At present, A is thrice as old as B. C is 5 years elder than B. The sum of the ages of A, B, C is 75 years. The ratio of the age of B five years ago to the age of A three years from now is:

Correct — C. Write every age in terms of B.

B = x, A = 3x, C = x + 5
Sum: 3x + x + (x + 5) = 75
5x = 70 → x = 14
So B = 14, A = 42, C = 19 (check: 42 + 14 + 19 = 75)

Now apply the two time shifts separately.
B five years ago = 14 − 5 = 9
A three years from now = 42 + 3 = 45
9 : 45 = 1 : 5 → option (c).

Why the others are wrong

• (a)
  1 : 4 — 1 : 4 would need A's age three years from now to be 4 × 9 = 36. A is 42 today and 45 then, so 36 matches no age in this problem.
• (b)
  2 : 3 — With B's past age fixed at 9, a ratio of 2 : 3 needs 13.5 as the second term — not a whole age, and nowhere near A's 45.
• (d)
  4 : 7 — 4 : 7 against a first term of 9 requires 15.75. Both surviving numbers here are whole, so any ratio whose first term does not divide 9 cleanly can be dropped.

An age problem of this shape is one linear equation wearing three sentences. "Thrice as old" multiplies; "5 years elder" adds. Put all three people in the same unknown, then solve once.

The time shifts come after the solve, not before. A past age subtracts, a future age adds — and the stem may move two people by two different amounts, as it does here.

Only when both numbers are on the table do you form the ratio and reduce it.

Key facts

• A = 42, B = 14 and C = 19 satisfy all three conditions: 42 = 3 × 14, 19 = 14 + 5, and the ages sum to 75.
• A past age subtracts from the present age.
• A future age adds to the present age.
• The gap between two people's ages never changes with time.
• 9 : 45 reduces to 1 : 5 on dividing both terms by 9.

Common traps

• Applying the −5 and the +3 to the wrong person — the stem moves B backwards and A forwards
• Reading "C is 5 years elder than B" as C = B − 5, which gives x = 16
• Answering the present-age ratio 14 : 42 = 1 : 3 and stopping before the shifts

No directly related past PYQ was found.