Ratio and Proportion
Ratio questions are guaranteed marks — tutors said you will “definitely have one or two questions based on this”. They are also where careless reading costs most, because the arithmetic is easy and the trap is which number you’re being asked for.
1. Simplifying ratios
Divide all parts by their highest common factor.
- 12 : 18 → divide by 6 → 2 : 3
- 20 : 30 : 50 → divide by 10 → 2 : 3 : 5
Two rules that are constantly broken:
Convert to the SAME UNITS first. 50 cm : 2 m is not 50 : 2. Convert: 50 cm : 200 cm = 1 : 4. Writing a ratio with mixed units was a recorded error and gives an answer wrong by a factor of 100.
Clear decimals and fractions before simplifying. For 0.5 : 1.5, multiply both by 2 → 1 : 3. For ½ : ⅓, multiply both by 6 → 3 : 2.
Ratio to fraction: in the ratio 3 : 4, the first part is 3/7 of the total — the denominator is the sum of the parts, not the other number.
3 : 4 means 3/7 and 4/7, not 3/4. Giving 5/2 where 3/7 was needed was a documented error, and it comes from using the wrong denominator.
2. Sharing in a given ratio
Example: share £90 in the ratio 2 : 3
- Add the parts: 2 + 3 = 5
- Divide by the total parts: 90 ÷ 5 = £18 per part
- Multiply out: 2 × 18 = £36 and 3 × 18 = £54
- Check: 36 + 54 = 90 ✓
Divide by the SUM of the parts, not by each part separately. Dividing 90 by 2 and by 3 was a specific recorded error. The sum tells you how many equal shares the whole splits into.
Always check your parts add back to the total. It catches nearly every error in this question type, and takes one line.
With three parts, the method is identical: for 2 : 3 : 5, divide by 10.
3. The three question types
Ratio questions come in three shapes, and mixing them up is the main source of lost marks.
Type 1 — the total is given
As above: divide by the sum of the parts.
Type 2 — ONE PART is given
Example: two people share money in the ratio 3 : 5. The first gets £24. How much in total?
- 3 parts = £24, so 1 part = £8
- Total = 8 parts = £64
- (The second person gets 5 × 8 = £40)
Find the value of ONE part first. That single number unlocks everything else. Dividing by the wrong part of the ratio was a recorded error — match the amount you’re given to its own number of parts.
Type 3 — the DIFFERENCE is given
Example: shared in the ratio 3 : 7, one person gets £40 more than the other. Find the total.
- The difference is 7 − 3 = 4 parts
- 4 parts = £40, so 1 part = £10
- Total = 10 parts = £100
Read carefully whether you are given the total, one part, or the difference. These three need different first steps, and the numbers often look similar.
4. Scales and maps
A scale of 1 : 50 000 means 1 unit on the map represents 50 000 units in real life.
Map to real life — multiply. Real life to map — divide.
Example: on a 1 : 50 000 map, 4 cm represents:
- 4 × 50 000 = 200 000 cm = 2 km
Convert units at the end. 200 000 cm → ÷100 → 2000 m → ÷1000 → 2 km. Failing to convert was a documented error, and an answer of “200 000” without units is meaningless.
A scale ratio has no units — both sides are in the same unit, which is why you can convert afterwards.
5. Direct and inverse proportion
Direct proportion
As one quantity increases, the other increases in the same ratio. y ∝ x, so y = kx.
The unitary method — find the value of one, then scale:
Example: 5 pens cost £3.50. What do 8 cost?
- One pen: 3.50 ÷ 5 = £0.70
- Eight pens: 0.70 × 8 = £5.60
With a formula: if y ∝ x and y = 12 when x = 3, then k = 4 and y = 4x.
Other direct forms: y ∝ x² gives y = kx², and y ∝ √x gives y = k√x.
Inverse proportion
As one quantity increases, the other decreases. y ∝ 1/x, so y = k/x — and xy = k is constant.
Example: 4 workers take 6 days. How long for 3 workers?
- Total work = 4 × 6 = 24 worker-days
- 24 ÷ 3 = 8 days
More workers means fewer days — if your answer moves the wrong way, you have used direct proportion where inverse was needed. Ask yourself first: should this go up or down?
Method for any proportion question:
- Write the relationship (y = kx or y = k/x)
- Substitute the given pair to find k
- Rewrite the formula with k in it
- Substitute the new value
6. Real-life contexts
Recipes: scaling ingredients — find the multiplier from the quantity you know.
Best value: work out the price per unit (or the amount per £1) for each option and compare.
Currency: multiply or divide by the exchange rate — check which direction you need.
“How many whole coaches are needed for 130 people at 45 per coach?” — 130 ÷ 45 = 2.89, so you need 3 coaches. Context questions like this need rounding up, whatever the decimal says. Miscounting these was a recorded error.
When you don’t know an amount, call it x and form an equation — the advice tutors gave for the harder word problems.
7. Mistakes that cost marks
Not converting to the same units before simplifying.
Not clearing decimals or fractions first.
Using the wrong denominator when converting a ratio to a fraction.
Dividing by each part instead of by their sum.
Matching a given amount to the wrong part of the ratio.
Treating a “difference” question as a “total” question.
Forgetting to convert map distances to sensible units.
Using direct proportion where inverse was needed.
Rounding down when the context requires rounding up.
Not checking that the parts add back to the total.
Frequently asked questions
How do I simplify a ratio? Divide every part by their highest common factor — after converting to the same units.
How do I share an amount in a ratio? Add the parts, divide the total by that sum to get one part, then multiply.
What if I’m only given one person’s share? Divide it by its own number of parts to find one part, then scale up.
What if I’m given the difference? The difference corresponds to the difference in parts — divide to find one part.
What fraction of the total is the first part of 3 : 4? 3/7 — the denominator is the sum of the parts.
What does a scale of 1 : 50 000 mean? 1 unit on the map represents 50 000 of the same units in reality.
How do I convert a map distance to km? Multiply by the scale, then convert cm → m → km.
What is direct proportion? Both quantities change in the same direction: y = kx.
What is inverse proportion? As one increases the other decreases: y = k/x, and xy is constant.
How do I solve a proportion question? Find k from the given pair, write the formula, then substitute the new value.
Quick revision checklist
- I convert to the same units before simplifying a ratio
- I clear decimals and fractions first
- I know a ratio’s fraction uses the sum of the parts
- I share by dividing by the sum of the parts
- I check the parts add back to the total
- I can handle “one part given” questions
- I can handle “difference given” questions
- I read carefully which of the three types I have
- I can use map scales in both directions
- I convert map answers to sensible units
- I can tell direct from inverse proportion
- I can find k and write the formula
- I use the unitary method confidently
- I round up when the context demands it
These notes cover ratio and proportion in the Cambridge IGCSE Mathematics (0580) syllabus, and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Maths lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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