Ratios and Proportions: Scaling and Comparison
A ratio compares two or more quantities. A proportion states that two ratios are equal. Together they form the backbone of scaling, recipe adjustment, map reading, finance, and data analysis.
Simplifying a Ratio
Divide all parts by their GCD
24:36 → GCD=12 → 2:3
15:25:35 → GCD=5 → 3:5:7Solving a Proportion (Cross-Multiplication)
a/b = c/d → a×d = b×c
Example: 3/4 = x/20 → x = (3×20)/4 = 15Dividing a Quantity in a Ratio
Share £360 in ratio 3:5:4
Total parts = 12
Each part = £360÷12 = £30
Shares: £90 : £150 : £120Scale Ratios (Maps and Models)
Map scale 1:50,000
2.4 cm on map → 2.4 × 50,000 = 120,000 cm = 1.2 kmDirect vs Inverse Proportion
- Direct: y = kx (double x → double y). Speed and distance at constant time.
- Inverse: y = k/x (double x → halve y). Speed and time at constant distance.
Calculate ratios and proportions: Free Ratio Calculator
Ratio vs Proportion
A ratio compares two quantities: 3:2 means "three of the first for every two of the second." A proportion states that two ratios are equal: 3/2 = 6/4. The key tool for solving proportions is cross-multiplication: if a/b = c/d, then ad = bc. This allows any one unknown to be found from three known values. Example: if 3 kg of coffee costs £24, how much do 7 kg cost? 3/24 = 7/x → x = 24×7/3 = £56.
Real-World Applications
- Cooking and baking: Scaling recipes up or down. If a recipe for 4 serves uses 200 g flour, a batch for 10 needs 200 × 10/4 = 500 g.
- Map reading: A map scale of 1:50,000 means 1 cm on the map = 50,000 cm = 500 m in reality.
- Scale models: Architects and engineers build models at fixed ratios (1:100 is common) and need to convert real dimensions.
- Mixing solutions: A 1:9 cleaning concentrate to water ratio means 100 mL concentrate in 900 mL water for 1 litre total.
- Gear and pulley systems: A 3:1 gear ratio means the driven gear rotates three times per rotation of the drive gear.
Frequently Asked Questions
What is the difference between direct and inverse proportion?
Direct proportion: as x increases, y increases at the same rate (y = kx). Inverse proportion: as x increases, y decreases (xy = k). Speed and journey time are inversely proportional at fixed distance. Hours worked and pay are directly proportional at fixed hourly rate.
How do I simplify a ratio?
Divide both parts by their greatest common divisor (GCD). The ratio 18:24: GCD(18,24) = 6, so 18:24 = 3:4. For three-part ratios like 12:18:24, divide all parts by GCD(12,18,24) = 6 to get 2:3:4.
What is the Golden Ratio?
The golden ratio φ ≈ 1.618 satisfies the proportion a:b = (a+b):a. It appears in art, architecture, and nature (shell spirals, leaf arrangements). The ratio of consecutive Fibonacci numbers (1,1,2,3,5,8,13...) converges to φ.