Ganesh invests an amount of Rs x in a fixed deposit scheme that offers 5 percent per annum for the first year and 15 percent per annum for the second year, with interest compounded annually. If he receives Rs 9660 after two years, what is the value of x in rupees?

Introduction / Context:
This question involves compound interest with different rates in different years. In the first year, the investment grows at 5 percent per annum, and in the second year it grows at 15 percent per annum. The interests are compounded annually, so the amount at the end of the first year becomes the principal for the second year. The final amount is given, and we must backtrack to find the original principal x.

Given Data / Assumptions:
- Initial principal = Rs x.
- Rate for first year R1 = 5 percent per annum.
- Rate for second year R2 = 15 percent per annum.
- Interest is compounded annually.
- Amount after 2 years, A = Rs 9660.
- We must determine x.

Concept / Approach:
Under compound interest with different rates each year, we apply the appropriate factor for each year successively. After the first year, the amount is x * (1 + R1/100). After the second year, we multiply by another factor (1 + R2/100). Thus, the total amount after 2 years is:
A = x * (1 + R1/100) * (1 + R2/100)We substitute the known values for A, R1, and R2 and solve for x.

Step-by-Step Solution:
Given R1 = 5 %, so 1 + R1/100 = 1 + 0.05 = 1.05.Given R2 = 15 %, so 1 + R2/100 = 1 + 0.15 = 1.15.Thus, after 2 years, A = x * 1.05 * 1.15.Compute the combined factor: 1.05 * 1.15 = 1.2075.Given A = 9660, so 9660 = x * 1.2075.Solve for x: x = 9660 / 1.2075.Compute x: 9660 / 1.2075 = 8000.Therefore, the value of x is Rs 8000.

Verification / Alternative check:
We verify by forward calculation. Start with x = 8000. After first year at 5 percent: amount A1 = 8000 * 1.05 = 8400. After second year at 15 percent: amount A2 = 8400 * 1.15 = 9660. This matches the final amount stated in the question, confirming that x = 8000 is correct.

Why Other Options Are Wrong:
If x were Rs 9000, then after 2 years the amount would be 9000 * 1.2075 = 10867.5, which is larger than 9660. If x were Rs 8500, the final amount would be 8500 * 1.2075 = 10263.75. For Rs 8200, the amount is 8200 * 1.2075 = 9901.5. For Rs 7500, the amount is 9056.25. None of these values equals 9660. Only Rs 8000 produces the exact amount given.

Common Pitfalls:
Some students mistakenly average the two rates (5 percent and 15 percent) to get 10 percent per annum and then apply a constant rate, which is wrong because compound interest with changing rates must be handled year by year. Another error is to treat the problem as simple interest, which ignores the effect of interest on the increased amount in the second year. Applying the compound interest factors separately for each year ensures accurate results.

Final Answer:
The value of x, the amount Ganesh invested initially, is Rs 8000.