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

Age problems: olympiad question 8

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 Dad, Mum, the older sister and the younger brother are 110 years old in all.

4 years ago the family's ages added up to 97 (anyone not yet born doesn't count).

Dad is 3 years older than Mum, and this year Dad is 3 times as old as the older sister.

How old is Dad this year?

Show the hints (3)
  1. Hint 1: If all four were already born 4 years ago, how much smaller should the total be? Compare with the story.
  2. Hint 2: The total is only 13 smaller, less than 4 × 4. Who wasn't born 4 years ago, and how old is that child now?
  3. Hint 3: The younger brother is 1, so Dad, Mum and the older sister are 109 in all. Add 3 to Mum to make her Dad, then use the 3-times link.

Worked solution

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

  1. 1.If all four were born 4 years ago, the total would drop by 4 × 4 = 16; it drops only 110 − 97 = 13. So someone wasn't born yet — it can only be the youngest, the younger brother.
  2. 2.Dad, Mum and the older sister each drop 4: 3 × 4 = 12; the other 13 − 12 = 1 is the younger brother's age now (1 is less than 4, as it must be).
  3. 3.Dad, Mum and the older sister are 110 − 1 = 109 together. Mum plus 3 equals Dad: two Dads + the older sister = 109 + 3 = 112.
  4. 4.Dad is 3 times the older sister, so two Dads + the older sister = 2 × 3 + 1 = 7 times the older sister: 112 ÷ 7 = 16.
  5. 5.Dad is 3 × 16 = 48.
  6. 6.Check: Mum is 45; 48 + 45 + 16 + 1 = 110 ✓; 4 years ago 44 + 41 + 12 = 97 ✓
  7. ⚠️Common mistake: taking 4 off all four and getting stuck when the numbers don't match. The total falling too little is exactly the sign that someone wasn't born.

Answer: 48 years old

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