How to Calculate Ratios & Proportions

From simplifying a 24:36 ratio to scaling a recipe for 10 people — ratios and proportions appear everywhere.

QUICK ANSWER

Simplify a ratio: divide both numbers by their GCD. 24:36 → GCD=12 → 2:3. Solve a proportion: cross-multiply. If 3/4 = x/20 → x = (3×20)/4 = 15. Scale a recipe: multiply every ingredient by (new servings ÷ old servings). Recipe for 4 → serving 10: multiply by 2.5.

Part 1: Simplifying Ratios

A ratio compares two (or more) quantities. 12:8 means "12 for every 8" — but 3:2 says the same thing more simply.

1
Find the GCD (Greatest Common Divisor)

The largest number that divides evenly into both. For 12 and 8: factors of 12 = 1,2,3,4,6,12. Factors of 8 = 1,2,4,8. GCD = 4.

2
Divide both numbers by the GCD

12 ÷ 4 = 3. 8 ÷ 4 = 2. Simplified ratio = 3:2.

3
Check you cannot simplify further

GCD of 3 and 2 = 1. Cannot simplify further. Done: 12:8 = 3:2.

Part 2: Solving Proportions (Cross-Multiplication)

A proportion is two equal ratios: a/b = c/d. Cross-multiply: a × d = b × c, then solve for the unknown.

Example: If 5 apples cost $3, how much do 8 apples cost?

  1. Set up: 5/3 = 8/x
  2. Cross-multiply: 5x = 3 × 8 = 24
  3. Solve: x = 24/5 = $4.80
  4. Check: 5/3 ≈ 1.667; 8/4.8 ≈ 1.667 ✓

Real-World Applications

🍳 Scaling Recipes

Multiply all ingredients by (new servings ÷ original servings). Recipe for 4, need 6: factor = 1.5. Every ingredient × 1.5.

🗺️ Map Scales

Scale 1:50,000 means 1 cm on map = 50,000 cm = 500 m in reality. 4 cm on map → 4 × 500 = 2,000 m = 2 km real distance.

💱 Currency Exchange

$1 = £0.80. How many pounds is $250? Proportion: 1/0.80 = 250/x → x = 250 × 0.80 = £200.

🏗️ Scale Models

A 1:24 scale model is 1/24th of actual size. Real car = 4.5 m long → model = 4,500 mm ÷ 24 = 187.5 mm.

Frequently Asked Questions

How do you simplify a ratio? ▾
Divide both numbers by their greatest common divisor (GCD). Example: 24:36. GCD = 12. 24÷12 = 2, 36÷12 = 3. Simplified ratio = 2:3. Another example: 15:25. GCD = 5. 15÷5 = 3, 25÷5 = 5. Simplified = 3:5.
How do you solve a proportion? ▾
Set up as a/b = c/d and cross-multiply: a × d = b × c, then solve for the unknown. Example: 3/4 = x/20. Cross-multiply: 3 × 20 = 4 × x → 60 = 4x → x = 15. Verify: 3/4 = 15/20 = 0.75. ✓
How do you scale a recipe using ratios? ▾
Find the scaling factor: new servings ÷ original servings. Multiply every ingredient amount by that factor. Example: recipe serves 4, you want to serve 10. Factor = 10/4 = 2.5. If the recipe needs 200g flour: 200 × 2.5 = 500g flour. Repeat for every ingredient.
How do you read a map scale? ▾
A scale of 1:50,000 means 1 cm on the map equals 50,000 cm (500 m) in reality. So 4 cm on the map = 4 × 500 = 2,000 m = 2 km of real distance. The same logic applies to scale models: a 1:24 model is 1/24th of actual size.
How do you use a proportion for currency exchange? ▾
Set up a proportion using the exchange rate. If $1 = £0.80, to convert $250 to pounds: 1/0.80 = 250/x → x = 250 × 0.80 = £200. To go the other way, divide by the rate instead of multiplying.

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