Lesson 5: Solve Proportions and their Applications

New Concept

A proportion is an equation stating two ratios are equal, like $\frac{a}{b} = \frac{c}{d}$. This lesson shows how to use this powerful tool to find unknown quantities in real-world scenarios and even solve percent problems.

What’s next

Next, you’ll work through examples and interactive practice problems to master solving proportions and applying them to real-world challenges.

Property

A proportion is an equation of the form $\frac{a}{b} = \frac{c}{d}$, where $b \neq 0, d \neq 0$. The proportion states two ratios or rates are equal. For any proportion of this form, its cross products are equal: $a \cdot d = b \cdot c$. Cross products can be used to test whether a proportion is true.

Examples

• The sentence "4 is to 9 as 20 is to 45" is written as the proportion $\frac{4}{9} = \frac{20}{45}$.
• To determine if $\frac{6}{11} = \frac{30}{55}$ is a proportion, we check the cross products. Since $6 \cdot 55 = 330$ and $11 \cdot 30 = 330$, the equation is a proportion.
• To check if $\frac{8}{10} = \frac{30}{40}$ is a proportion, we find the cross products. $8 \cdot 40 = 320$ and $10 \cdot 30 = 300$. Since the products are not equal, it is not a proportion.

Explanation

A proportion is a statement that two ratios are equal, like a balanced scale. The cross-product rule is a quick check: if the products of the numbers on the diagonal are equal, the ratios form a true proportion.

Property

To solve a proportion for a variable, you can use the property of equal cross products. For a proportion like $\frac{a}{b} = \frac{c}{d}$, set the cross products equal, $a \cdot d = b \cdot c$, and then solve the resulting equation for the unknown variable.

Examples

• To solve $\frac{x}{63} = \frac{4}{7}$, set cross products equal: $7 \cdot x = 63 \cdot 4$. This gives $7x = 252$, so $x = 36$.
• To solve $\frac{144}{a} = \frac{9}{4}$, set cross products equal: $144 \cdot 4 = 9 \cdot a$. This gives $576 = 9a$, so $a = 64$.
• To solve $\frac{52}{91} = \frac{-4}{y}$, set cross products equal: $52 \cdot y = 91 \cdot (-4)$. This gives $52y = -364$, so $y = -7$.

Explanation

When a number in a proportion is missing, we use algebra to find it. The cross-product method is the fastest way to turn the proportion into a simple equation that you can solve for the unknown value.

Property

To solve a word problem using proportions, first identify the two quantities being compared and set up two equal ratios. Let a variable represent the unknown quantity. Ensure that the units are placed consistently in the numerators and denominators of the ratios.
$\frac{\text{unit A quantity 1}}{\text{unit B quantity 1}} = \frac{\text{unit A quantity 2}}{\text{unit B quantity 2}}$

Examples

• A doctor prescribes 5 ml of medicine for every 25 pounds of weight. For a child weighing 80 pounds, we set up $\frac{5 \text{ ml}}{25 \text{ lbs}} = \frac{a}{80 \text{ lbs}}$. Solving gives $25a = 400$, so $a = 16$ ml.
• A bag of popcorn with 3.5 servings has 120 calories per serving. To find total calories, set up $\frac{120 \text{ calories}}{1 \text{ serving}} = \frac{c}{3.5 \text{ servings}}$. Solving gives $c = 120 \cdot 3.5 = 420$ calories.
• If 1 U.S. dollar equals 12.54 Mexican pesos, to find how many pesos is 325 dollars, set up $\frac{1 \text{ dollar}}{12.54 \text{ pesos}} = \frac{325 \text{ dollars}}{p}$. Solving gives $p = 325 \cdot 12.54 = 4075.5$ pesos.

Explanation

Proportions are fantastic for real-world problems like scaling a recipe or converting units. The key is to set up your ratios carefully, making sure the corresponding units are in the same position on both sides of the equation.

Property

A percent proportion is an equation where a percent is equal to an equivalent ratio. The amount is to the base as the percent is to 100.
$\frac{\text{amount}}{\text{base}} = \frac{\text{percent}}{100}$
We can restate this as: The amount out of the base is the same as the percent out of one hundred.

Examples

• To solve "What number is 45% of 80?", set up the proportion $\frac{n}{80} = \frac{45}{100}$. Cross-multiply to get $100n = 3600$, so $n=36$.
• To solve "6.5% of what number is 1.56 dollars?", set up $\frac{1.56}{n} = \frac{6.5}{100}$. Cross-multiply to get $156 = 6.5n$, so $n=24$.
• To solve "What percent of 72 is 9?", set up $\frac{9}{72} = \frac{p}{100}$. Cross-multiply to get $900 = 72p$, so $p=12.5$. The answer is 12.5%.

Explanation

This special proportion is a powerful tool for any percent problem. It turns sentences like "What is 25% of 80?" into an equation you can easily solve by finding the missing piece. Just fill in what you know!