Ratio

2-part and 3-part ratios, equivalent ratios, dividing a quantity in a given ratio, and the ratio–fraction relationship.

a:b:c ratiosdivide in ratiofraction link P2 · calc
Ratio a:b:c → parts are a, b, c out of (a+b+c) Equivalent ratios: scale up or down Ratio ↔ fraction: a:b means a/(a+b)
Question 1 — Ratio changes when one quantity changes
Structured (3–4 marks) Paper 2 · calculator 2-part ratio · before & after

Ali and Ben had marbles in the ratio 3 : 5. After Ali gave 12 marbles to Ben, the ratio became 1 : 3. How many marbles did Ali have at first?

Before Ali: 3 units Ben: 5 units After Ali: 1u Ben: 3 units −12 →
Let Alihave 3u marbles before. After giving 12: Ali has 3u − 12.
After ratioAli : Ben = 1 : 3, so Ali = ¼ of total.
Totalis unchanged: 3u + 5u = 8u.
After Ali= ¼ × 8u = 2u
Equation3u − 12 = 2u → u = 12
Ali at first= 3 × 12 = 36 marbles
Ali had 36 marbles at first
Common trap
Assuming the total changes after the transfer — it doesn't. Marbles are only moved between Ali and Ben, so the total stays at 8u throughout.
Question 2 — 3-part ratio with a known difference
Structured (4 marks) Paper 2 · calculator 3-part ratio · find each share

A sum of money is shared among Ahmad, Betty and Calvin in the ratio 4 : 7 : 9. Calvin receives $150 more than Ahmad. How much does Betty receive?

Ahmad 4 units Betty 7 units Calvin 9 units difference = 5 units = $150
DifferenceCalvin − Ahmad = 9u − 4u = 5 units
5 units= $150, so 1 unit = $150 ÷ 5 = $30
Betty= 7 units = 7 × $30 = $210
Betty receives $210
Common trap
Students sometimes subtract the ratio numbers directly: 9 − 4 = 5, but forget to use the difference to find the value of 1 unit first. Always find 1 unit before scaling up to Betty's 7 units.
Question 3 — Ratio changes after a transaction (3-part, hardest type)
Long answer (5 marks) Paper 2 · calculator 3-part ratio · before & after Distinction level

Priya, Quinn and Ravi had savings in the ratio 5 : 3 : 2. Priya gave $60 to Ravi. The ratio of their savings then became 3 : 3 : 4.

(a) How much did each person have at first?
(b) What percentage of Priya's original savings did she give away?

Before Priya: 5 units Quinn: 3u Ravi: 2u After Priya: 3u Quinn: 3u Ravi: 4u $60 Total unchanged
Key insightQuinn's amount is unchanged (only Priya → Ravi). So Quinn's ratio part tells us the unit value.
QuinnBefore: 3 units. After: 3 units. ✓ Consistent — unit size has not changed.
PriyaBefore: 5u. After: 3u. She lost 2 units = $60 → 1 unit = $30
(a) Priya= 5 × $30 = $150
Quinn= 3 × $30 = $90
Ravi= 2 × $30 = $60
(b) %= $60 ÷ $150 × 100% = 40%
(a) Priya $150, Quinn $90, Ravi $60  |  (b) 40%
Common trap
Trying to equate ratios directly (5:3:2 vs 3:3:4) without first checking whether the unit size is the same in both ratios. Quinn's amount doesn't change — this anchors the unit value and makes the problem solvable. Always look for the unchanged quantity.