A hard question on surplus and shortage for P4–P5, with a step-by-step solution, diagrams and the answer. Surplus-and-shortage questions appear in the Hua Xia Cup's P1 to P5 papers; harder ones become questions about distance and time.
The question
At Mid-Autumn a community centre shares out mooncakes among the elderly.
With 2 each, 18 are left over.
With 3 each, the last person gets only 2, and everyone else gets 3.
How many mooncakes are there?
Show the hints (3)
💡Hint 1: Put "the last gets only 2" another way: to give the last person 3 too, how many more are needed?
💡Hint 2: So: 2 each leaves 18 over; 3 each is 1 short.
💡Hint 3: Each gets 1 more, needing 18 + 1 = 19 more in all, so there are 19 people.
Worked solution
Have a go first, then read the solution step by step.
1.Look at the second way first: the last person gets only 2, so 3 − 2 = 1 more are needed for 3 each.
2.So: 2 each leaves 18 over; 3 each is 1 short. The second way gives everyone 3 − 2 = 1 more.
3.Going from 18 over to 1 short, everyone together got 18 + 1 = 19 more.
4.At 1 more each, there are 19 ÷ 1 = 19 people.
5.Mooncakes: 19 × 2 + 18 = 56.
6.Check: 18 people × 3 = 54, plus 2 for the last = 56 ✓
⚠️Common mistake: reading "the last gets only 2" as 2 over or 2 short. It is 1 short.
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