Training Ratios and Proportions Setting Up and Solving Proportions
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Setting Up and Solving Proportions

20 min Ratios and Proportions

Proportions

A proportion is a statement that two ratios are equal. Setting up and solving proportions is one of the most versatile problem-solving techniques in all of mathematics, used in everything from cooking to engineering.

The key tool for solving proportions is cross-multiplication: if a/b equals c/d, then a times d equals b times c. This converts the proportion into a simple equation that you can solve in one or two steps.

This lesson walks you through setting up proportions from word problems, cross-multiplying, and solving for the unknown — skills that will reappear in geometry, trigonometry, and statistics.

Definition

A proportion states that two ratios are equal: $\;\dfrac{a}{b} = \dfrac{c}{d}$

Cross-Multiplication

If $\dfrac{a}{b} = \dfrac{c}{d}$, then $a \times d = b \times c$.

Example 1

Solve: $\dfrac{x}{12} = \dfrac{5}{8}$

$$8x = 60 \quad\Longrightarrow\quad x = 7.5$$

Example 2

Solve: $\dfrac{3}{7} = \dfrac{15}{x}$

$$3x = 105 \quad\Longrightarrow\quad x = 35$$

Example 3

A car travels 180 miles in 3 hours. How far in 5 hours at constant speed?

$$\frac{180}{3} = \frac{x}{5} \quad\Longrightarrow\quad 3x = 900 \quad\Longrightarrow\quad x = 300 \text{ miles}$$

Example 4

Solve: $\dfrac{x + 2}{9} = \dfrac{4}{3}$

$$3(x + 2) = 36 \quad\Longrightarrow\quad 3x + 6 = 36 \quad\Longrightarrow\quad x = 10$$

Verifying a Proportion

Example 5

Is $\dfrac{6}{15} = \dfrac{10}{25}$ true?

$6 \times 25 = 150$ and $15 \times 10 = 150$. Cross-products equal → true. ✓

Practice Problems

1. Solve: $\dfrac{x}{6} = \dfrac{10}{15}$
2. Solve: $\dfrac{4}{x} = \dfrac{12}{27}$
3. Is $\dfrac{8}{14} = \dfrac{12}{21}$ a true proportion?
4. Solve: $\dfrac{x}{8} = \dfrac{21}{24}$
5. If 5 notebooks cost $\$8.75$, what do 12 cost?
6. Solve: $\dfrac{9}{x} = \dfrac{3}{11}$
7. Solve: $\dfrac{x - 1}{5} = \dfrac{3}{10}$
8. If 3 kg costs $\$7.50$, what does 8 kg cost?
9. Solve: $\dfrac{2.5}{4} = \dfrac{x}{10}$
10. Is $\dfrac{9}{24} = \dfrac{15}{40}$ a true proportion?
11. If you drive 210 mi in 3.5 hr, how far in 5 hr?
12. Solve: $\dfrac{7}{x + 3} = \dfrac{14}{10}$
Show Answer Key

1. $x = 4$

2. $x = 9$

3. Yes ($168 = 168$)

4. $x = 7$

5. $\$21$

6. $x = 33$

7. $x = 2.5$

8. $\$20$

9. $x = 6.25$

10. Yes ($360 = 360$)

11. $300$ mi

12. $x = 2$

Ratios and Rates Applications of Proportions