How to Solve Ratio and Proportion Word Problems

Why Ratio Problems Trip Students Up

Ratio and proportion questions are among the most commonly examined topics at IGCSE, appearing on both Core and Extended papers. The maths itself is not complicated, but students often lose marks because the problems are presented as word problems, and it is not always obvious which method to use.

The good news is that a small number of reliable techniques cover almost every ratio question you will face. This guide walks through each type with worked examples.

This is the most common ratio question at IGCSE. You are given a total and asked to divide it.

The Method

Add up the parts of the ratio.
Divide the total by the sum of parts to find the value of one part.
Multiply each ratio number by the value of one part.

Worked Example 1: Share RM 450 in the ratio 2 : 3 : 4.

Total parts = 2 + 3 + 4 = 9

One part = 450 ÷ 9 = RM 50

The three shares are: 2 × 50 = RM 100, 3 × 50 = RM 150, 4 × 50 = RM 200.

Check: 100 + 150 + 200 = 450. Correct.

Worked Example 2: Ali and Bala share some money in the ratio 5 : 3. Ali receives RM 40 more than Bala. Find the total amount shared.

The difference in ratio parts is 5 − 3 = 2 parts.

2 parts = RM 40, so 1 part = RM 20.

Total parts = 5 + 3 = 8.

Total = 8 × 20 = RM 160.

Type 2: Finding a Missing Quantity Using Equivalent Ratios

Sometimes you know one of the quantities and need to find the other.

Worked Example 3: The ratio of boys to girls in a class is 3 : 5. If there are 15 boys, how many girls are there?

Set up the proportion: 3/5 = 15/x

Cross-multiply: 3x = 75

x = 25. There are 25 girls.

Alternatively, notice that 15 ÷ 3 = 5, so each part represents 5 students. Girls = 5 × 5 = 25.

Type 3: Simplifying Ratios

You may be asked to simplify a ratio or express quantities as a ratio in its simplest form.

Worked Example 4: Write 45 minutes : 2 hours as a ratio in its simplest form.

First, convert to the same units. 2 hours = 120 minutes.

45 : 120

Divide both by 15: 3 : 8.

Key point: Always convert to the same units before simplifying. This catches out many students who write 45 : 2 and try to simplify from there.

Type 4: Direct Proportion

Two quantities are in direct proportion if when one doubles, the other doubles too. The ratio between them stays constant.

Worked Example 5: A recipe for 4 people requires 300 g of flour. How much flour is needed for 10 people?

Method: Find the amount per person first.

300 ÷ 4 = 75 g per person

For 10 people: 75 × 10 = 750 g.

Alternatively, use a scaling factor: 10/4 = 2.5. So 300 × 2.5 = 750 g.

Worked Example 6: If 5 litres of paint covers 40 m², how much paint is needed to cover 100 m²?

Scaling factor: 100 ÷ 40 = 2.5

Paint needed: 5 × 2.5 = 12.5 litres.

Type 5: Inverse Proportion

Two quantities are inversely proportional if when one increases, the other decreases by the same factor. Their product stays constant.

Worked Example 7: 6 workers can complete a job in 10 days. How long would it take 4 workers?

The product stays constant: 6 × 10 = 60 worker-days.

With 4 workers: 60 ÷ 4 = 15 days.

More workers means less time, fewer workers means more time — this confirms our answer is sensible.

Type 6: Map Scales and Scale Drawings

Map scales are ratios in disguise.

Worked Example 8: A map has a scale of 1 : 50 000. Two towns are 3.6 cm apart on the map. Find the actual distance in kilometres.

Actual distance = 3.6 × 50 000 = 180 000 cm

Convert to km: 180 000 ÷ 100 000 = 1.8 km.

Worked Example 9: On a map with scale 1 : 25 000, what length on the map represents an actual distance of 2 km?

2 km = 200 000 cm

Map distance = 200 000 ÷ 25 000 = 8 cm.

Type 7: Percentage and Ratio Combined

Worked Example 10: In a school, the ratio of students who walk to school to those who take the bus is 3 : 7. What percentage of students walk?

Total parts = 3 + 7 = 10

Fraction who walk = 3/10

Percentage = 3/10 × 100 = 30%.

Common Mistakes

Forgetting to convert units before comparing quantities. Minutes and hours, centimetres and kilometres — always make units match first.
Mixing up direct and inverse proportion. Ask yourself: if one quantity goes up, does the other go up (direct) or down (inverse)?
Adding instead of multiplying when scaling up. If 3 parts = 150, then 5 parts = 5 × 50 = 250, not 150 + something.
Not checking your answer adds up to the total. For sharing problems, the parts must sum to the original amount.

Exam Strategy

Read the question twice. Identify what type of ratio problem it is before starting.
Always show your working. Even if you can see the answer, the method marks require visible steps.
For proportion questions, write “1 part = …” or “1 person needs …” as an intermediate step. This keeps your logic clear and earns method marks.
Check that your answer makes sense in context. If 4 workers take longer than 6 workers, your answer should be a larger number of days.

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