A hard question on working backwards for P4–P5, with a step-by-step solution, diagrams and the answer. Working-backwards questions appear in the Hua Xia Cup's P1 to P5 first rounds, qualifying rounds and finals.
The question
There is a tin of biscuits on the table.
Big brother takes half and 1 more.
Big sister takes half of what's left and 1 more.
Ming takes half of what's left and 5 more.
Now 2 are left.
How many biscuits were in the tin at first?
Show the hints (3)
💡Hint 1: Work backwards: how many were there before Ming took his?
💡Hint 2: After "half and 5 more" there were 2 left, so half was 5 more than 2.
💡Hint 3: Before big sister took hers, there were 30.
Worked solution
Have a go first, then read the solution step by step.
1.Before Ming: half was 2 + 5 = 7, so there were 7 × 2 = 14.
2.Before big sister: half was 14 + 1 = 15, so there were 30.
3.Before big brother: half was 30 + 1 = 31, so there were 62.
4.Check: 62 → take 32, leaving 30 → take 16, leaving 14 → take 12, leaving 2 ✓
⚠️Common mistake: doubling before adding 5 back. The extra was taken after the half, so add it back first, then double.
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