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

Working backwards: olympiad question 7

An expert question on working backwards for P5–P6, 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

Big Brother, Big Sister and Ming play a card game.

In round 1, Big Brother hands out a fifth of the cards in hand, split evenly between the other two.

In round 2, Big Sister hands out a quarter of the cards in hand then, split evenly between the other two.

In round 3, Ming does the same with half of the cards in hand.

In the end all three have 192 cards each.

How many cards did Big Brother have at first?

Show the hints (3)
  1. Hint 1: Work back from the end. After handing out half in round 3, what fraction does Ming keep? That is 192 cards.
  2. Hint 2: Find how many Ming handed out; each of the others got half, so before round 3 they had that much less. Then work back rounds 2 and 1 the same way.
  3. Hint 3: After round 2 (before round 3) they had 96, 96, 384.

Worked solution

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

  1. 1.Cards only change hands, so the total stays 3 × 192 = 576. Work back from the last round.
  2. 2.Before round 3: after handing out half, Ming kept 1/2 of the cards, 192, so had 192 ÷ 1 × 2 = 384 before it and handed out 192, 96 to each of the others. Big Brother had 192 − 96 = 96, Big Sister 192 − 96 = 96.
  3. 3.Before round 2: after handing out a quarter, Big Sister kept 3/4 of the cards, 96, so had 96 ÷ 3 × 4 = 128 before it and handed out 32, 16 to each of the others. Big Brother had 96 − 16 = 80, Ming 384 − 16 = 368.
  4. 4.Before round 1: after handing out a fifth, Big Brother kept 4/5 of the cards, 80, so had 80 ÷ 4 × 5 = 100 before it and handed out 20, 10 to each of the others. Big Sister had 128 − 10 = 118, Ming 368 − 10 = 358.
  5. 5.So Big Brother had 100 at first (the three had 100, 118, 358, 576 in all ✓).
  6. 6.Check (forwards): after round 1: 80, 128, 368; after round 2: 96, 96, 384; after round 3: 192, 192, 192 ✓
  7. ⚠️Common mistake: undoing "hand out half" as "add half back": 192 × (1 + 1/2). What's left is 1/2, so divide: 192 ÷ 1 × 2.

Answer: 100 cards

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