The average age of father and elder son is 35 years, the average age of father and younger son is 32 years and the average age of the two sons is 17 years. What is the average age of the father and his two sons?
- (a)30 years
- (b)27 years
- (c)28 years
- (d)29 years
Correct — C, 28 years. Turn every average into a total. Father plus elder son is 2 × 35 = 70; father plus younger son is 2 × 32 = 64; the two sons together are 2 × 17 = 34. Add the three totals: (F + S₁) + (F + S₂) + (S₁ + S₂) = 70 + 64 + 34 = 168, and the left side is exactly 2(F + S₁ + S₂). So F + S₁ + S₂ = 84, and the average of the three is 84 ÷ 3 = 28 years.
- (a)30 years — Thirty is what you get by averaging the three given averages after mishandling the pair sums; the correct total of the three ages is 84, not 90.
- (b)27 years — This needs a three-person total of 81. Adding the three pair totals gives 168, which is twice 84, so 81 cannot arise.
- (d)29 years — A total of 87 would be needed. The arithmetic is exact and leaves no room to round: 168 ÷ 2 ÷ 3 = 28.
Every average statement is a disguised total, and totals add. When each condition names a different pair drawn from the same three people, adding all the pair totals counts every person exactly twice, so halving the sum gives the grand total in one step.
There is no need to solve for the individual ages, and trying to do so wastes time. If you do want them: subtract the son-pair total from the grand total to get the father's age, 84 − 34 = 50; the elder son is 70 − 50 = 20 and the younger is 64 − 50 = 14, which checks against every condition.
- Average × count = total; converting to totals first is the standard opening move in every average problem.
- Summing all pair totals from a group of three counts each member twice, so grand total = (sum of pair totals) ÷ 2.
- For n people the same identity generalises: each member appears in (n − 1) of the pair sums.
- Ages here work out to father 50, elder son 20 and younger son 14 — worth computing once as a check.
Each person is counted twice on the left, so one division by two finishes the problem.
- Averaging the three given averages, which double-counts the father and gives 30.
- Solving for the three ages one at a time when the sum is all that is asked for.
- Reading 'average age of the two sons' as involving the father as well.
Asked as several overlapping pairwise averages with the group average as the target, sometimes with one individual's age asked for instead.
The average age of the boys in a class is 12 years. The average age of the girls in the class is 11 years. There are 50% more girls than boys in the class. Which one of the following is the average age of the class (in years)?
- (a) 11.2 years
- (b) 11.4 years
- (c) 11.6 years
- (d) 11.8 years
Answer(b) 11.4 years
CAPF's own averages question from two years earlier, built on the same first move — convert each average into a total before combining. There the weights differ between the two groups; here the groups overlap.
- practice — not a real PYQ
The average of A and B is 20, the average of B and C is 25 and the average of A and C is 27. The average of A, B and C is
- (a)22
- (b)24
- (c)26
- (d)28
Answer(b) 24 — the pair totals are 40, 50 and 54; their sum 144 equals twice (A + B + C), so A + B + C = 72 and the average is 24.
- practice — not a real PYQ
The average age of a family of five is 24 years. If the youngest member is 8 years old, the average age of the family at the time of that member's birth was
- (a)16 years
- (b)18 years
- (c)20 years
- (d)22 years
Answer(a) 16 years — the present total is 120; eight years ago the other four were each 8 years younger, giving 120 − 8 − 32 = 64 for four people, an average of 16.