In how much time will ₹5,000 at 4% per annum simple interest produce the same interest as ₹8,000 in 2 1⁄2 years at 8% per annum simple interest?

Correct — B. Find the interest that has to be matched, then the time that produces it.

SI = P × R × T ⁄ 100
Second sum: 8,000 × 8 × 2.5 ⁄ 100 = ₹1,600
First sum, one year: 5,000 × 4 ⁄ 100 = ₹200
Time = 1,600 ⁄ 200 = 8 years

That is option (b).

Why the others are wrong

• (a)
  2 years — 2 years on ₹5,000 at 4% earns 5,000 × 4 × 2 ⁄ 100 = ₹400, a quarter of the ₹1,600 that has to be matched.
• (c)
  5 years — 5 years earns ₹1,000. It comes from scaling 2.5 years by the rate ratio 8 : 4 alone, forgetting that the principal drops from ₹8,000 to ₹5,000 as well.
• (d)
  4 years — 4 years earns ₹800, exactly half the target. This is 2.5 years scaled by the principal ratio 8,000 : 5,000 = 1.6, with the rate ratio left out.

Simple interest is linear in all three of principal, rate and time: SI = PRT ⁄ 100. Nothing compounds, so a ratio in one factor trades directly against a ratio in another.

To match ₹1,600 with a smaller principal at a lower rate, the time has to absorb both ratios:
2.5 × (8,000 ⁄ 5,000) × (8 ⁄ 4) = 2.5 × 1.6 × 2 = 8 years

The two-step route is safer under pressure — get the target interest in rupees, then divide by the yearly interest of the other scheme.

Key facts

• SI = P × R × T ⁄ 100, with T in years and R in per cent per annum.
• ₹8,000 at 8% for 2.5 years yields ₹1,600 of simple interest.
• ₹5,000 at 4% yields ₹200 a year, so ₹1,600 of interest takes 8 years.

Common traps

• Reading 2½ years as 2 years, which puts the target interest at ₹1,280.
• Scaling the time by the rate ratio or the principal ratio but not by both.

No directly related past PYQ was found.