The difference between the simple interest and the compound interest (compounded annually) on a certain sum of money for 3 years at 10% per annum is Rs 93. What is the value of the sum (principal)?

Introduction / Context:
Here we compare simple interest (SI) and compound interest (CI) on the same sum at the same rate for 3 years. The rate is 10% per annum with annual compounding. We are given the numerical difference between CI and SI and asked to recover the principal. For 3 years, there is a standard expression for the difference between CI and SI, which helps solve the problem quickly without separately computing full SI and CI values from trial and error.

Given Data / Assumptions:

• Principal (sum) = P rupees (unknown).
• Rate of interest r = 10% per annum.
• Time period t = 3 years.
• Interest for CI is compounded annually.
• Difference between CI and SI for 3 years is Rs 93.

Concept / Approach:
The simple interest for 3 years is SI = P * r * t / 100. The compound interest for 3 years is CI = P * [(1 + r/100)^3 - 1]. For 3 years, there is a known shortcut for the difference between CI and SI: Difference = P * [ (r^2 / 100^2) * (100/2) + (r^3 / 100^3) ], but this is less convenient in practice. Instead, we can directly write expressions for SI and CI, subtract, and simplify. Alternatively, we can find a simpler expression for the difference using the expansion of (1 + r/100)^3, which is efficient for standard exam rates like 10%.

Step-by-Step Solution:
Let P be the principal. Simple interest for 3 years at 10% per annum is SI = P * 10 * 3 / 100 = 0.30P. Amount with compound interest after 3 years is A = P * (1.10)^3 = P * 1.331. So CI = A - P = P * 1.331 - P = 0.331P. Difference between CI and SI is CI - SI = 0.331P - 0.30P = 0.031P. We are told this difference is Rs 93, so 0.031P = 93. Therefore, P = 93 / 0.031 = 3000. Hence, the required sum is Rs 3,000.

Verification / Alternative check:
We can check the answer by plugging back P = 3000. Simple interest: SI = 3000 * 10 * 3 / 100 = 900. Amount with CI: A = 3000 * 1.1^3 = 3000 * 1.331 = 3993. So CI = 3993 - 3000 = 993. Difference CI - SI = 993 - 900 = 93, matching the given difference exactly. This confirms that the derived principal is correct. Any other principal would change both SI and CI proportionally and would not preserve the exact difference of Rs 93.

Why Other Options Are Wrong:
If P were Rs 30,000, the difference would be ten times larger, Rs 930, not Rs 93. For Rs 3,030, the difference would be slightly above Rs 93. For Rs 5,000 or Rs 2,500, the difference would scale to different values and would not equal Rs 93. Only Rs 3,000 precisely satisfies the equation 0.031P = 93.

Common Pitfalls:
Some candidates mistakenly use the 2-year shortcut formula for a 3-year problem, which leads to an incorrect factor in the difference expression. Others compute CI and SI separately but make arithmetic mistakes or forget to subtract properly. Working systematically with the ratio 1.1^3 = 1.331 helps keep calculations clean and reduces the chance of errors. Always ensure you subtract SI from CI, not the other way around, since CI is larger for positive interest rates over more than one year.

Final Answer:
The principal (original sum) is Rs 3,000.