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Ratio & Proportion GCSE Level

Grade 10 · Algebra & Number · Cambridge IGCSE · Age 14–15

Welcome to Ratio & Proportion!

Ratio and proportion are fundamental tools in mathematics and everyday life — from mixing cement to reading maps to engineering gear ratios. This topic bridges arithmetic and algebra, and appears heavily in IGCSE exams both in computation and problem-solving contexts.

Ratio compares quantities  |  Direct: y = kx  |  Inverse: y = k/x  |  Variation: y ∝ xⁿ

Learning Objectives

  • Simplify ratios including those with different units
  • Divide a quantity in a given ratio (two-way and three-way)
  • Use map scales and combine ratios
  • Set up and use direct proportion equations (y = kx, y = kx², y = k√x)
  • Set up and use inverse proportion equations (y = k/x, y = k/x²)
  • Solve word problems involving both types of proportion
  • Recognise and sketch graphs of proportional relationships

Simplifying Ratios

HCF, unit conversion, three-way

Dividing in a Ratio

Parts method, £360 in 4:5

Map Scales

1:50000, cm → km conversions

Direct Proportion

y = kx, y = kx², y = k√x

Inverse Proportion

y = k/x, y = k/x², hyperbola

Combining Ratios

A:B and B:C → A:B:C

Learn 1 — Ratio

Simplifying Ratios

A ratio compares two (or more) quantities of the same kind. To simplify, divide all parts by their highest common factor (HCF). If units differ, convert to the same unit first.

Example: Simplify 45 cm : 2 m
Step 1 — Convert to same unit: 45 cm : 200 cm
Step 2 — HCF of 45 and 200: HCF = 5
Step 3 — Divide both by 5: 9 : 40
Example: Simplify 1.5 : 4.5
Multiply both by 2 to remove decimals: 3 : 9
HCF = 3 → 1 : 3
When you see mixed units (cm and m, g and kg, minutes and hours), ALWAYS convert to the smaller unit before simplifying.

Dividing a Quantity in a Ratio

The "parts" method: add the ratio numbers to get total parts, divide the quantity by total parts to find one part, then multiply.

Example: Share £360 in the ratio 4 : 5
Total parts = 4 + 5 = 9
One part = £360 ÷ 9 = £40
First share: 4 × £40 = £160
Second share: 5 × £40 = £200
Check: £160 + £200 = £360 ✓
Three-way ratio: Share 540 g in ratio 2 : 3 : 4
Total parts = 2 + 3 + 4 = 9
One part = 540 ÷ 9 = 60 g
Shares: 120 g : 180 g : 240 g
Always check your answer by adding the shares. They must equal the original total.

Map Scales

A scale of 1 : n means every 1 unit on the map represents n units in reality. The units must be the same on both sides.

Scale 1 : 50 000
1 cm on map = 50 000 cm in real life = 500 m = 0.5 km
If two points are 7.4 cm apart on map: real distance = 7.4 × 50 000 = 370 000 cm = 3.7 km
Working backwards: A road is 12 km long. Find its map length with scale 1 : 200 000
12 km = 1 200 000 cm
Map length = 1 200 000 ÷ 200 000 = 6 cm
Common error: not converting km to cm before dividing. The scale ratio uses the same units (cm on both sides).

Combining Ratios

If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C. You need B to be the same number in both ratios.

Method: Make the shared quantity equal
A : B = 2 : 3    B : C = 4 : 5
LCM of 3 and 4 is 12. Scale up:
A : B = 8 : 12    B : C = 12 : 15
Combined: A : B : C = 8 : 12 : 15
Think of it like equivalent fractions — multiply each ratio by whatever makes the shared term equal in both ratios.

Learn 2 — Direct Proportion

We say y is directly proportional to x, written y ∝ x, when doubling x doubles y, tripling x triples y, etc. This means y = kx for some constant k.

y ∝ x  ⟹  y = kx   |   Find k by substituting a known (x, y) pair
Example: y is directly proportional to x. When x = 4, y = 20. Find y when x = 7.
Step 1 — Write equation: y = kx
Step 2 — Find k: 20 = k × 4 → k = 5
Step 3 — Write full equation: y = 5x
Step 4 — Find y: y = 5 × 7 = 35

y ∝ x² (proportional to the square)

When y ∝ x², the equation is y = kx². Doubling x quadruples y.

