Ratio and Proportion

Understanding comparisons and scale

Ratio is a key number topic in British secondary schools. It helps students compare quantities, share amounts fairly, use scale drawings, solve recipe questions and understand direct proportion. Students usually meet simple comparison work in primary school, then develop it in KS3. At GCSE and IGCSE it becomes more demanding because questions are often written as real-life problems rather than direct calculations. The main idea is that a ratio compares parts. If red counters and blue counters are in the ratio 2:3, then for every 2 red counters there are 3 blue counters. The total number of parts is 5. This does not mean there are only 5 counters; it means the quantities follow the same pattern. There could be 10 red and 15 blue, or 20 red and 30 blue. The relationship stays the same. Proportion is closely connected. If two quantities increase or decrease in the same pattern, they are in direct proportion. For example, if 4 notebooks cost £8, then 8 notebooks cost £16, because the number of notebooks and the price both double. Students need to recognise when they should multiply, divide, scale up or scale down.

Useful formulas and methods

Total parts method: add the parts of the ratio, then divide the total amount by that number of parts. Value of one part = total amount divided by total number of parts. Share for each part = value of one part multiplied by the number of parts in that share. Scale factor = new amount divided by original amount. Direct proportion can be written as y = kx, where k is the constant of proportion. If y = 5x, then every time x increases by 1, y increases by 5. A good strategy is to write the ratio clearly, add the parts, find one part and then build each share. This avoids guesswork and helps students show exam working clearly.

Worked examples

Example 1: Share 60 in the ratio 2:3. The total number of parts is 2 + 3 = 5. One part is 60 divided by 5 = 12. The first share is 2 multiplied by 12 = 24. The second share is 3 multiplied by 12 = 36. Example 2: A map has a scale of 1 cm to 4 km. A distance on the map is 7 cm. The real distance is 7 multiplied by 4 = 28 km. Example 3: If 5 pencils cost £2, find the cost of 15 pencils. The number of pencils has been multiplied by 3, so the cost is also multiplied by 3. The answer is £6.

Practice exercises and answers

Exercise 1: Share 84 in the ratio 3:4. Answer: total parts = 7, one part = 12, so the shares are 36 and 48. Exercise 2: A recipe uses 200 g of flour for 8 cakes. How much flour is needed for 20 cakes? Answer: 20 is 2.5 times 8, so 200 g multiplied by 2.5 = 500 g. Exercise 3: y is directly proportional to x. When x = 6, y = 24. Find y when x = 10. Answer: k = 24 divided by 6 = 4, so y = 4x. When x = 10, y = 40.

Common mistakes and exam focus

A common mistake is treating a ratio as the actual quantity rather than a pattern. If a question says the ratio is 2:5, the total is not automatically 7 items; 7 is the number of parts. The real total may be 28, 70 or another value. Another mistake is sharing by subtracting instead of finding the value of one part. GCSE and IGCSE exam questions often combine ratio with geometry, money, rates, density or algebra. A student may need to form an equation from a ratio or use a scale factor with area and volume. This is why the topic needs flexible understanding rather than one memorised method. Clear working is also important because many marks are awarded for the method, not only the final answer.

Why one-to-one online lessons help

Online maths lessons are very beneficial for this topic because ratio problems can look different even when the same method is needed. A tutor can help the student identify the structure of the question, choose the correct scale factor and check whether the answer is realistic. In a one-to-one setting, lessons can move from simple sharing questions to harder exam-style applications at the right pace. For future study, strong proportional reasoning supports science, economics, engineering, cooking, design, finance and many practical careers. It also helps students with graphs, similarity, trigonometry, probability and rates of change. When students understand ratio properly, they become more confident with multi-step maths problems and can apply the same reasoning in new situations.