Ratio and Proportion

Ratios compare two quantities by division. Proportions equate two ratios. Together they describe how quantities scale relative to each other — recipes, maps, mixing ratios, currency conversion, share prices, statistical odds, mechanical leverage. Anywhere you reason about ‘this many per that many’, you’re using ratio and proportion. The mathematics is straightforward, but the applications stretch from elementary arithmetic to chemistry stoichiometry and finance.

Ratios

A ratio compares two quantities of the same kind, usually written \( a : b \) or as a fraction \( a/b \). Examples:

• Cooking. 2 cups of flour to 1 cup of sugar — ratio 2 : 1.
• Class composition. 18 girls to 12 boys — ratio 18 : 12 = 3 : 2 (simplified).
• Map scale. 1 cm on a map represents 100,000 cm on the ground — ratio 1 : 100,000.

Ratios are scale-free: 2 : 1 and 4 : 2 and 100 : 50 all describe the same proportion. The simplified form (using the GCD) is preferred.

Proportions

A proportion is an equation of two ratios:

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

This says ‘a is to b as c is to d’. The cross-multiplication identity is:

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

This is the single most useful tool for solving proportions: if three of \( a, b, c, d \) are known, you can solve for the fourth.

Direct and Inverse Proportion

Two quantities are in direct proportion if doubling one doubles the other. \( y = kx \) for some constant \( k \). Examples: cost and quantity at a fixed price, distance traveled at a fixed speed, weight of a uniform material.

Two quantities are in inverse proportion if doubling one halves the other. \( y = k/x \) for some constant \( k \). Examples: time taken and speed for a fixed distance, pressure and volume of a gas at fixed temperature (Boyle’s law), number of workers and time taken to complete a task.

Worked Examples

Direct: scaling a recipe. A recipe for 4 people uses 200 g of pasta. How much for 7 people? \( \frac{200}{4} = \frac{x}{7} \Rightarrow x = \frac{200 \cdot 7}{4} = 350 \) g.

Inverse: workers and days. 6 workers finish a wall in 10 days. How many days for 4 workers (assuming same total work)? Direct ratio gives wrong answer; this is inverse proportion: \( 6 \cdot 10 = 4 \cdot x \Rightarrow x = 15 \) days.

Currency conversion. 100 USD = 8,300 INR. So 250 USD = \( \frac{8300}{100} \cdot 250 = 20{,}750 \) INR.

Compound Ratios and Percentages

A percentage is a ratio with denominator 100. ‘Tax is 18%’ means a ratio of 18 to 100 between tax and base amount. Compound ratios — multiplying two ratios together, e.g., for converting square units or for combining two scale changes — work by ordinary fraction multiplication.

Example. A map’s scale is 1 : 50,000. Areas on the map scale as the square of the linear ratio: \( 1^2 : 50{,}000^2 = 1 : 2.5 \times 10^9 \). So 1 cm² on the map represents 2.5 km² on the ground.

Applications

• Cooking and baking. Scaling recipes up or down preserves taste only if you respect the ingredient ratios.
• Chemistry. Stoichiometry — the calculation of reactant quantities needed for a given reaction — is just ratio and proportion applied to molecular and molar quantities.
• Map and engineering drawings. A scale of 1 : 100 means 1 cm on the drawing = 100 cm in reality.
• Finance. Price-to-earnings ratios, debt-to-equity ratios, return on equity — every fundamental analysis ratio is a comparison of two financial quantities.
• Mechanical advantage. Gear ratios, pulley ratios, and lever arms all express force or speed multiplications as ratios.
• Statistics. Odds ratios in epidemiology, signal-to-noise ratios in engineering, conversion ratios in marketing — proportional reasoning is everywhere.

Related study notes: Percentages, Golden Ratio, Molarity, Compound Interest.

Frequently Asked Questions