Ratios and Proportions

A lesson on comparing quantities with ratios, simplifying ratios, setting up and solving proportions with cross-multiplication, and dividing quantities in a given ratio.

Ratios

A ratio compares two quantities of the same kind. It can be written three ways:

$3 \text{ to } 5 \quad = \quad 3:5 \quad = \quad \frac{3}{5}$

Ratios are simplified like fractions by dividing by the GCD:

$12:8 = 3:2$

Note Ratios must compare the same units. Convert if needed: 30 cm : 1.2 m = 30 : 120 = 1 : 4.

Proportions

A proportion is a statement that two ratios are equal:

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

This is true when $a \times d = b \times c$ (cross-multiplication).

Solving Proportions

1
Write the proportion
$\frac{5}{8} = \frac{x}{40}$
2
Cross-multiply
$8x = 5 \times 40 = 200$
3
Solve
$x = 200 \div 8 = 25$

Dividing in a Given Ratio

To divide 120 in the ratio 3:5:

Total parts = 8 → each part = 15

$3 \times 15 = 45, \quad 5 \times 15 = 75$

Direct Proportion

If $y$ is directly proportional to $x$ : $y = kx$ (as $x$ increases, $y$ increases).

Inverse Proportion

If $y$ is inversely proportional to $x$ : $y = k/x$ (as $x$ increases, $y$ decreases).

A car travels 150 km in 2 hours. How far does it travel in 5 hours (constant speed)?

$\frac{150}{2} = \frac{d}{5}$

$d = \frac{150 \times 5}{2} = 375 \text{ km}$

Real world: map scales

A map has a scale of 1 : 50 000. A measured distance of 4 cm on the map represents $4 \times 50\,000 = 200\,000\text{ cm} = 2\text{ km}$ on the ground. Hikers and engineers use map scales every day to plan routes.

Real world: mixing paint colours

A decorator mixes blue and white paint in the ratio 3 : 7 to get a sky-blue tint. For 5 litres total: blue $= \frac{3}{10} \times 5 = 1.5\text{ L}$ , white $= \frac{7}{10} \times 5 = 3.5\text{ L}$ . Ratios ensure the exact same shade can be reproduced.

Real world: recipe scaling

A bread recipe for 2 loaves needs 600 g of flour, 10 g of yeast, and 400 mL of water (ratio 60 : 1 : 40). For 5 loaves: multiply every ingredient by $\frac{5}{2} = 2.5$ . Proportions keep the recipe balanced regardless of batch size.

Tip Before solving a proportion, check that both ratios use the same units in the same order (e.g., kilometres per hour : kilometres per hour).

Practice Problems

Simplify 45:30.

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GCD = 15. Simplified: 3:2.

Divide 200 in the ratio 3:7.

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Total parts = 10. Each part = 20. Shares: 60 and 140.

Solve: 4/x = 6/15.

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Cross-multiply: 60 = 6x → x = 10.