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Maths olympiad question · P4–P5★★★ Hard

Working backwards: olympiad question 5

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

The teacher asks Tsz-ching to use a calculator: take a number, multiply by 3, add 1, then divide by 4.

Tsz-ching pressed the wrong key, dividing by 3 instead of multiplying, did the other steps correctly, and got 2.

What should the right answer have been?

Show the hints (3)
  1. Hint 1: First work back from Tsz-ching's wrong answer to find the number.
  2. Hint 2: Tsz-ching actually did: divide by 3, add 1, divide by 4, getting 2. Undo each with its opposite.
  3. Hint 3: The number was 21.

Worked solution

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

  1. 1.What Tsz-ching actually did: ÷ 3, + 1, ÷ 4, giving 2.
  2. 2.Undo: before ÷ 4, 2 × 4 = 8; before + 1, 8 − 1 = 7; before ÷ 3, 7 × 3 = 21.
  3. 3.The right way: 21 × 3 = 63, + 1 = 64, ÷ 4 = 16.
  4. ⚠️Common mistake: trying to "fix" 2 directly, say by multiplying by 3 twice. The "+ 1" in the middle gets in the way, so find the number first.

Answer: 16

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