The difference between simple interest and compound interest, compounded annually, on a certain sum of money for 2 years at 16% per annum is Rs. 320. What is the value of the principal (sum invested) in rupees?

Introduction / Context:
This problem compares simple interest and compound interest on the same principal, at the same rate and time. The difference between compound interest and simple interest over 2 years has a neat formula that can be used to find the principal quickly when the rate and difference are known.

Given Data / Assumptions:

Rate of interest r = 16% per annum.
Time t = 2 years.
Difference between compound interest and simple interest = Rs. 320.
Interest is compounded annually for the CI calculation.
Principal is P and must be found.

Concept / Approach:
For 2 years at rate r% per annum, the difference between compound interest (CI) and simple interest (SI) on principal P is: CI − SI = P * (r^2 / 100^2). This arises because CI for 2 years equals SI for 2 years plus an extra interest on the first year's interest. Knowing r and the difference, we can directly solve for P.

Step-by-Step Solution:
Step 1: Write the formula for the difference. CI − SI = P * r^2 / 100^2. Given CI − SI = 320 and r = 16. Step 2: Substitute values. 320 = P * 16^2 / 100^2. 16^2 = 256 and 100^2 = 10,000. So 320 = P * 256 / 10,000. Step 3: Solve for P. P = 320 * 10,000 / 256. P = 3,200,000 / 256 = 12,500.

Verification / Alternative check:
Compute SI and CI explicitly. SI for 2 years: SI = P * r * t / 100 = 12,500 * 16 * 2 / 100 = 4,000. CI amount: Amount A = 12,500 * (1.16)^2 = 12,500 * 1.3456 = 16,820. CI = A − P = 16,820 − 12,500 = 4,320. Difference CI − SI = 4,320 − 4,000 = 320, matching the given value.

Why Other Options Are Wrong:
If P were 25,000, the difference would be double, 640. For 37,500 it would be 960, and for 50,000 it would be 1,280. None of these match the specified difference of 320, so those options must be rejected.

Common Pitfalls:
Some candidates try to compute SI and CI with an unknown P separately, leading to more algebra than necessary. Others forget or mishandle the r^2 / 100^2 term. Remembering the direct formula for the difference over 2 years saves time and reduces the likelihood of algebraic errors.

Final Answer:
The principal (sum invested) was Rs. 12,500.