Hema is 20 years old and her sister Neha is 30 years old. How many years ago were their ages in the ratio of 3 : 5?

Correct — A. Let it be x years ago. Both sisters were younger by the same x, so the ages then were 20 − x and 30 − x.

(20 − x) ⁄ (30 − x) = 3⁄5
Cross-multiply: 5(20 − x) = 3(30 − x)

100 − 5x = 90 − 3x
10 = 2x → x = 5

Check: 15 and 25 are in the ratio 3 : 5 → option (a).

Why the others are wrong

• (b)
  3 — Three years ago the ages were 17 and 27, a ratio of 17 : 27. For 3 : 5 the younger age would have to be 27 × 3⁄5 = 16.2.
• (c)
  2 — Two years ago they were 18 and 28, i.e. 9 : 14, which is larger than 3 : 5. Going back too little leaves the ratio above the target.
• (d)
  7 — Seven years ago the ages were 13 and 23, a ratio of 13 : 23. That is below 3 : 5, so seven years overshoots.

The difference between two ages never changes — here it is 30 − 20 = 10 years, yesterday, today and forever.

That gives the one-line route. In a 3 : 5 ratio the difference is 2 parts. Two parts equal 10 years, so one part is 5 years, making the ages 15 and 25 at that moment — five years before 20 and 30.

The equation route reaches the same x = 5, and is safer when the ratio is given at two different times.

Key facts

• The difference between two people's ages is constant, so it can be equated across any two dates.
• In a 3 : 5 ratio the difference is 2 parts, and a real difference of 10 years makes one part 5 years.
• Ages 15 and 25 are exactly 5 years before ages 20 and 30.

Common traps

• Subtracting x from only one of the two ages.
• Writing the ratio as 5 : 3 by putting the elder sister first.
• Assuming an age ratio stays fixed over time.

No directly related past PYQ was found.