A hard question on ratios for P6, with a step-by-step solution, diagrams and the answer. Ratio questions appear in the Hua Xia Cup's P4 to P6 first rounds, qualifying rounds and finals, such as linked ratios and ratios that change.
The question
Chi-keung and Hiu-ching each have some stickers; the numbers they have are in the ratio 7 :
2 (Chi-keung :
Hiu-ching).
After Chi-keung gives Hiu-ching 164 stickers, the ratio becomes 4 :
7.
How many stickers did Chi-keung have at first?
Show the hints (3)
💡Hint 1: When Chi-keung gives Hiu-ching some stickers, what stays the same?
💡Hint 2: Before, the total is 9 parts; after, 11 parts — different-sized parts. Rewrite both ratios with a total of 99 parts.
💡Hint 3: Then Chi-keung goes from 77 parts to 36; the parts lost are the 164 given away.
Worked solution
Have a go first, then read the solution step by step.
1.Chi-keung gives to Hiu-ching: the stickers only change hands, so their total stays the same — the two bars are the same length.
2.But before, the total is cut into 7 + 2 = 9 parts, and after into 4 + 7 = 11: different-sized parts. Cut the total again into 99 parts (the lowest common multiple of 9 and 11).
4.Chi-keung goes from 77 parts to 36: 77 − 36 = 41 parts, and that is the 164 given to Hiu-ching. 1 part = 164 ÷ 41 = 4.
5.Chi-keung had 77 parts: 77 × 4 = 308.
6.Check: at first Chi-keung 308, Hiu-ching 88, and 308 : 88 = 7 : 2; after, Chi-keung 144, Hiu-ching 252, and 144 : 252 = 4 : 7 ✓
⚠️Common mistake: subtracting 4 parts from 7 parts straight away. The two ratios have different totals (9 and 11), so their parts differ in size; make the totals match first.
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