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Maths olympiad question · P3–P4★★★★ Expert

Age problems: olympiad question 7

An expert question on age problems for P3–P4, with a step-by-step solution, diagrams and the answer. Age problems are common in the Hua Xia Cup's P1 to P4 first and qualifying rounds, and rarer in the P5 and P6 papers.

The question

This year Grandpa's age is a whole multiple of his grandson's.

Next year and in 2 years, Grandpa's age will be a whole multiple of his grandson's then, too.

His grandson is 4 to 12 this year, and Grandpa is 55 to 99.

How old is Grandpa this year?

Show the hints (3)
  1. Hint 1: The gap between their ages never changes. If Grandpa's age is a whole multiple of the child's, how is the gap related to the child's age?
  2. Hint 2: Gap = Grandpa − child = (multiple − 1) × child, so the gap must be a multiple of the child's age at all three times. Try each age the child could be now.
  3. Hint 3: Only a child of 4 now works: 4, 5, 6 at the three times, so the gap is a multiple of 60.

Worked solution

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

  1. 1.The gap never changes. In a year when Grandpa's age is a whole multiple of the child's, the gap = (multiple − 1) × the child's age — a multiple of it.
  2. 2.So the gap is a multiple of the child's ages this year, next year, in 2 years — a common multiple. And Grandpa is 55 to 99 now, which bounds the gap.
  3. 3.Try each age for the child now: 4: LCM 60, gap 60, Grandpa 64 ✓; 5: LCM 210, too big; 6: LCM 168, too big; 7: LCM 504, too big; 8: LCM 360, too big; 9: LCM 990, too big; 10: LCM 660, too big; 11: LCM 1716, too big; 12: LCM 1092, too big.
  4. 4.Only one works: the child is 4, the gap 60, so Grandpa is 4 + 60 = 64.
  5. 5.Check: this year 64 ÷ 4 = 16; next year 65 ÷ 5 = 13; in 2 years 66 ÷ 6 = 11 ✓
  6. ⚠️Common mistake: using only this year's multiple, or thinking the multiple stays the same. The multiple changes; the gap doesn't. And check every age in the range.

Answer: 64 years old

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