Example: y ∝ x². When x = 3, y = 36. Find y when x = 5.
y = kx²  →  36 = k × 9  →  k = 4
y = 4x²  →  y = 4 × 25 = 100

y ∝ √x (proportional to the square root)

Example: y ∝ √x. When x = 16, y = 12. Find y when x = 25.
y = k√x  →  12 = k × 4  →  k = 3
y = 3√x  →  y = 3 × 5 = 15

Graph Shapes

y = kx: Straight line through the origin. Gradient = k.
y = kx²: Parabola through the origin, opening upward (if k>0).
y = k√x: Curved line through the origin, increasing but getting shallower.
The METHOD is always the same: (1) Write the proportionality equation, (2) substitute the given pair to find k, (3) write the full equation, (4) use it to answer the question.

Learn 3 — Inverse Proportion

We say y is inversely proportional to x, written y ∝ 1/x, when doubling x halves y. The product xy stays constant: y = k/x.

y ∝ 1/x  ⟹  y = k/x   |   Equivalently: xy = k (constant)
Example: y ∝ 1/x. When x = 4, y = 15. Find y when x = 12.
y = k/x  →  15 = k/4  →  k = 60
y = 60/x  →  y = 60/12 = 5

y ∝ 1/x² (inversely proportional to the square)

Example: y ∝ 1/x². When x = 2, y = 25. Find y when x = 5.
y = k/x²  →  25 = k/4  →  k = 100
y = 100/x²  →  y = 100/25 = 4

Classic Word Problem: Workers and Days

5 workers take 12 days to complete a job. How long would 8 workers take?
More workers → fewer days (inverse proportion)
Step 1 — Find k (total worker-days): k = 5 × 12 = 60
Step 2 — For 8 workers: days = 60 ÷ 8 = 7.5 days
Speed × Time = Distance is a classic inverse relationship.
Same distance: if speed doubles, time halves.
Constant: Speed × Time = Distance (the "k" here is the distance).

Graph Shape

y = k/x: Hyperbola — two curved branches in quadrants 1 and 3 (for k>0). Never touches the axes.
y = k/x²: Similar shape but approaches axes faster.
To distinguish direct from inverse in a word problem: ask "if one quantity goes up, does the other go up (direct) or down (inverse)?" Workers going up → days going down → inverse.

Example 1 — Ratio sharing with three-way ratio

Q: Alice, Ben and Cara share £1080 in the ratio 3 : 4 : 5. Find each person's share.
M1: Total parts = 3 + 4 + 5 = 12
M1: One part = £1080 ÷ 12 = £90
A1: Alice = 3 × £90 = £270  |  Ben = 4 × £90 = £360  |  Cara = 5 × £90 = £450
Check: 270 + 360 + 450 = 1080 ✓

Example 2 — Map scale problem

Q: A map has scale 1 : 25 000. Two towns are 8.4 cm apart on the map. Find the actual distance in km.
M1: Actual distance = 8.4 × 25 000 = 210 000 cm
M1: Convert to km: 210 000 ÷ 100 000 = 2.1 km A1

Example 3 — Direct proportion, find k then find value

Q: y is proportional to x². When x = 3, y = 45. Find y when x = 5.
B1: Write equation: y = kx²
M1: Substitute: 45 = k × 9 → k = 5
M1: y = 5 × 5² = 5 × 25 = 125 A1

Example 4 — Inverse proportion

Q: y is inversely proportional to x. When x = 6, y = 8. Find x when y = 12.
B1: Write equation: y = k/x
M1: Find k: 8 = k/6 → k = 48
M1: 12 = 48/x → x = 48/12 = 4 A1

