Understand how to simplify ratios, divide quantities, and solve direct and inverse proportion problems — with fully worked exam examples and real-life applications.
A ratio compares two or more quantities of the same kind. It tells you how much of one thing there is relative to another. If a bag contains 3 red sweets and 5 green sweets, the ratio of red to green is 3 : 5. You read this as "three to five."
Ratios do not carry units — you are simply comparing numbers. However, the quantities being compared must be in the same unit before you write the ratio. You cannot directly compare 500 g to 2 kg without first converting: 500 g : 2000 g, which simplifies to 1 : 4.
The order of a ratio matters. The ratio 3 : 5 is not the same as 5 : 3. Always write your ratio in the order the question states.
A ratio is in its simplest form when the only common factor of all parts is 1. Divide every part by their highest common factor (HCF).
When a ratio includes fractions, multiply every part by the lowest common denominator first, then simplify as normal.
Equivalent ratios express the same relationship but with different numbers — just as equivalent fractions do. Multiply or divide all parts by the same number to move between equivalent ratios.
Example: 2 : 3 is equivalent to 4 : 6, 6 : 9, and 10 : 15. All describe the same comparison; they just use bigger or smaller numbers.
This is one of the most frequently tested ratio skills. The method has three steps:
Question: Share £240 between Ali and Bella in the ratio 3 : 5.
Step 1: Total shares = 3 + 5 = 8
Step 2: Value of one share = £240 ÷ 8 = £30
Step 3: Ali gets 3 × £30 = £90; Bella gets 5 × £30 = £150
Check: £90 + £150 = £240 ✓ Always add your answers to confirm they equal the original total.
Question: 360 students are split into three groups in the ratio 2 : 3 : 4. How many students are in each group?
Step 1: Total shares = 2 + 3 + 4 = 9
Step 2: One share = 360 ÷ 9 = 40 students
Step 3: Group A = 2 × 40 = 80; Group B = 3 × 40 = 120; Group C = 4 × 40 = 160
Check: 80 + 120 + 160 = 360 ✓
Sometimes one part of a ratio is given and you must find another part or the total.
Question: Paint is mixed in the ratio red : white = 2 : 7. If you use 14 litres of red, how much white do you need?
Red part = 2 shares = 14 litres, so 1 share = 7 litres. White part = 7 shares = 7 × 7 = 49 litres.
Alternatively, set up a proportion equation: 2/7 = 14/w → w = (14 × 7) ÷ 2 = 49 litres. Both methods give the same result; use whichever you find faster.
Proportion describes a relationship between two quantities. When two quantities are in proportion, their ratio stays constant as both values change. There are two types you must know: direct proportion and inverse proportion.
Two quantities are in direct proportion when they increase or decrease by the same factor at the same time. If you double one, the other doubles too.
The symbol ∝ means "is proportional to." If y ∝ x, then y = kx where k is the constant of proportionality. On a graph, direct proportion always gives a straight line through the origin.
Question: 5 identical pens cost £3.50. Find the cost of 8 pens.
Cost of 1 pen = £3.50 ÷ 5 = £0.70
Cost of 8 pens = 8 × £0.70 = £5.60
This method — find the value of one unit first, then multiply — is called the unitary method. It works reliably on every direct proportion question.
Two quantities are in inverse proportion when one increases as the other decreases by the same factor. If you double one, the other halves.
If y is inversely proportional to x, then y = k/x. The product of the two quantities always equals the same constant k.
Question: 6 workers can build a wall in 10 days. How long would 4 workers take?
Step 1 — Find total work: 6 × 10 = 60 worker-days
Step 2 — Divide by new number of workers: 60 ÷ 4 =
15 days
Always find the total work (or total units) first in inverse proportion problems. This "total" stays constant; what changes is how it is split.
A map scale is a ratio between a distance on the map and the actual distance on the ground. A scale of 1 : 50 000 means 1 cm on the map represents 50 000 cm (500 m or 0.5 km) in real life.
Question: On a 1 : 25 000 map, two towns are 8 cm apart. What is the real distance?
Real distance = 8 × 25 000 = 200 000 cm = 2 km
To go from real distance to map distance, divide instead: if two places are 3 km apart on a 1 : 50 000 map, the map distance = 300 000 ÷ 50 000 = 6 cm.
Ratios and proportion appear across almost every topic in GCSE and O-Level maths — from geometry to probability. Mastering the three-step division method and the unitary method now makes every later topic easier.