Training Ratios and Proportions Applications of Proportions
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Applications of Proportions

20 min Ratios and Proportions

Applications of Proportions

Proportions are far more than a classroom exercise — they are one of the most widely used mathematical tools in the real world. Map scales, recipe adjustments, unit conversions, and scale models all depend on proportional reasoning.

This lesson focuses on applications of proportions. You will learn to set up the correct proportion from a word problem, solve for the unknown, and verify that your answer makes sense in context.

Similar Figures

Definition

Two figures are similar if their corresponding angles are equal and their corresponding sides are proportional.

Example 1

Triangle A has sides 3, 4, 5. Triangle B is similar with shortest side 9. Find all sides of B.

Scale factor: $\dfrac{9}{3} = 3$. Sides: $9$, $12$, $15$.

Scale Drawings and Maps

Example 2

Scale: $1$ in $= 50$ mi. Two cities are 3.5 in apart on the map. Actual distance?

$$\frac{1}{50} = \frac{3.5}{x} \quad\Longrightarrow\quad x = 175 \text{ miles}$$

Recipe Scaling

Example 3

Recipe for 4 servings uses 3 cups rice. How much for 10 servings?

$$\frac{3}{4} = \frac{x}{10} \quad\Longrightarrow\quad x = 7.5 \text{ cups}$$

Direct and Inverse Proportion

Key Distinction
  • Direct: Both quantities increase/decrease together. $y = kx$.
  • Inverse: One increases as the other decreases. $xy = k$.
Example 4

5 workers finish a job in 12 hours. How long for 3 workers?

Inverse proportion: $5 \times 12 = 3 \times x \;\Longrightarrow\; x = 20$ hours.

Example 5

A model car is $\dfrac{1}{24}$ scale. Real car is 15 ft long. Model length?

$\dfrac{x}{15} = \dfrac{1}{24} \;\Longrightarrow\; x = 0.625$ ft $= 7.5$ inches.

Practice Problems

1. Triangles are similar. Sides of A: 5, 8, 10. Shortest side of B is 15. Find all sides of B.
2. Map scale $1$ cm $= 25$ km. Distance is 7.2 cm. Actual distance?
3. Recipe for 6 serves needs 2 cups sugar. How much for 15 servings?
4. 8 workers paint in 6 days. How long for 4 workers?
5. A 6-ft pole casts a 4-ft shadow. Tree casts a 20-ft shadow. Height?
6. Blueprint scale: 1 in = 8 ft. Room is 3.5 in × 4 in on blueprint. Actual dimensions?
7. Gear A (20 teeth) meshes with Gear B (50 teeth). A turns 100 times. How many turns for B?
8. Model train scale $1 : 87$. Actual car is 50 ft. Model length in inches?
9. 3 pumps fill a tank in 8 hours. Time for 6 pumps?
10. Similar rectangles: small is 4 × 6. Large has length 15. Width?
Show Answer Key

1. $15, 24, 30$

2. $180$ km

3. $5$ cups

4. $12$ days

5. $30$ ft

6. $28$ ft × $32$ ft

7. $40$ turns

8. $\dfrac{50 \times 12}{87} \approx 6.9$ inches

9. $4$ hours

10. $10$

Setting Up and Solving Proportions Proportion Word Problems