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
A scout troop goes camping with a number of tents.
With 4 to a tent, 12 people have nowhere to sleep.
With 5 to a tent, 3 tents are left empty and one more has only 2 in it; all the others are full.
How many people are there?
Show the hints (3)
💡Hint 1: In the second plan, how many more people would it take to fill the empty tents and the part-full one?
💡Hint 2: 5 × 3 + (5 − 2) = 18 short. So one plan has 12 over, the other 18 short.
💡Hint 3: Each tent takes 1 more, 12 + 18 = 30 in all, so there are 30 tents.
Worked solution
Have a go first, then read the solution step by step.
1.Look at the second plan first, as if every tent were full: the empty 3 need 5 × 3 = 15, the part-full one needs 5 − 2 = 3, so 15 + 3 = 18 short.
2.So: 4 each leaves 12 over; 5 each is 18 short. The second plan puts 5 − 4 = 1 more in each tent.
3.Going from 12 over to 18 short, all the tents together take 12 + 18 = 30 more.
4.At 1 more each, there are 30 ÷ 1 = 30 tents.
5.People: 4 × 30 + 12 = 132.
6.Check: 26 full tents of 5 plus 2: 5 × 26 + 2 = 132 ✓
⚠️Common mistake: counting only the empty tents and forgetting the part-full one is 3 short too.
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