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Maths olympiad question · P4–P5★★★ Hard

Clever sums: olympiad question 6

A hard question on clever sums for P4–P5, with a step-by-step solution, diagrams and the answer. The first few questions of a competition paper are often clever sums, and the Hua Xia Cup has them in every year from P1 to P6.

The question

Work this out the clever way.

19 × 185 + 55 × 20 + 15 × 19 − 20 × 45 = ?

What is the answer?

Show the hints (3)
  1. Hint 1: Which of the four products share a factor?
  2. Hint 2: Put the two products with 20 together, and the two with 19. Keep the minus sign with its product.
  3. Hint 3: 55 − 45 = 10, and 185 + 15 = 200.

Worked solution

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

  1. 1.Put the two with 20 together, and the two with 19; the minus sign moves with its product.
  2. 2.The two with 20: 55 × 20 − 45 × 20 = (55 − 45) × 20 = 10 × 20 = 200.
  3. 3.The two with 19: 185 × 19 + 15 × 19 = (185 + 15) × 19 = 200 × 19 = 3800.
  4. 4.200 + 3800 = 4000.
  5. ⚠️Common mistake: moving the terms but leaving the minus sign behind, so the wrong product gets subtracted.

Answer: 4000

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