Solve Proportions and their Applications

Use the definition of proportion

In Ratios and Rate we saw some ways ratios and rates are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a proportion.

The equation $\tfrac{1}{2} = \tfrac{4}{8}$ is a proportion because the two fractions are equal. The proportion $\tfrac{1}{2} = \tfrac{4}{8}$ 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 $\tfrac{20\ \text{students}}{1\ \text{teacher}} = \tfrac{60\ \text{students}}{3\ \text{teachers}}$ we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.

Example. Write each sentence as a proportion: (a) $3$ is to $7$ as $15$ is to $35$. (b) $5$ hits in $8$ at-bats is the same as $30$ hits in $48$ at-bats. (c) $\text{\textdollar}1.50$ for $6$ ounces is equivalent to $\text{\textdollar}2.25$ for $9$ ounces.

(a) $\tfrac{3}{7} = \tfrac{15}{35}$.

(b) Write each fraction to compare hits to at-bats: $\tfrac{\text{hits}}{\text{at-bats}} = \tfrac{\text{hits}}{\text{at-bats}}$, so $\tfrac{5}{8} = \tfrac{30}{48}$.

(c) Write each fraction to compare dollars to ounces: $\tfrac{\text{\textdollar}}{\text{ounces}} = \tfrac{\text{\textdollar}}{\text{ounces}}$, so $\tfrac{1.50}{6} = \tfrac{2.25}{9}$.

Look at the proportions $\tfrac{1}{2} = \tfrac{4}{8}$ and $\tfrac{2}{3} = \tfrac{6}{9}$. 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. Determine whether each equation is a proportion: (a) $\tfrac{4}{9} = \tfrac{12}{28}$ (b) $\tfrac{17.5}{37.5} = \tfrac{7}{15}$.

(a) Find the cross products: $28 \cdot 4 = 112$ and $9 \cdot 12 = 108$. Since the cross products are not equal, $28 \cdot 4 \neq 9 \cdot 12$, the equation is not a proportion.

(b) Find the cross products: $15 \cdot 17.5 = 262.5$ and $37.5 \cdot 7 = 262.5$. Since the cross products are equal, $15 \cdot 17.5 = 37.5 \cdot 7$, the equation is 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. When the variable is in a numerator, we can solve the proportion by multiplying both sides by the Least Common Denominator (LCD), using the Multiplication Property of Equality.

Example. Solve: $\tfrac{x}{63} = \tfrac{4}{7}$.

To isolate $x$, multiply both sides by the LCD, $63$: $63\left(\tfrac{x}{63}\right) = 63\left(\tfrac{4}{7}\right)$. Simplify: $x = \tfrac{9 \cdot 7 \cdot 4}{7}$. Divide the common factors: $x = 36$. Check: substitute $x = 36$ into the original proportion — $\tfrac{36}{63} \overset{?}{=} \tfrac{4}{7}$; showing common factors, $\tfrac{4 \cdot 9}{7 \cdot 9} \overset{?}{=} \tfrac{4}{7}$; simplify: $\tfrac{4}{7} = \tfrac{4}{7}\ \checkmark$.

When the variable is in a denominator, we’ll use the fact that the cross products of a proportion are equal to solve the proportion. 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. Solve: $\tfrac{144}{a} = \tfrac{9}{4}$.

Find the cross products and set them equal: $4 \cdot 144 = a \cdot 9$. Simplify: $576 = 9a$. Divide both sides by $9$: $\tfrac{576}{9} = \tfrac{9a}{9}$. Simplify: $64 = a$. Check: substitute $a = 64$ — $\tfrac{144}{64} \overset{?}{=} \tfrac{9}{4}$; showing common factors, $\tfrac{9 \cdot 16}{4 \cdot 16} \overset{?}{=} \tfrac{9}{4}$; simplify: $\tfrac{9}{4} = \tfrac{9}{4}\ \checkmark$.

Example. Solve: $\tfrac{52}{91} = \tfrac{-4}{y}$.

Find the cross products and set them equal: $y \cdot 52 = 91(-4)$. Simplify: $52y = -364$. Divide both sides by $52$: $\tfrac{52y}{52} = \tfrac{-364}{52}$. Simplify: $y = -7$. Check: substitute $y = -7$ — $\tfrac{52}{91} \overset{?}{=} \tfrac{-4}{-7}$; showing common factors, $\tfrac{13 \cdot 4}{13 \cdot 7} \overset{?}{=} \tfrac{-4}{-7}$; simplify: $\tfrac{4}{7} = \tfrac{4}{7}\ \checkmark$.

