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§ Ratios & Proportions

Ratios & Proportions

§ Ratios & Proportions

Ratios & Proportions

CCSS.6.RPCCSS.7.RP3 min read

A ratio expresses the relationship between two or more quantities by comparing their sizes, written as a:b or a/b. Proportions occur when two ratios are equal, forming equations that can be solved through cross-multiplication. These mathematical tools appear throughout Year 10 GCSE mathematics, particularly in real-world problem-solving contexts.

§ 01

Why it matters

Ratios and proportions form the foundation for countless real-world calculations. Map reading relies on scale ratios like 1:25,000 to convert 4 cm measurements into 1 km distances. Recipe scaling uses proportions when cooking for 8 people instead of 4, doubling ingredient quantities proportionally. Financial literacy involves ratio comparisons, such as determining that £3.60 for 12 biscuits offers better value than £2.50 for 8 biscuits. Construction workers use ratios for concrete mixing, combining cement, sand, and gravel in fixed proportions like 1:2:4. Fashion designers scale patterns proportionally across different clothing sizes. These skills directly support GCSE topics including similar triangles, percentage calculations, and algebraic problem-solving, making ratios essential for mathematical progression into A-levels and beyond.

§ 02

How to solve ratios & proportions

Ratios & Proportions

  • A ratio compares two quantities (a:b or a/b).
  • To solve a proportion a/b = c/d: cross-multiply (a×d = b×c).
  • Simplify ratios by dividing both by their GCF.

Example: 23 = x/12 → 2×12 = 3x → x = 8.

§ 03

Worked examples

Beginner§ 01

Simplify the ratio 8:10.

Answer: 4:5

  1. Find GCF of 8 and 10 → GCF = 2 — Divide both by the GCF.
  2. Divide → 8÷2 : 10÷2 = 4:5 — Simplified ratio.
Easy§ 02

A map has a scale of 1:25,000. A road measures 3 cm on the map. How long is the road in real life (in km)?

Answer: 0.75 km

  1. Multiply by scale → 3 cm × 25,000 = 75,000 cm — Map distance times scale gives real distance in cm.
  2. Convert to km → 75,000 cm ÷ 100,000 = 0.75 km — 100,000 cm = 1 km.
Medium§ 03

You need 3 eggs to make 36 cookies. How many eggs do you need for 44 cookies?

Answer: 3.7

  1. Set up proportion → 336 = ?/44 — Eggs to cookies ratio.
  2. Cross-multiply and solve → ? = 3 × 44 ÷ 36 = 3.7 — Solve for the unknown.
§ 04

Common mistakes

  • Adding ratios incorrectly, writing 2:3 + 4:5 = 6:8 instead of finding a common denominator first
  • Cross-multiplying incorrectly in proportions, calculating 3/4 = x/8 as 3×8 = x×4 but then solving x = 24÷8 = 3 instead of x = 6
  • Confusing ratio order, writing speed ratios as distance:time instead of time:distance, leading to inverted calculations
§ 05

Frequently asked questions

What is the difference between a ratio and a proportion?
A ratio compares two quantities (like 3:4), whilst a proportion states that two ratios are equal (like 3:4 = 6:8). Ratios show relationships; proportions create equations that can be solved for unknown values.
How do you simplify ratios with different units?
Convert both quantities to the same unit first. For example, to simplify 2 metres:50 centimetres, convert to 200 cm:50 cm, then divide both by 50 to get 4:1.
When do you use cross-multiplication for proportions?
Use cross-multiplication when solving equations like a/b = c/d for an unknown variable. Multiply the numerator of one fraction by the denominator of the other: a×d = b×c.
How do you check if ratios are equivalent?
Cross-multiply or convert to decimal form. For 4:6 and 2:3, either check 4×3 = 6×2 (both equal 12) or divide 4÷6 = 0.667 and 2÷3 = 0.667.
What does a 1:50,000 map scale actually mean?
Every 1 unit on the map represents 50,000 units in reality. So 1 cm on the map equals 50,000 cm (or 500 metres) in real life. This scale helps calculate actual distances from map measurements.
§ 06

See also

RatioPercentageProportionDenominatorVariableNumeratorFractionMean
§ 06

Where to next?

Prerequisites
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