Example 5 — Combining ratios

Q: A : B = 3 : 5 and B : C = 2 : 7. Find A : B : C.
M1: Make B equal in both ratios. LCM of 5 and 2 is 10.
A : B = 6 : 10    B : C = 10 : 35
A1: A : B : C = 6 : 10 : 35

Example 6 — Real-world inverse proportion

Q: 6 identical machines fill 2400 bottles per hour. How many bottles per hour would 10 machines fill?
M1: Direct proportion: more machines → more bottles.
M1: Rate per machine = 2400 ÷ 6 = 400 bottles/hr
A1: 10 machines: 10 × 400 = 4000 bottles/hr

Common Mistakes in Ratio & Proportion

These are the errors examiners see most often. Understanding WHY they're wrong helps you avoid them.

Mistake 1 — Not converting units before simplifying a ratio

✗ Wrong: Simplify 30 cm : 2 m  →  HCF of 30 and 2 is 2  →  15 : 1
✓ Correct: Convert first! 30 cm : 200 cm  →  HCF = 10  →  3 : 20

The units must be the same before you can simplify. 30 and 2 are in different units so their HCF is meaningless here.

Mistake 2 — Confusing direct and inverse proportion

✗ Wrong: "y ∝ 1/x so y = kx" — using the wrong equation
✓ Correct: y ∝ 1/x means y = k/x (as x increases, y decreases)

Read the proportionality statement carefully. ∝ x means y = kx. ∝ 1/x means y = k/x. These have completely opposite behaviours.

Mistake 3 — Forgetting to find k before answering

✗ Wrong: y ∝ x², x=3, y=18. Find y when x=5. Answer: y = 5² = 25
✓ Correct: Find k first: 18 = k×9 → k=2. Then y = 2×25 = 50

You cannot just substitute x into the proportion relationship without knowing the constant k. Always find k first.

Mistake 4 — Confusing "ratio 3:5" with "3/5 of total"

✗ Wrong: Share £80 in ratio 3:5. First share = 3/5 × 80 = £48
✓ Correct: Total parts = 8. First share = 3/8 × 80 = £30

In ratio 3:5, the first part gets 3 out of 8 total parts (not 3 out of 5). Always divide by the sum of the ratio parts.

Mistake 5 — Map scale: confusing direction of conversion

✗ Wrong: Scale 1:50000, map distance = 3cm. Actual = 3 ÷ 50000 = 0.00006 cm
✓ Correct: Actual = 3 × 50000 = 150000 cm = 1.5 km

Map → real: multiply by the scale factor. Real → map: divide by the scale factor. The real distance is always larger.

Key Formulas — Ratio & Proportion

Given in ExamMust Memorise
Scale / ratio given in questionRatio simplification: divide by HCF
Proportionality type stated (e.g. y ∝ x²)y ∝ x  ⟹  y = kx
—y ∝ x²  ⟹  y = kx²
—y ∝ √x  ⟹  y = k√x
—y ∝ 1/x  ⟹  y = k/x
—y ∝ 1/x²  ⟹  y = k/x²
—Workers × Days = constant (if work is fixed)
—Speed × Time = Distance
—Map: actual = map distance × scale factor
Parts method: One part = Total ÷ (sum of ratio)  |  Share = ratio number × one part
Combining ratios: make shared term equal (use LCM)

Graph Recognition

y = kx: Straight line through origin (positive gradient)
y = kx²: Parabola through origin (U-shape)
y = k√x: Increasing curve from origin, gets shallower
y = k/x: Hyperbola — never touches axes, in quadrants 1 and 3
y = k/x²: Steeper hyperbola, approaches axes faster

Proportion Explorer

Select a proportion type, enter a known point (x₁, y₁) to find k. The graph will be drawn and you can find y for any x value.

Enter values and click "Find k & Plot" to begin.

