{\textdollar 13.53}$ is $\mathrm{8.25\%}$ of what number?

1. 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 $\text{\textdollar 3.09}$ per gallon, what is the total cost of the gas for the trip?

2. 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?

3. 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?

4. What number is $\mathrm{35\%}$ of $250?$

5. What number is $\mathrm{110\%}$ of $47?$

1. What percent of $56$ is $14?$

2. What percent of $96$ is $12?$

3. 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?

4. To solve “what number is $\mathrm{45\%}$ 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.