If the simple interest on a sum at 4% per annum for 2 years is Rs. 80, then…

Concept: For a principal P at rate r% per annum, Simple Interest for time t years is SI = P × r × t / 100, and Compound Interest is CI = P[(1 + r/100)t − 1]. For exactly 2 years, this simplifies to CI − SI = P × (r/100)2 — the extra interest is exactly one year's interest earned on the first year's interest.

1. Recover the principal from the given simple interest: P = SI × 100 / (r × t) = 80 × 100 / (4 × 2) = 8000 / 8 = Rs. 1000.

2. Find the amount after 2 years compounded annually at 4%: A = P(1 + r/100)t = 1000 × (1.04)2 = 1000 × 1.0816 = Rs. 1081.60.

3. Compound Interest = A − P = 1081.60 − 1000 = Rs. 81.60.

Cross-check using the identity CI − SI = P × (r/100)2: 1000 × (4/100)2 = 1000 × 0.0016 = Rs. 1.60. Adding this to the simple interest gives CI = 80 + 1.60 = Rs. 81.60, confirming the result independently.

So, the compound interest on the same sum for the same period is Rs. 81.60.