Exercise 1 — Simplifying Ratios (convert units first)

1. Simplify 24 : 36

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2. Simplify 45 cm : 1 m (answer as a : b in simplest form)

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3. Simplify 2.4 kg : 600 g (answer a : b)

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4. Simplify 3 hours : 45 minutes (answer a : b)

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5. Simplify 80 cm : 1.2 m (answer a : b)

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6. Simplify 1.5 : 2.5 (answer a : b, integers)

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7. Simplify 500 ml : 2 litres (answer a : b)

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8. Simplify 75 cm : 1.5 m (answer a : b)

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Exercise 2 — Dividing a Quantity in a Ratio

1. Share £240 in ratio 3 : 5. Find the larger share (£).

2. Share 360 g in ratio 2 : 7. Find the smaller share (g).

3. Share £540 in ratio 4 : 5. Find the larger share (£).

4. Share 720 ml in ratio 1 : 2 : 3. Find the largest share (ml).

5. Share £1200 in ratio 3 : 4 : 5. Find the middle share (£).

6. A rectangle's length and width are in ratio 5 : 2. Width = 8 cm. Find the length (cm).

7. Two numbers are in ratio 7 : 3. Their sum is 150. Find the larger number.

8. Share £2520 in ratio 3 : 5 : 4. Find the smallest share (£).

Exercise 3 — Direct Proportion y = kxⁿ (find k, then find value)

1. y ∝ x. When x = 5, y = 30. Find y when x = 9.

2. y ∝ x. When x = 7, y = 42. Find x when y = 18.

3. y ∝ x². When x = 3, y = 27. Find y when x = 5.

4. y ∝ x². When x = 4, y = 48. Find y when x = 6.

5. y ∝ √x. When x = 9, y = 12. Find y when x = 25.

6. y ∝ √x. When x = 16, y = 20. Find x when y = 30.

7. y ∝ x². When x = 2, y = 20. Find y when x = 7.

8. y ∝ x. When x = 8, y = 56. Find y when x = 11.

Exercise 4 — Inverse Proportion

1. y ∝ 1/x. When x = 3, y = 20. Find y when x = 12.

2. y ∝ 1/x. When x = 5, y = 8. Find x when y = 20.

3. y ∝ 1/x². When x = 2, y = 25. Find y when x = 5.

4. 4 workers take 15 days. How many days would 12 workers take?

5. A car travels at 60 km/h and takes 3 hours. How long at 90 km/h?

6. y ∝ 1/x. When x = 6, y = 5. Find y when x = 15.

7. y ∝ 1/x². When x = 3, y = 12. Find y when x = 6.

8. 8 pumps drain a tank in 9 hours. How many pumps needed to drain it in 6 hours?

Exercise 5 — Combined & Harder Problems

1. A : B = 2 : 5 and B : C = 3 : 4. Find A : C as a simplified ratio (enter A, then C).

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2. Scale 1 : 40 000. Map distance = 5.5 cm. Actual distance in km?

3. A real road is 18 km long. Find map length in cm with scale 1 : 300 000.

4. y ∝ x². x increases by 50%. By what percentage does y increase? (%)

5. y ∝ 1/x. x doubles. What fraction of the original is y?

6. P is directly proportional to Q² and inversely proportional to R. When Q=2, R=4, P=3. Find P when Q=6, R=3.

7. Two numbers are in ratio 5 : 8. The larger is increased by 12. New ratio is 5 : 10. Find the original larger number.

8. Map scale 1:25000. Two towns measure 12.4 cm apart. Actual distance in km?

Practice — 25 Mixed Questions

🔵 = Non-calculator    🟢 = Calculator allowed

🔵 1. Simplify 12 : 18

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🔵 2. Share £180 in ratio 2 : 7. Larger share (£)?

🔵 3. Simplify 30 min : 2 hours (a : b)

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🟢 4. y ∝ x. When x=4, y=28. Find y when x=9.

🟢 5. y ∝ 1/x. When x=3, y=16. Find y when x=12.

🔵 6. A : B = 4 : 3 and B : C = 6 : 5. Find A : B : C. (Enter A, B, C)

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🟢 7. Scale 1 : 50 000. Map = 6 cm. Actual distance in km?

🟢 8. y ∝ x². When x=5, y=75. Find y when x=8.

🔵 9. Share 450 g in ratio 1 : 2 : 3. Smallest share (g)?

