An expert question on fraction problems for P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P5 and P6 qualifying rounds and finals; harder ones work backwards through fractions, or need clever shortcuts with fractions.
The question
At the school open day, the maths stall had a batch of small presents for visitors who answered a question correctly.
In the first hour they gave away half of the presents; in the second hour, a third of what was left; in the third hour, a quarter of what was left; in the fourth hour, a fifth of what was left.
In hour 3 they gave away 4 more presents than in hour 4.
How many presents did the stall have at the start?
Show the hints (3)
💡Hint 1: Don't work out each hour's amount yet. Ask instead: after each hour, what fraction of the whole is left?
💡Hint 2: After hour k, 1/(k + 1) of the whole is left. So what fraction of the whole did hour k give away? (What was left before it, less what was left after.)
💡Hint 3: Hour 3 took 1/12 of the whole and hour 4 took 1/20. The difference is 4.
Worked solution
Have a go first, then read the solution step by step.
1.First, what fraction of the whole is left after each hour: after the first 1/2; after the second 1/2 × 2/3 = 1/3; after the third 1/3 × 3/4 = 1/4… after hour k, 1/(k + 1).
2.So hour k gave away what was left before less what was left after: 1/k − 1/(k + 1) = 1/(k × (k + 1)) of the whole.
3.Hour 3: 1/(3 × 4) = 1/12 of the whole; hour 4: 1/(4 × 5) = 1/20.
4.They differ by 1/12 − 1/20 = 8/240 = 1/30 of the whole, which is 4.
5.The whole = 4 × 30 = 120 presents.
6.Check: each hour gives away 60, 20, 10, 6; hour 3 is 10 − 6 = 4 more than hour 4 ✓
⚠️Common mistake: thinking hour k takes 1/(k + 1) of the whole and working with 1/4 − 1/5, answering 80. After the first hour, each fraction is of what was left then.
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