The difference between the compound interest and the simple interest on an amount of Rs 15,000 for 2 years is Rs 96. What is the annual rate of interest (in percent)?

Introduction / Context:
In this problem, we compare compound interest (CI) and simple interest (SI) on the same principal and time, and the difference between them is given. We must find the annual rate of interest. There is a useful shortcut for the difference between CI and SI for 2 years at the same rate, which depends on the square of the rate and the principal. Using this shortcut makes the problem very quick and is commonly tested in aptitude exams.

Given Data / Assumptions:

• Principal P = Rs 15,000.
• Time t = 2 years.
• Difference between CI and SI over 2 years is Rs 96.
• Interest is compounded annually for the CI part.
• Annual rate of interest r (in percent) is unknown and needs to be found.

Concept / Approach:
For 2 years, the difference between compound interest and simple interest at rate r% per annum on principal P is given by Difference = P * r^2 / 100^2. This formula comes from comparing P * (1 + r/100)^2 - P (which is CI) with P * r * 2 / 100 (which is SI). Applying this formula, we substitute P = 15,000 and the known difference and solve the resulting quadratic relationship for r. Because standard exam questions usually use neat numeric values, r will turn out to be a simple integer percentage.

Step-by-Step Solution:
Use the formula for 2 years: Difference = P * r^2 / 100^2. We have Difference = 96 and P = 15,000. So 96 = 15,000 * r^2 / 10,000. Simplify the factor: 15,000 / 10,000 = 1.5, so 96 = 1.5 * r^2. Therefore, r^2 = 96 / 1.5 = 64. Taking the positive square root (since rate is positive), r = 8. Hence, the annual rate of interest is 8% per annum.

Verification / Alternative check:
To verify, we can compute SI and CI separately for r = 8%. Simple interest for 2 years: SI = 15,000 * 8 * 2 / 100 = 15,000 * 0.16 = 2,400. Amount under CI: A = 15,000 * (1.08)^2 = 15,000 * 1.1664 = 17,496. So CI = 17,496 - 15,000 = 2,496. Difference CI - SI = 2,496 - 2,400 = 96, which matches the given difference exactly, confirming that 8% is correct.

Why Other Options Are Wrong:
If r = 10%, then r^2 = 100 and the difference would be 15,000 * 100 / 10,000 = 150, not 96. For r = 12%, r^2 = 144 and the difference becomes 15,000 * 144 / 10,000 = 216, again not 96. For r = 14% or 6%, the differences would be 294 and 54 respectively. None of these match the given Rs 96 difference, so they cannot be correct.

Common Pitfalls:
Some test-takers mistakenly use the 3-year difference formula or attempt to derive SI and CI separately without noticing the 2-year shortcut, which makes the process longer and prone to arithmetic errors. Others forget to square the rate when applying the shortcut, which leads to a very different value of r. Being aware of and correctly using the special formula for 2 years is crucial for speed and accuracy in such questions.

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