The average age of Raj and his father is 45 years. If the ages of the father and the grandfather of Raj are respectively two and three times that of Raj, then the age of Raj's grandfather is

Q: What is the answer to CAPF 2021 Paper I Q77?

B. Correct — B, 90 years. An average of 45 across two people means their ages add to 90, so Raj plus his father is 90. The father is twice Raj's age, which turns the sum into Raj plus 2 × Raj, or 3 × Raj, equal to 90. Raj is therefore 30 and his father 60, and the check holds since their average is indeed 45. The grandfather is three times Raj's age, that is 3 × 30 = 90 years. Notice that the grandfather's age never enters the average — it is attached to Raj alone, so the whole question is solved the moment Raj's age is known. (Provisional — pending the official UPSC key.)

(a)75 years
(b)90 years
(c)81 years
(d)84 years

Answer

Why

Correct — B, 90 years. An average of 45 across two people means their ages add to 90, so Raj plus his father is 90. The father is twice Raj's age, which turns the sum into Raj plus 2 × Raj, or 3 × Raj, equal to 90. Raj is therefore 30 and his father 60, and the check holds since their average is indeed 45. The grandfather is three times Raj's age, that is 3 × 30 = 90 years. Notice that the grandfather's age never enters the average — it is attached to Raj alone, so the whole question is solved the moment Raj's age is known.

Why the others are wrong

(a)75 years — Would need Raj to be 25, which makes the father 50 and their average 37·5 rather than 45.
(c)81 years — Not a multiple of three times any age consistent with the sum of 90; it would require Raj to be 27 and the father 54, whose average is 40·5.
(d)84 years — Implies Raj is 28 and the father 56, an average of 42. Only Raj at 30 satisfies the stated average of 45.

Concept

An average is a total in disguise. The first move in any average problem is to convert the mean back into a sum by multiplying by the number of items, because sums can be added, subtracted and equated while means cannot. Age problems then add one further step: express every person's age in terms of a single unknown before writing the equation.

The trap in this item is a second one hidden in the ratio. Both the father's and the grandfather's ages are given as multiples of Raj's, not of each other, so the tempting move of taking the grandfather to be one and a half times the father is fine here by coincidence but is the wrong instinct in general. Set the youngest person as the unknown, express the others in terms of him, and the sentence about the average supplies the single equation you need.

Key facts

An average of 45 for two people means their ages total 90.
Father = 2 × Raj, so Raj + 2Raj = 90 and Raj is 30.
The father is 60 and the grandfather 3 × 30 = 90.
The grandfather is not part of the average given in the stem.
Convert averages into sums before doing anything else.

From an average to three ages

Average of Raj and his father→45, so their ages total 90

Father in terms of Raj→2 × Raj

Equation→Raj + 2Raj = 90, so Raj = 30

Father→60 — and the average of 30 and 60 is 45, as stated

Grandfather→3 × 30 = 90 years

Every wrong option corresponds to a different value for Raj, none of which returns an average of 45.

Study next

Averages and the conversion of a mean into a totalAge problems with present, past and future agesWeighted averages and the rule of alligationRatios expressed against a common baseAverages of an arithmetic progression, where the mean is the middle term

Common traps

Assuming the grandfather is included in the average of 45.
Reading 'three times' as three times the father's age instead of Raj's.
Solving for Raj and then reporting his father's age.

A short average-and-ratio item. Turning the average into a sum solves it in one line.

Related PYQs

CAPF_GAI_2022_Q302022Open & attempt →

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

Answer(c) 28 years

The same family and the same first move, with three averages instead of one. Converting each average into a total and adding them gives twice the father plus twice each son, which is why it resolves without ever finding an individual age.

Practice

practice — not a real PYQ
The average age of a father and his son is 28 years. If the father is three times as old as the son, the son's age is
- (a)12 years
- (b)14 years
- (c)16 years
- (d)18 years
Answer(b) 14 years — the ages total 56 and son plus 3 × son equals 56.
practice — not a real PYQ
The average of five numbers is 20. If one number is removed, the average of the remaining four is 18. The number removed is
- (a)26
- (b)28
- (c)30
- (d)32
Answer(b) 28 — the totals are 100 and 72, and the difference is the number removed.