The difference between the compound interest and the simple interest on an amount of Rs 15000 for 2 years is Rs 96, when interest is compounded annually. What is the rate of interest per annum?

Introduction / Context:
Here we are asked to determine the annual interest rate from the difference between compound interest and simple interest over a two year period. This type of question is conceptually rich because it uses a known relationship between simple and compound interest for short periods. Instead of directly applying both formulas independently, we can use a shortcut expression for the difference when the principal, rate, and time are the same and the time is exactly two years.

Given Data / Assumptions:

• Principal P = Rs 15000
• Time t = 2 years
• Difference between compound interest and simple interest = Rs 96
• Interest is compounded annually for the compound interest calculation
• Simple interest is calculated at the same rate r

Concept / Approach:
For 2 years at rate r (percent per annum), the simple interest is P * r * 2 / 100. The compound interest for 2 years is P * [(1 + r / 100)^2 - 1]. The difference between compound and simple interest over 2 years can be shown to be P * r^2 / 100^2. This is a well known result derived from expanding (1 + r / 100)^2. Therefore, if we know the difference and the principal, we can solve for r using the relation Difference = P * r^2 / 10000.

Step-by-Step Solution:
Difference between CI and SI for 2 years = P * r^2 / 100^2 Given difference = 96 and P = 15000 So 96 = 15000 * r^2 / 10000 Simplify 15000 / 10000 = 1.5 Therefore, 96 = 1.5 * r^2 r^2 = 96 / 1.5 r^2 = 64 r = square root of 64 = 8 So the rate of interest is 8% per annum

Verification / Alternative check:
We can verify by computing both simple and compound interest at 8%. Simple interest for 2 years at 8% is SI = 15000 * 8 * 2 / 100 = 15000 * 16 / 100 = 2400. Compound amount after 2 years at 8% is A = 15000 * (1.08)^2 = 15000 * 1.1664 = 17496. Compound interest is CI = 17496 - 15000 = 2496. Difference CI - SI = 2496 - 2400 = 96, which matches the given value, confirming that 8% is correct.

Why Other Options Are Wrong:
If the rate were 10%, then r^2 would be 100 and the difference would be 15000 * 100 / 10000 = 150, which is not 96. For 12%, r^2 = 144 and the difference would be 15000 * 144 / 10000 = 216. For 13%, r^2 = 169 giving an even larger difference. Therefore none of these options match the required difference of 96.

Common Pitfalls:
A frequent mistake is to attempt to compute full simple and compound interest expressions for an unknown r, which leads to more complex algebra. Using the known shortcut formula for the difference saves time. Another error is to forget that the difference formula P * r^2 / 100^2 is valid specifically for a period of two years and may not be directly used for other time periods without adjustment.

Final Answer:
The required rate of interest is 8% per annum.