🟢 10. 10 workers take 6 days. How many days for 4 workers?

🟢 11. y ∝ √x. When x=4, y=10. Find y when x=100.

🔵 12. Two numbers in ratio 3:7, sum = 200. Smaller number?

🟢 13. y ∝ 1/x². When x=2, y=50. Find y when x=5.

🔵 14. Simplify 250 m : 2 km (a : b)

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🟢 15. Real distance = 4.5 km. Map scale 1:90000. Map distance (cm)?

🟢 16. y ∝ x². When x=2, y=12. Find x when y=108.

🔵 17. Share £900 in ratio 2 : 3 : 4. Middle share (£)?

🟢 18. y ∝ 1/x. When x=8, y=3. Find y when x=2.

🔵 19. A : B = 5 : 2. A = 35. Find B.

🟢 20. y ∝ x. y=45 when x=9. Find y when x=13.

🟢 21. y ∝ 1/x². When x=4, y=2. Find y when x=1.

🔵 22. Simplify 1.8 : 2.4 (a : b, integers)

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🟢 23. 6 taps fill a tank in 8 hours. How long for 4 taps?

🟢 24. y ∝ √x. When x=25, y=15. Find y when x=64.

🔵 25. Share £2000 in ratio 3 : 5. Larger share (£)?

Challenge — 12 Questions (IGCSE Extended Level)

1. y ∝ x³. When x=2, y=24. Find y when x=5.

2. y ∝ 1/x². When x=3, y=4. Find y when x=9.

3. y ∝ x². x is increased by 20%. By what % does y increase?

4. y ∝ 1/x. y is decreased by 50%. By what % does x increase?

5. P ∝ Q² and P ∝ 1/R. When Q=3, R=4, P=9. Find P when Q=6, R=1.

6. A : B = 3 : 4, B : C = 5 : 6, C : D = 7 : 8. Find A : D. (enter A, then D — simplify)

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7. A map has scale 1:25000. A lake has area 1.44 cm² on the map. Find its actual area in km².

8. y ∝ √x. When x=4, y=6. Find y when x=9.

9. T workers complete a job in D days. How many days for (T+4) workers? Express answer in terms of T and D. Enter the NUMERATOR of your fraction (TD).

10. Three quantities split in ratio 2:3:5 give a middle quantity of 48. What is the total?

11. y ∝ x². The difference between y when x=5 and y when x=3 is 64. Find k.

12. Speed ∝ 1/time (same distance). A journey at 80 km/h takes 2.5 hours. Find the speed needed to complete the journey in 2 hours (km/h).

Exam Style Questions

Mark-scheme style. Show working in your book. Enter final answers for self-marking.

Question 1 — Map Scale [4 marks]

A map has scale 1 : 25 000.

(a) Two towns are 8.4 cm apart on the map. Find the actual distance in km.
(b) A road is 15 km long. Find its length on the map in cm.

Question 2 — Direct Proportion [5 marks]

y is directly proportional to the square of x. When x = 4, y = 80.

(a) Find an equation for y in terms of x.
Enter k (the constant of proportionality):
(b) Find y when x = 7.
(c) Find x when y = 245.

Question 3 — Ratio Sharing [4 marks]

Three friends Alice, Ben and Cara invest in a business in the ratio 5 : 3 : 2. The total investment is £6400.

(a) Find Alice's investment (£).
(b) After one year, the profit is £3200, shared in the same ratio. Find Ben's profit share (£).

Question 4 — Inverse Proportion [5 marks]

The time T (hours) taken to complete a task is inversely proportional to the number of workers W.

(a) When W = 5, T = 12. Find T when W = 8.
(b) How many workers are needed to complete the task in 4 hours?
(c) The number of workers is increased by 50% from 5. Find the new time (hours).

Question 5 — Combined Ratio & Proportion [5 marks]

P is proportional to Q² and inversely proportional to R.

(a) Write down the equation connecting P, Q and R, using k.
(Just enter k when Q=2, R=3, P=8)
(b) Find P when Q = 4, R = 2.
(c) Find Q when P = 54, R = 6.