Ratio and Proportion Word Problems

Understanding Ratios in Context

A ratio compares two quantities and can be applied to real-world situations.

Ways to write ratios:

• 3 to 4
• 3:4
• 3/4

Types of Ratio Problems

Part-to-part: Comparing two parts of a whole

• "The ratio of boys to girls is 3:2"

Part-to-whole: Comparing a part to the total

• "3 out of 5 students walk to school" (3:5)

Rate: Comparing different units

• "60 miles in 2 hours" (60:2 or 30 miles per hour)

Setting Up Proportions

When two ratios are equal, we have a proportion.

Format: a/b = c/d

Cross-multiply to solve: a × d = b × c

Example 1: Recipe Scaling

Problem: A recipe for 4 servings needs 2 cups of flour. How much flour for 10 servings?

Set up proportion:

4 servings     10 servings
─────────  =  ─────────
  2 cups        x cups

Cross-multiply:

• 4 × x = 2 × 10
• 4x = 20
• x = 5

Answer: 5 cups of flour

Example 2: Finding Unknown Quantity

Problem: The ratio of cats to dogs at a shelter is 5:3. If there are 15 cats, how many dogs?

Set up proportion:

5 cats     15 cats
──────  =  ───────
3 dogs     x dogs

Cross-multiply:

• 5 × x = 3 × 15
• 5x = 45
• x = 9

Answer: 9 dogs

Example 3: Scale Drawings

Problem: On a map, 2 inches represents 50 miles. How many miles does 7 inches represent?

Set up proportion:

2 inches     7 inches
────────  =  ────────
50 miles     x miles

Cross-multiply:

• 2 × x = 50 × 7
• 2x = 350
• x = 175

Answer: 175 miles

Example 4: Unit Price Comparison

Problem: Store A sells 3 apples for $2. Store B sells 5 apples for $3. Which is the better deal?

Find unit price (price per apple):

Store A: $2 ÷ 3 ≈ $0.67 per apple Store B: $3 ÷ 5 = $0.60 per apple

Answer: Store B is the better deal ($0.60 < $0.67)

Example 5: Mixture Problems

Problem: A paint mixture uses blue and yellow in a ratio of 2:3. If you use 6 cups of blue, how much yellow do you need?

Set up proportion:

2 blue      6 blue
──────  =  ─────────
3 yellow   x yellow

Cross-multiply:

• 2 × x = 3 × 6
• 2x = 18
• x = 9

Answer: 9 cups of yellow

Example 6: Part-to-Whole

Problem: In a class, the ratio of students who walk to school to total students is 3:8. If 24 students walk, how many total students?

Set up proportion:

3 walk      24 walk
──────  =  ─────────
8 total    x total

Cross-multiply:

• 3 × x = 8 × 24
• 3x = 192
• x = 64

Answer: 64 total students

Example 7: Similar Figures

Problem: Two similar triangles have a ratio of sides 4:7. If the smaller triangle has a side of 12 cm, what's the corresponding side on the larger triangle?

Set up proportion:

4 (small)     12 cm
─────────  =  ───────
7 (large)     x cm

Cross-multiply:

• 4 × x = 7 × 12
• 4x = 84
• x = 21

Answer: 21 cm

Example 8: Speed/Rate Problems

Problem: A car travels 120 miles in 2 hours. At this rate, how far will it travel in 5 hours?

Set up proportion:

120 miles     x miles
─────────  =  ───────
 2 hours      5 hours

Cross-multiply:

• 120 × 5 = 2 × x
• 600 = 2x
• x = 300

Answer: 300 miles

Example 9: Finding Both Parts

Problem: The ratio of red marbles to blue marbles is 3:5. If there are 40 marbles total, how many of each color?

Method:

• Parts: 3 + 5 = 8 parts total
• Each part: 40 ÷ 8 = 5 marbles

Red: 3 parts × 5 = 15 marbles Blue: 5 parts × 5 = 25 marbles

Check: 15 + 25 = 40 ✓

Example 10: Percent as Ratio

Problem: 15% of students play soccer. If 60 students play soccer, how many students total?

Think: 15% means 15:100 ratio

Set up:

  15        60
──────  =  ─────
  100       x

Cross-multiply:

• 15 × x = 100 × 60
• 15x = 6,000
• x = 400

Answer: 400 students total

Problem-Solving Strategy

Step 1: Identify what you're comparing

Step 2: Write the known ratio

Step 3: Set up a proportion with the unknown

Step 4: Cross-multiply and solve

Step 5: Check if your answer makes sense

Practice