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, that is, the original principal invested?

Introduction / Context:
This question involves compound interest with different rates in each year. Ganesh invests an amount x, which earns 5 percent in the first year and 15 percent in the second year, with annual compounding. The final amount after 2 years is given, and we must find the original principal x. This is a practical type of banking and finance question that tests your understanding of compounding with changing yearly rates.

Given Data / Assumptions:

• Initial principal = Rs. x.
• Rate in first year r₁ = 5 percent.
• Rate in second year r₂ = 15 percent.
• Interest is compounded annually, with the rate changing year by year.
• Final amount after 2 years A = Rs. 9660.

Concept / Approach:
With annual compounding and different rates each year, the amount after 2 years is obtained by multiplying the principal by successive growth factors. After the first year, the amount becomes x * (1 + 5/100). After the second year, 15 percent is applied on this new amount. Therefore, the overall growth factor across 2 years is (1 + 5/100) * (1 + 15/100) = 1.05 * 1.15. Setting x * 1.05 * 1.15 equal to 9660 allows us to solve for x by division.

Step-by-Step Solution:
Step 1: After the first year, amount A₁ = x * (1 + 5/100) = x * 1.05. Step 2: After the second year, interest at 15 percent is applied on A₁, giving A₂ = A₁ * (1 + 15/100) = x * 1.05 * 1.15. Step 3: We know that A₂ = 9660. Step 4: So x * 1.05 * 1.15 = 9660. Step 5: Compute the combined factor: 1.05 * 1.15 = 1.2075. Step 6: Therefore, x * 1.2075 = 9660. Step 7: Solve for x: x = 9660 / 1.2075. Step 8: Perform the division to get x = 8000. Step 9: Hence, the original principal invested by Ganesh is Rs. 8000.

Verification / Alternative check:
We can verify by directly recomputing the amount starting from x = 8000. After 1 year at 5 percent: A₁ = 8000 * 1.05 = 8400. After 2 years at 15 percent on A₁: A₂ = 8400 * 1.15 = 9660. This matches the given final amount, confirming that the principal must have been Rs. 8000.

Why Other Options Are Wrong:
For Rs. 7500, the final amount would be 7500 * 1.2075 = 9056.25, which is less than 9660. For Rs. 8200, the final amount would be 8200 * 1.2075 ≈ 9901.5, larger than 9660. Similarly, Rs. 8500 and Rs. 9000 give final amounts that do not match 9660. Only Rs. 8000, when multiplied by the combined factor 1.2075, reproduces the given final amount exactly.

Common Pitfalls:
Some students mistakenly average the two rates (5 percent and 15 percent) to 10 percent per year and then treat the situation as a 2 year investment at 10 percent compound interest, which is incorrect. Others apply 5 percent and 15 percent as simple interest instead of compounding. Forgetting to multiply the factors sequentially (1.05 and 1.15) is another frequent error. Working carefully with the compound growth factors avoids these mistakes.

Final Answer:
The original principal that Ganesh invested in the fixed deposit scheme is Rs. 8000.