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. When pediatricians prescribe acetaminophen to children, they prescribe $5$ milliliters (ml) of acetaminophen for every $25$ pounds of the child’s weight. If Zoe weighs $80$ pounds, how many milliliters of acetaminophen will her doctor prescribe?

Let $a =$ ml of acetaminophen. If $5$ ml is prescribed for every $25$ pounds, how much will be prescribed for $80$ pounds? Translate into a proportion, being careful of the units: $\tfrac{\text{ml}}{\text{pounds}} = \tfrac{\text{ml}}{\text{pounds}}$, so $\tfrac{5}{25} = \tfrac{a}{80}$. Multiply both sides by $80$: $80 \cdot \tfrac{5}{25} = 80 \cdot \tfrac{a}{80}$. Multiply and show common factors: $\tfrac{16 \cdot 5 \cdot 5}{5 \cdot 5} = \tfrac{80a}{80}$. Simplify: $16 = a$. Check: since $80$ is about $3$ times $25$, the medicine should be about $3$ times $5$, and this checks out. The pediatrician would prescribe $16$ ml of acetaminophen to Zoe.

Example. One brand of microwave popcorn has $120$ calories per serving. A whole bag of this popcorn has $3.5$ servings. How many calories are in a whole bag of this microwave popcorn?

Let $c =$ number of calories. If there are $120$ calories per serving, how many calories are in a whole bag with $3.5$ servings? Translate into a proportion: $\tfrac{120}{1} = \tfrac{c}{3.5}$. Multiply both sides by $3.5$: $(3.5)\left(\tfrac{120}{1}\right) = (3.5)\left(\tfrac{c}{3.5}\right)$. Multiply: $420 = c$. Check: since $3.5$ is between $3$ and $4$, the total calories should be between $360$ ($3 \cdot 120$) and $480$ ($4 \cdot 120$), and $420$ fits. The whole bag of microwave popcorn has $420$ calories.

Example. Josiah went to Mexico for spring break and changed $\text{\textdollar}325$ dollars into Mexican pesos. At that time, the exchange rate had $\text{\textdollar}1$ U.S. equal to $12.54$ Mexican pesos. How many Mexican pesos did he get for his trip?

Let $p =$ number of pesos. If $\text{\textdollar}1$ U.S. is equal to $12.54$ Mexican pesos, then $\text{\textdollar}325$ is how many pesos? Translate into a proportion: $\tfrac{1}{12.54} = \tfrac{325}{p}$. The variable is in the denominator, so find the cross products and set them equal: $p \cdot 1 = 12.54(325)$. Simplify: $p = 4{,}075.5$. Check: since $\text{\textdollar}100$ would be $1{,}254$ pesos, and $\text{\textdollar}325$ is a little more than $3$ times this amount, this is reasonable. Josiah has $4{,}075.5$ pesos for his spring break trip.

Write percent equations as proportions

Previously, we solved percent equations by applying the properties of equality. 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, $60\% = \tfrac{60}{100}$, and we can simplify $\tfrac{60}{100} = \tfrac{3}{5}$. Since the equation $\tfrac{60}{100} = \tfrac{3}{5}$ shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier for percent equations:

If we restate the problem in the words of a proportion, it may be easier to set up: 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.

Example. Translate to a proportion: What number is $75\%$ of $90$?

If you look for the word “of”, it may help you identify the base. Identify the parts of the percent proportion: “what number” is the amount, $75\%$ is the percent, and $90$ (following “of”) is the base. Restate as a proportion: what number out of $90$ is the same as $75$ out of $100$? Set up the proportion, letting $n =$ number: $\tfrac{n}{90} = \tfrac{75}{100}$.

Example. Translate to a proportion: $19$ is $25\%$ of what number?

Identify the parts of the percent proportion: $19$ is the amount, $25\%$ is the percent, and “what number” is the base. Restate as a proportion: $19$ out of what number is the same as $25$ out of $100$? Set up the proportion, letting $n =$ number: $\tfrac{19}{n} = \tfrac{25}{100}$.

