The present ages (in years) of A and B are in the ratio of 3 : 4. Five years back, the ratio of their ages was 2 : 3. What is the present age (in years) of A?

Correct — C. Write both present ages with a single unknown scale factor.

A = 3x and B = 4x
Five years back: (3x − 5)⁄(4x − 5) = 2⁄3
Cross-multiply: 3(3x − 5) = 2(4x − 5)
9x − 15 = 8x − 10

x = 5
A = 3x = 15 → option (c)

Check it: five years back A was 10 and B was 15, and 10 : 15 is exactly 2 : 3.

Why the others are wrong

• (a)
  28 — Option (a), 28, forces B = 4⁄3 × 28 = 37⅓. Five years back that pair reads 23 : 32⅓, about 0.71, not the required 2 : 3 ≈ 0.67.
• (b)
  21 — Option (b), 21, forces B = 28, and five years back the pair is 16 : 23 ≈ 0.70. Close, but the ratio has to be exactly 2 : 3.
• (d)
  20 — Option (d), 20, is B's present age, not A's. Solving correctly gives A = 15 and B = 20, so 20 is what you reach by answering for the wrong letter.

Two ratios at two different dates need exactly one unknown, not two.

Set the present ages as 3x and 4x so the first ratio is built in, then impose the second condition to pin x. Subtracting 5 from each actual age is legitimate. Subtracting 5 from each ratio term is not, because 3 : 4 five years ago is not (3 − 5) : (4 − 5).

The structural check is the gap. A difference of ages never changes with time. Here B − A = 4x − 3x = x = 5, and five years back the ages were 10 and 15, still 5 apart.

Key facts

• Present ages in the ratio 3 : 4 can always be written as 3x and 4x for one positive x.
• Here x = 5, so A is 15 years old and B is 20.
• Five years back the ages were 10 and 15, which is exactly the stated 2 : 3.
• The difference between two people's ages is constant, so B − A = 5 at every date in this question.

Common traps

• Answering 20, which is B's age rather than A's.
• Subtracting 5 from the ratio terms instead of from the ages.
• Treating x as an age when it is only a scale factor.

No directly related past PYQ was found.