Ratios and Proportions

A ratio compares two quantities. A proportion states that two ratios are equal. Proportions are solved using cross-multiplication.

What Is a Ratio?

A ratio compares two quantities. The ratio of 3 apples to 5 oranges is written:

$3 : 5 \quad \text{or} \quad \frac{3}{5}$

Ratios can be simplified like fractions: $6:10 = 3:5$ .

Note Order matters! The ratio 3:5 is different from 5:3.

A proportion says two ratios are equal:

$\frac{a}{b} = \frac{c}{d}$

\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c

Cross-multiply to solve for an unknown in a proportion.

Find $x$ :

$\frac{3}{4} = \frac{x}{12}$

Cross-multiply: $3 \times 12 = 4 \times x$

$36 = 4x \implies x = 9$

Divide 60 in the ratio 2:3:

Total parts = 2 + 3 = 5

Each part = 60 ÷ 5 = 12

$2 \times 12 = 24, \quad 3 \times 12 = 36$

A recipe uses flour and sugar in the ratio 5:2. If you use 350 g of flour, how much sugar do you need?

$\frac{5}{2} = \frac{350}{x}$

$5x = 700 \implies x = 140 \text{ g}$

Real world: reading a map scale

A hiking map has a scale of 1 : 50,000. A trail measures 4.6 cm on the map.

Real distance $= 4.6 \times 50{,}000 = 230{,}000 \text{ cm} = 2.3 \text{ km}$

Hikers and orienteers use this proportion to plan routes and estimate walking time before setting off.

Real world: architect's blueprint

A building plan is drawn at a scale of 1 : 100. A wall appears as 3.5 cm on the drawing.

Real length $= 3.5 \times 100 = 350 \text{ cm} = 3.5 \text{ m}$

Architects, builders, and interior designers all use scale ratios to translate drawings into real structures.

Real world: mixing concrete

A concrete mix requires cement, sand, and gravel in the ratio 1 : 2 : 4. A builder needs 70 kg of gravel.

Total parts $= 1 + 2 + 4 = 7$ . One part $= 70 \div 4 = 17.5 \text{ kg}$

Cement: $17.5 \text{ kg}$ — Sand: $35 \text{ kg}$ — Gravel: $70 \text{ kg}$

Using the wrong ratio weakens the concrete — structural engineers specify exact ratios for safety.

Simplify the ratio 24:36.

Show answer

GCD(24,36) = 12. Simplified: 2:3.

Solve: 7/x = 14/10.

Show answer

Cross-multiply: 70 = 14x → x = 5.