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

Identify the parts of the percent proportion: “what percent” is the percent, $27$ (following “of”) is the base, and $9$ is the amount. Restate as a proportion: $9$ out of $27$ is the same as what number out of $100$? Set up the proportion, letting $p =$ percent: $\tfrac{9}{27} = \tfrac{p}{100}$.

Translate and solve percent proportions

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

Example. Translate and solve using proportions: What number is $45\%$ of $80$?

Identify the parts of the percent proportion: “what number” is the amount, $45\%$ is the percent, $80$ is the base. Restate as a proportion: what number out of $80$ is the same as $45$ out of $100$? Set up the proportion, letting $n =$ number: $\tfrac{n}{80} = \tfrac{45}{100}$. Find the cross products and set them equal: $100 \cdot n = 80 \cdot 45$. Simplify: $100n = 3{,}600$. Divide both sides by $100$: $n = 36$. Check: $45$ is a little less than half of $100$, and $36$ is a little less than half of $80$, so this is reasonable. $36$ is $45\%$ of $80$.

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. Translate and solve using proportions: $125\%$ of $25$ is what number?

Identify the parts of the percent proportion: $125\%$ is the percent, $25$ is the base, “what number” is the amount. Restate as a proportion: what number out of $25$ is the same as $125$ out of $100$? Set up the proportion: $\tfrac{n}{25} = \tfrac{125}{100}$. Find the cross products and set them equal: $100 \cdot n = 25 \cdot 125$. Simplify: $100n = 3{,}125$. Divide both sides by $100$: $n = 31.25$. Check: $125$ is more than $100$ and $31.25$ is more than $25$, so this is reasonable. $125\%$ of $25$ is $31.25$.

Percents with decimals and money are also used in proportions.

Example. Translate and solve: $6.5\%$ of what number is $\text{\textdollar}1.56$?

Identify the parts of the percent proportion: $6.5\%$ is the percent, “what number” is the base, $\text{\textdollar}1.56$ is the amount. Restate as a proportion: $\text{\textdollar}1.56$ out of what number is the same as $6.5$ out of $100$? Set up the proportion, letting $n =$ number: $\tfrac{1.56}{n} = \tfrac{6.5}{100}$. Find the cross products and set them equal: $100(1.56) = n \cdot 6.5$. Simplify: $156 = 6.5n$. Divide both sides by $6.5$: $\tfrac{156}{6.5} = \tfrac{6.5n}{6.5}$. Simplify: $24 = n$. Check: $6.5\%$ is a small amount and $\text{\textdollar}1.56$ is much less than $\text{\textdollar}24$, so this is reasonable. $6.5\%$ of $\text{\textdollar}24$ is $\text{\textdollar}1.56$.

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

Identify the parts of the percent proportion: “what percent” is the percent, $72$ is the base, $9$ is the amount. Restate as a proportion: $9$ out of $72$ is the same as what number out of $100$? Set up the proportion, letting $n =$ number: $\tfrac{9}{72} = \tfrac{n}{100}$. Find the cross products and set them equal: $72 \cdot n = 100 \cdot 9$. Simplify: $72n = 900$. Divide both sides by $72$: $n = 12.5$. Check: $9$ is $\tfrac{1}{8}$ of $72$, and $\tfrac{1}{8}$ is $12.5\%$, so this checks out. $12.5\%$ of $72$ is $9$.

Key terms

proportion — an equation of the form $\tfrac{a}{b} = \tfrac{c}{d}$, where $b \neq 0, d \neq 0$, stating that two ratios or rates are equal. cross products — the products found by multiplying each denominator of a proportion with the opposite numerator; equal cross products indicate a true proportion. percent proportion — a proportion of the form $\tfrac{\text{amount}}{\text{base}} = \tfrac{\text{percent}}{100}$.

Practice

Use the definition of proportion

Solve proportions

Solve applications using proportions

Write percent equations as proportions

Translate and solve percent proportions

This section is adapted from Prealgebra 2e, Section 6.5: Solve Proportions and their Applications by Lynn Marecek, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis, © OpenStax, licensed under CC BY-NC-SA 4.0. Access the original for free at openstax.org. Changes: omitted the Be Prepared quiz, Self Check checklist, Writing Exercises, and media links; converted the practice problems (“Try Its”) into interactive exercises with instant feedback, including entering proportion setups directly as equations; and adapted selected end-of-section exercises into the interactive Practice block, presenting the proportion-setup and is-it-a-proportion items as multiple choice.