Proportions
Proportions are equations that say two ratios are equal, like \frac{2}{3} = \frac{8}{12}. To solve a proportion with a missing number, cross multiply. Multiply each top number by the bottom number on the other side, set the two products equal, then divide. In \frac{4}{5} = \frac{x}{10}, you get 5x = 40, so x = 8.
Before this lesson: You should know how to write equivalent ratios and how to find a unit rate.
What Is a Proportion?
A proportion says that two ratios make the same comparison. For example, 2 cups of flour for every 3 cups of sugar is the same comparison as 8 cups of flour for every 12 cups of sugar. So \frac{2}{3} = \frac{8}{12} is a proportion.
In a proportion problem, one number is often missing. We use a letter, like x, to stand for it. A letter that stands for a number is called a variable.
How to Solve a Proportion
Use cross multiplication. The two products along the X are always equal in a true proportion.
\frac{a}{b} = \frac{c}{d} \quad \text{means} \quad a \times d = b \times c
- Write the proportion so the units match. If miles are on top on the left, miles go on top on the right.
- Cross multiply.
- Divide both sides by the number next to the variable.
- Check your answer by putting it back into the proportion.
Worked Examples
Example 1: Test a proportion
Is \frac{6}{8} = \frac{9}{12} a true proportion?
Step 1. Cross multiply: 6 \times 12 = 72 and 8 \times 9 = 72.
Step 2. The cross products are equal.
Answer: Yes. Both ratios simplify to \frac{3}{4}, so the proportion is true.
Example 2: Missing number on top
Solve \frac{x}{6} = \frac{10}{15}.
Step 1. Cross multiply.
\begin{aligned}
15 \times x &= 6 \times 10 \\
15x &= 60
\end{aligned}Step 2. Divide both sides by 15: x = 60 \div 15 = 4.
Answer: x = 4. Check: \frac{4}{6} and \frac{10}{15} both equal \frac{2}{3}.
Example 3: Missing number on the bottom
Solve \frac{3}{7} = \frac{12}{x}.
Step 1. Cross multiply: 3 \times x = 7 \times 12, so 3x = 84.
Step 2. Divide both sides by 3: x = 28.
Answer: x = 28.
Example 4: A map scale
On a map, 1 inch stands for 25 miles. Two towns are 3.5 inches apart on the map. How far apart are they in real life?
Step 1. Write the proportion with inches on top and miles on the bottom.
\frac{1 \text{ inch}}{25 \text{ miles}} = \frac{3.5 \text{ inches}}{x \text{ miles}}Step 2. Cross multiply: 1 \times x = 25 \times 3.5, so x = 87.5.
Answer: The towns are 87.5 miles apart.
Example 5: Word problem
Five notebooks cost $8.75. At the same price, how much do 8 notebooks cost?
Step 1. Set up the proportion with notebooks on top and dollars on the bottom: \frac{5}{8.75} = \frac{8}{x}.
Step 2. Cross multiply: 5x = 8 \times 8.75 = 70.
Step 3. Divide both sides by 5: x = 14.
Answer: Eight notebooks cost $14.00. You can also use the unit rate: $8.75 ÷ 5 = $1.75 each, and 8 × $1.75 = $14.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Mixing up the units | \frac{5 \text{ notebooks}}{8.75 \text{ dollars}} = \frac{x \text{ dollars}}{8 \text{ notebooks}} | Keep the same unit on top of both ratios. |
| Multiplying straight across | 4 \times x = 5 \times 10 for \frac{4}{5} = \frac{x}{10} | Multiply along the X: 4 \times 10 = 5 \times x. |
| Adding instead of multiplying | \frac{2}{5} = \frac{4}{x}, so x = 7 | Ratios grow by multiplying. Cross multiply: 2x = 20, so x = 10. |
| Stopping at the cross product | 5x = 40, so x = 40 | Divide by the number next to x: x = 8. |
Proportions, Percents and Scale
Many problems are proportions in disguise:
- Percents: \frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}. See percentages.
- Scale drawings and maps: drawing length over real length is the same for every part of the drawing.
- Recipes and prices: if one amount doubles, the other doubles too.
When two amounts always change by the same factor, they are in a proportional relationship. In Algebra 1, this becomes direct variation (coming soon).
Practice Problems
For problems 1–2, tell whether the proportion is true. For problems 3–7, solve for x.
- \frac{3}{4} = \frac{15}{20}
- \frac{2}{5} = \frac{5}{12}
- \frac{x}{9} = \frac{4}{12}
- \frac{5}{x} = \frac{20}{36}
- \frac{7}{2} = \frac{21}{x}
- \frac{12}{x} = \frac{3}{5}
- \frac{2.5}{x} = \frac{5}{8}
- On a map, 1 inch stands for 40 miles. How many miles do 2.25 inches stand for?
- A recipe uses 3 eggs for every 2 cups of flour. How many eggs do you need for 4 cups of flour?
- A car goes 84 miles on 3 gallons of gas. How far can it go on 10 gallons?
Show answers
- 3 \times 20 = 60 and 4 \times 15 = 60. True.
- 2 \times 12 = 24 and 5 \times 5 = 25. Not true, because 24 is not 25.
- 12x = 36, so x = 3.
- 20x = 180, so x = 9.
- 7x = 42, so x = 6.
- 3x = 60, so x = 20.
- 5x = 20, so x = 4.
- \frac{1}{40} = \frac{2.25}{x}, so x = 40 \times 2.25 = 90 miles.
- \frac{3}{2} = \frac{x}{4}, so 2x = 12 and x = 6 eggs.
- \frac{84}{3} = \frac{x}{10}, so 3x = 840 and x = 280 miles.
Proportions FAQ
What is the difference between a ratio and a proportion?
A ratio compares two amounts, like 3:4. A proportion is an equation that says two ratios are equal, like \frac{3}{4} = \frac{6}{8}.
Why does cross multiplying work?
Multiplying both sides of \frac{a}{b} = \frac{c}{d} by b and by d clears both fractions. What is left is a \times d = b \times c.
How do you know if two ratios are proportional?
Cross multiply. If the two products are equal, the ratios are proportional. You can also simplify both ratios and see if they match.
Summary
- A proportion says two ratios are equal.
- Cross multiply to solve: \frac{a}{b} = \frac{c}{d} means a \times d = b \times c.
- Divide by the number next to the variable to finish.
- Keep the same units on top of both ratios.
- Check your answer by putting it back in the proportion.
Previous lesson: Unit Rates
Next lesson: Unit Conversions
Back to the Pre-Algebra Index