If the simple interest on a sum of Rs. A at 6% per annum for 2 years is equal to the simple interest on Rs. B at 4% per annum for 3 years, then what is the ratio of A to B?

Correct — A.
Simple interest = P × R × T ⁄ 100. Set the two interests equal.

Write both sides: A × 6 × 2 ⁄ 100 = B × 4 × 3 ⁄ 100
Cancel the 100 and multiply out: 12A = 12B
Divide by 12B: A⁄B = 1

So A : B = 1 : 1 → option (a)

Why the others are wrong

• (b)
  2:3 — 2 : 3 puts the sums in the inverse ratio of the rates alone, 4 : 6, and drops the years. Once time is included, R × T is 12 on both sides.
• (c)
  2:1 — 2 : 1 fails a check: with A = 100 and B = 50, A earns 100 × 12 ⁄ 100 = 12 but B earns 50 × 12 ⁄ 100 = 6. Not equal.
• (d)
  1:2 — 1 : 2 fails the same check the other way: A = 50 earns 6 and B = 100 earns 12. Equal interest here needs equal sums.

When two simple interests are equal, P × R × T is the same on both sides, because the ⁄100 cancels.

So the sums are in the inverse ratio of R × T: P₁ : P₂ = R₂T₂ : R₁T₁.

Here R × T is 6 × 2 = 12 for A and 4 × 3 = 12 for B. Equal products mean equal sums, however the rates and years differ.

Key facts

• SI = P × R × T ⁄ 100
• If SI₁ = SI₂, then P₁R₁T₁ = P₂R₂T₂
• Equal SI puts the sums in the inverse ratio of their R × T products

Common traps

• Comparing the rates alone (6% against 4%) and forgetting the years
• Writing the sums in the direct ratio of R × T instead of the inverse ratio

No directly related past PYQ was found.