A ratio compares parts. A proportion is a statement that two things change at the same rate. They show up in recipes, maps, currency conversion, and most "real world" Year 8 problems. This guide walks the four problem types you'll see in KS3 exams.

What ratio means

A ratio like 2:3 is two numbers separated by a colon. It says: "for every 2 of one thing, there are 3 of the other."

Examples:

  • Boys to girls in a class: 12:18 simplifies to 2:3
  • Flour to sugar in a cake: 300g : 200g = 3:2
  • Map scale: 1:50,000 (1cm on the map = 50,000cm in reality = 500m)

Simplifying ratios

Divide both sides by the highest common factor (HCF):

  • 12:18 → divide by 6 → 2:3
  • 25:35 → divide by 5 → 5:7
  • 8:12:20 → divide by 4 → 2:3:5

Check: the simplified form should have no common factor between any pair.

Problem type 1: Share in a ratio

Question: Share £20 in the ratio 2:3.

Total parts = 2 + 3 = 5
One part = £20 ÷ 5 = £4
2 parts = 2 × £4 = £8
3 parts = 3 × £4 = £12
Check: £8 + £12 = £20  ✓

Question: Share 60 sweets in the ratio 1:2:3.

Total parts = 1 + 2 + 3 = 6
One part = 60 ÷ 6 = 10
1 part = 10, 2 parts = 20, 3 parts = 30
Check: 10 + 20 + 30 = 60  ✓

Problem type 2: Scale up / scale down (recipes, maps)

Question: A recipe for 4 people uses 200g flour. How much for 6 people?

Think of it as "per person, then multiply":

Flour per person = 200g ÷ 4 = 50g
For 6 people: 50g × 6 = 300g

Or as a ratio: people-to-flour is 4:200 = 1:50. So 6 people → 6 × 50 = 300g.

Problem type 3: Direct proportion

Two quantities are in direct proportion if doubling one doubles the other (and so on).

Question: 3 pencils cost £1.50. What do 8 pencils cost?

Cost per pencil = £1.50 ÷ 3 = £0.50
8 pencils cost = 8 × £0.50 = £4.00

Algebraically

If y is directly proportional to x, then y = kx for some constant k. Find k from one pair, then use it.

For pencils: cost = 0.5 × pencils. The 0.5 is k.

Problem type 4: Given one part of a ratio

Question: A money jar holds coins in the ratio 5p:10p of 3:5. There are 15 five-pence coins. How many 10p coins?

3 parts represent 15 five-pence coins
1 part = 15 ÷ 3 = 5
5 parts (ten-pence) = 5 × 5 = 25 coins

The two common mistakes

Mistake 1: Treating ratio as "out of"

❌ "Boys to girls is 2:3 so 2/3 of the class are boys." ✅ Boys are 2/5 of the class (2 parts out of 5 total).

The "out of" is the total parts (2 + 3 = 5), not the other side of the colon.

Mistake 2: Not checking proportion direction

When scaling: are you scaling up (more people, more flour) or down (less)? Direct proportion goes the same way. Make sure your final answer is bigger if your "going up", smaller if "going down". 30 second sanity check.

Maps and scale

A 1:50,000 map means 1 unit on the map = 50,000 units in reality. So:

  • 2 cm on the map = 2 × 50,000 cm = 100,000 cm = 1 km in reality

Currency conversion

If £1 = €1.20, then £5 = 5 × 1.20 = €6. And €12 = 12 ÷ 1.20 = £10.

Direct proportion: pounds and euros change in step at a fixed rate.

How Professor Pi teaches this

Pi never gives the final number. The hint ladder for "Share £20 in 2:3":

  • L1: "How many parts does the ratio have in total?"
  • L2: "Right — 5 parts. So what's one part worth?"
  • L3: "£4. Now how many parts go to each share?"
  • L4: worked twin for Share £30 in 2:3, then you do £20.

FAQ

What's the difference between ratio and proportion?

Ratio compares parts. Proportion is a statement that two ratios are equal, or that one quantity changes in step with another.

How do I share £20 in the ratio 2:3?

Total parts = 5. One part = £4. So £8 and £12. Check sums to £20.

What's direct proportion?

When two quantities increase or decrease at the same rate. Doubling one doubles the other.

Pedagogy from Professor Pi at aitutors.me. Updated 20 May 2026.