The amount received on a certain principal at 10% per annum compound interest after 3 years is Rs 10,648. What was the principal amount (in rupees)?

Introduction / Context:
This is a standard reverse compound interest question where the final amount, rate, and time are known, and we must determine the original principal. The rate of interest is 10% per annum and the interest is compounded annually for 3 years. Such questions reinforce familiarity with the compound interest formula and the idea of reversing the growth process by dividing by the growth factor instead of multiplying.

Given Data / Assumptions:

• Amount after 3 years, A = Rs 10,648.
• Rate r = 10% per annum.
• Time t = 3 years.
• Interest is compounded annually.
• Principal P is unknown and needs to be found.

Concept / Approach:
Under compound interest with annual compounding, the amount after t years is given by A = P * (1 + r/100)^t. When A, r, and t are known, we rearrange to get P = A / (1 + r/100)^t. For common percentages like 10% and small integer time periods like 3 years, it is easy to compute powers like (1.10)^3 exactly. Here, we use the fact that (1.10)^3 = 1.331 to quickly find P and then check against the given amount.

Step-by-Step Solution:
Use the formula A = P * (1 + r/100)^t. We have A = 10,648, r = 10 and t = 3. So 10,648 = P * (1.10)^3. Compute (1.10)^3 = 1.331. Therefore, P = 10,648 / 1.331. By calculation, 8,000 * 1.331 = 10,648, so P = 8,000. Hence, the principal amount was Rs 8,000.

Verification / Alternative check:
We can verify by forward calculation. Starting from Rs 8,000 at 10% compound interest: After 1 year, amount = 8,000 * 1.10 = 8,800. After 2 years, amount = 8,800 * 1.10 = 9,680. After 3 years, amount = 9,680 * 1.10 = 10,648. This matches the given amount exactly. Any other principal from the options would not yield exactly 10,648 after 3 years at 10% per annum, confirming that Rs 8,000 is uniquely correct.

Why Other Options Are Wrong:
If P were Rs 7,500, the amount after 3 years would be 7,500 * 1.331 = 9,982.5, which is lower than 10,648. For Rs 9,000, the amount becomes 9,000 * 1.331 = 11,979, too high. For Rs 8,500, the amount is 8,500 * 1.331 = 11,313.5. For Rs 10,000, we get 13,310. None of these matches the given amount, so they cannot be correct.

Common Pitfalls:
Some learners attempt to use simple interest formulas, which will not produce the correct exponential growth. Others miscalculate (1.10)^3, often treating it as 1.30 instead of 1.331. It is important to remember that compounding multiplies the factor each year, not just adds the percentage linearly. Careful handling of the power and division step avoids these errors.

Final Answer:
The principal amount invested was Rs 8,000.