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Maths olympiad question · P5–P6★★★★ Expert

Surplus and shortage: olympiad question 10

An expert question on surplus and shortage for P5–P6, 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 school camp books big and small rooms at a holiday camp.

A big room sleeps 6 and a small room sleeps 5.

With every one full, 2 people have nowhere to sleep.

If every small room were a big room instead (the same number in all), there would be 14 spare places once everyone is in.

If the numbers of big and small were swapped (as many big as there are small now, and the other way round), there would be 11 spare places once everyone is in.

How many people are there?

Show the hints (3)
  1. Hint 1: Compare the first and second ways: how many more places does the second have? Which small rooms make the difference?
  2. Hint 2: Each small room turned big gives 1 more place. When the numbers swap, only the small rooms beyond the number of big rooms really add places.
  3. Hint 3: All big gives 2 + 14 = 16 more places; swapping gives 2 + 11 = 13 more.

Worked solution

Have a go first, then read the solution step by step.

  1. 1.From 2 left out to 14 spare, the places grow by 2 + 14 = 16.
  2. 2.They all come from turning small rooms big, 6 − 5 = 1 more each, so there are 16 ÷ 1 = 16 small rooms.
  3. 3.Swapping: 2 + 11 = 13 more places. After the swap there are as many more big rooms as there were extra small rooms, each (with one small room fewer) giving 1 more place.
  4. 4.So there are 13 ÷ 1 = 13 more small rooms than big rooms, and 16 − 13 = 3 big rooms.
  5. 5.People: 6 × 3 + 5 × 16 + 2 = 100.
  6. 6.Check: all big, 6 × 19 = 114, 14 spare ✓; swapped, 6 × 16 + 5 × 3 = 111, 11 spare ✓
  7. ⚠️Common mistake: working with 14 − 2 or 11 − 2. Going from too few places to spare ones, the growth is the sum. Another: thinking every small room adds places in the swap — only the 13 extra ones do.

Answer: 100 people

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