Page 1 · Introduction
Ratios and proportion
A ratio compares amounts — like 2 parts juice to 3 parts water. Proportion is about keeping that comparison the same when the amounts grow or shrink. Once you can find the value of one part, almost every ratio question becomes easy.
The big ideaAdd the parts to find the total number of parts, divide to find the value of one part, then multiply back up. That one method solves most ratio questions.
💡We build from simple sharing up to scaling and reverse problems. Take it one step at a time.
On the real testWhere this appearsRatio and proportion come up throughout Mathematical Reasoning (25% of your mark, no calculator from 2025) — in recipes, maps, mixtures and money-sharing problems.
Page 2 · The part method
Sharing in a ratio
To share an amount in a ratio, add the parts, divide to find one part, then multiply.
Share $20 in the ratio 2:3.
Parts: 2 + 3 = 5. One part: 20 ÷ 5 = $4. So the shares are 2×4 = $8 and 3×4 = $12.
$8 and $12
Share 24 sweets in the ratio 1:2:3.
Parts: 1 + 2 + 3 = 6. One part: 24 ÷ 6 = 4. Shares: 4, 8 and 12.
4, 8 and 12
✅Always check: your shares should add back up to the total. 8 + 12 = 20. ✓
Page 3 · Scaling up and down
Keeping things in proportion
If a recipe or mix keeps the same ratio, find the value of one unit first (the "unitary method"), then scale.
3 apples cost $6. How much do 5 apples cost?
One apple: 6 ÷ 3 = $2. Five apples: 5 × 2 = $10.
$10
A recipe for 4 people uses 200 g of rice. How much for 6 people?
One person: 200 ÷ 4 = 50 g. Six people: 6 × 50 = 300 g.
300 g
💡The trick is always the same: get down to one first, then multiply up to what you need.
Page 4 · Working backwards
When you're given one share
Sometimes you know one person's share, not the total. Use it to find one part, then work out the rest.
Money is shared 2:5. The smaller share is $10. What is the total?
The smaller share is 2 parts = $10, so one part = $5. Total parts = 2 + 5 = 7, so total = 7 × 5 = $35.
$35
A ratio 3:4 gives one person 12 more than the other. Find both shares.
The difference is 4 − 3 = 1 part = 12. So shares are 3×12 = 36 and 4×12 = 48.
36 and 48
✅Ask yourself: how many parts does the number I'm given represent? That unlocks everything else.
Level 1 · Easy
Warm up
Simple sharing and one-step scaling. Tap to check.
1. Share $10 in the ratio 1:1.
A $4 and $6
B $5 and $5
C $3 and $7
D $2 and $8
Equal parts: 10 ÷ 2 = $5 each.
2. Share 12 in the ratio 1:2.
A 4 and 8
B 3 and 9
C 5 and 7
D 6 and 6
Parts 1+2 = 3. One part 12 ÷ 3 = 4. Shares 4 and 8.
3. 2 pens cost $6. How much is 1 pen?
6 ÷ 2 = $3.
Level 2 · Medium
Building up
Three-part ratios and unitary scaling.
1. Share 30 in the ratio 2:3.
A 10 and 20
B 12 and 18
C 15 and 15
D 14 and 16
Parts 2+3 = 5. One part 30 ÷ 5 = 6. Shares 12 and 18.
2. 4 tickets cost $32. How much do 7 tickets cost?
One ticket 32 ÷ 4 = $8. Seven: 7 × 8 = $56.
3. Share 36 in the ratio 1:2:3.
A 6, 12, 18
B 4, 12, 20
C 9, 12, 15
D 6, 10, 20
Parts 1+2+3 = 6. One part 36 ÷ 6 = 6. Shares 6, 12, 18.
Level 3 · Hard
Getting tougher
Reverse problems — you're given a share or a difference.
1. Money shared 3:5. The smaller share is $18. What is the total?
Smaller = 3 parts = $18, so one part = $6. Total parts 3+5 = 8, total = 8 × 6 = $48.
2. A ratio 4:7 gives one person 15 more than the other. What is the larger share?
Difference 7−4 = 3 parts = 15, so one part = 5. Larger = 7 × 5 = 35.
3. 5 workers build a wall in 12 days. How long for 3 workers (same rate)?
A 20 days
B 15 days
C 18 days
D 24 days
Total work = 5 × 12 = 60 worker-days. With 3 workers: 60 ÷ 3 = 20 days.
Level 4 · Challenge
Challenge
Multi-step ratio reasoning. Take your time.
1. A bag has red and blue counters in ratio 2:3. There are 12 red counters. How many counters in total?
Red = 2 parts = 12, so one part = 6. Total parts 2+3 = 5, total = 5 × 6 = 30.
2. Two numbers are in ratio 5:8. Their sum is 78. What is the larger number?
Parts 5+8 = 13. One part 78 ÷ 13 = 6. Larger = 8 × 6 = 48.
3. A recipe uses flour and sugar in ratio 5:2. If 350 g of flour is used, how much sugar?
A 140 g
B 175 g
C 100 g
D 125 g
Flour = 5 parts = 350 g, so one part = 70 g. Sugar = 2 × 70 = 140 g.
Page 9 · Well done
You’ve finished Ratios & Proportion
You can now share amounts in a ratio, scale up and down with the unitary method, and work backwards from a single share. These same moves appear all over the Mathematical Reasoning section.
Remember the one methodAdd the parts → find one part → multiply back up. Get to one first, every time.
🎉Great work getting through all four tiers. Try another module next — every one builds a different skill.
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