Solving Proportions

A proportion is a statement that two ratios are equal. Proportions let you find unknown quantities when you know that two things are in the same ratio — which happens constantly in recipes, construction, medicine, and science. If you can set up and solve a proportion, you can handle a huge range of practical problems.

What Is a Proportion?

A proportion says that two ratios have the same value:

$\frac{a}{b} = \frac{c}{d}$

For example, $\frac{2}{3} = \frac{8}{12}$ is a proportion because both fractions simplify to the same value.

You can verify a proportion by simplifying both sides or by cross-multiplying (which we will cover next).

Example 1: Is This a Proportion?

Is $\frac{4}{6} = \frac{10}{15}$ a true proportion?

Method 1 — Simplify both sides:

$\frac{4}{6} = \frac{2}{3} \qquad \frac{10}{15} = \frac{2}{3}$

Both simplify to $\frac{2}{3}$, so yes, this is a true proportion.

Method 2 — Cross-multiply:

$4 \times 15 = 60 \qquad 6 \times 10 = 60$

The cross products are equal, so it is a true proportion.

Cross-Multiplication

Cross-multiplication is the standard method for solving proportions. Given:

$\frac{a}{b} = \frac{c}{d}$

Cross-multiply to get:

$a \times d = b \times c$

This works because multiplying both sides of the equation by $b \times d$ clears the fractions.

Example 2: Solve for the Unknown

Solve $\frac{3}{5} = \frac{x}{20}$.

Step 1: Cross-multiply.

$3 \times 20 = 5 \times x$

$60 = 5x$

Step 2: Divide both sides by 5.

$x = \frac{60}{5} = 12$

Step 3: Check. $\frac{12}{20}$ simplifies to $\frac{3}{5}$. Checks out.

Answer: $x = 12$

Example 3: Unknown in the Denominator

Solve $\frac{7}{x} = \frac{21}{30}$.

Step 1: Cross-multiply.

$7 \times 30 = x \times 21$

$210 = 21x$

Step 2: Divide both sides by 21.

$x = \frac{210}{21} = 10$

Step 3: Check. $\frac{7}{10} = \frac{21}{30}$. Simplify the right side: $\frac{21}{30} = \frac{7}{10}$. Checks out.

Answer: $x = 10$

Example 4: Unknown That Produces a Decimal

Solve $\frac{5}{8} = \frac{x}{14}$.

Step 1: Cross-multiply.

$5 \times 14 = 8 \times x$

$70 = 8x$

Step 2: Divide both sides by 8.

$x = \frac{70}{8} = 8.75$

Step 3: Check. $\frac{5}{8} = 0.625$ and $\frac{8.75}{14} = 0.625$. Checks out.

Answer: $x = 8.75$

Solving Proportion Word Problems

The key to word problems is setting up the proportion correctly. Follow these steps:

1. Identify the two ratios the problem gives you (or one ratio and a partial second ratio).
2. Keep consistent units — the same quantity must be in the same position on both sides.
3. Cross-multiply and solve.
4. Check your answer by substituting back.

Example 5: Recipe Scaling

A recipe calls for 3 cups of flour to make 24 cookies. How much flour is needed for 40 cookies?

Step 1: Set up the proportion with flour on top and cookies on the bottom.

$\frac{3 \text{ cups}}{24 \text{ cookies}} = \frac{x \text{ cups}}{40 \text{ cookies}}$

Step 2: Cross-multiply.

$3 \times 40 = 24 \times x$

$120 = 24x$

Step 3: Solve.

$x = \frac{120}{24} = 5$

Answer: You need 5 cups of flour for 40 cookies.

Example 6: Map Distance

On a map, 2 inches represents 50 miles. Two cities are 7 inches apart on the map. What is the actual distance?

$\frac{2 \text{ in}}{50 \text{ mi}} = \frac{7 \text{ in}}{x \text{ mi}}$

Cross-multiply:

$2x = 50 \times 7 = 350$

$x = 175$

Answer: The actual distance is 175 miles.

Real-World Application: Nursing — Dosage Calculation

A doctor orders 250 mg of a medication. The available liquid concentration is 100 mg per 4 mL. How many mL should the nurse administer?

Step 1: Set up the proportion.

$\frac{100 \text{ mg}}{4 \text{ mL}} = \frac{250 \text{ mg}}{x \text{ mL}}$

Step 2: Cross-multiply.

$100 \times x = 4 \times 250$

$100x = 1{,}000$

Step 3: Solve.

$x = \frac{1{,}000}{100} = 10$

Answer: The nurse administers 10 mL of the medication.

Real-World Application: Carpentry — Lumber Estimation

A carpenter knows that a small deck project used 14 boards for an 84 square foot area. Now the carpenter is building a larger deck that is 210 square feet. How many boards are needed?

$\frac{14 \text{ boards}}{84 \text{ sq ft}} = \frac{x \text{ boards}}{210 \text{ sq ft}}$

Cross-multiply:

$14 \times 210 = 84 \times x$

$2{,}940 = 84x$

$x = \frac{2{,}940}{84} = 35$

Answer: The carpenter needs 35 boards for the larger deck.

Common Mistakes to Avoid

1. Mismatched positions. If cups are in the numerator on the left, they must be in the numerator on the right too. Writing $\frac{\text{cups}}{\text{cookies}} = \frac{\text{cookies}}{\text{cups}}$ produces the wrong answer.

2. Forgetting to check. Always substitute your answer back into the original proportion to verify.

3. Setting up the wrong ratio. Read the problem carefully to decide which quantities are being compared. Drawing a simple table (left column = known, right column = unknown) helps.

4. Confusing cross-multiplication with cross-addition. You multiply diagonally, not add. $\frac{a}{b} = \frac{c}{d}$ gives $ad = bc$, not $a + d = b + c$.

Practice Problems

Key Takeaways

• A proportion states that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$.
• Cross-multiplication ($ad = bc$) is the standard solving technique.
• When setting up word problems, keep the same units in the same position on both sides of the equation.
• Always check your answer by substituting back into the original proportion.
• Proportions work for direct relationships — when one quantity increases, the other increases at the same rate.

Return to Pre-Algebra for more topics in this section.