A certain bank offers 8% rate of interest in the first year and 9% in the second year on a fixed deposit scheme. If Rs. 17,658 is received after investing for 2 years in this scheme, what was the initial amount (in rupees) invested?

Introduction / Context:
In this question, a fixed deposit earns interest at different rates in different years. Specifically, the bank offers 8% in the first year and 9% in the second year. This is essentially compound interest with varying rates each year. The final amount after 2 years is given, and we must determine the initial investment. This type of question reinforces understanding of compound growth with non-uniform rates and shows how to reverse the process to recover the principal from the final amount.

Given Data / Assumptions:

• Let the initial principal be P rupees.
• Rate in the first year R1 = 8% per annum.
• Rate in the second year R2 = 9% per annum.
• Final amount after 2 years A = Rs. 17,658.
• Interest is applied yearly and compounded year by year based on the current balance.

Concept / Approach:
When interest rates differ each year, we multiply the principal successively by (1 + R1/100) in the first year and then by (1 + R2/100) in the second year. Therefore, the amount after 2 years is A = P * (1 + R1/100) * (1 + R2/100). Here, that becomes A = P * 1.08 * 1.09. We know A and must solve for P by dividing the final amount by the product of the growth factors. This approach is an extension of the standard compound interest formula for variable rates.

Step-by-Step Solution:
Step 1: Let the initial principal be P. Step 2: After the first year at 8%, the amount becomes P1 = P * (1 + 8/100) = P * 1.08. Step 3: In the second year, 9% is applied on P1, giving final amount A = P1 * (1 + 9/100) = P * 1.08 * 1.09. Step 4: So A = P * 1.08 * 1.09 = P * 1.1772. Step 5: It is given that A = 17658. Step 6: Therefore, 17658 = P * 1.1772. Step 7: Solve for P: P = 17658 / 1.1772. Step 8: Perform the division: P = 15000. Step 9: Hence, the initial investment was Rs. 15000.

Verification / Alternative check:
Check by forward calculation. Starting with P = 15000, after the first year at 8%: P1 = 15000 * 1.08 = 16200. After the second year at 9%: A = 16200 * 1.09 = 17658. This matches the given final amount exactly, confirming that the initial principal is Rs. 15000.

Why Other Options Are Wrong:
If the principal were 16000, the final amount would be 16000 * 1.08 * 1.09, which is higher than 17658. Similarly, 15500 or 16500 would produce final amounts that do not match 17658 when multiplied by 1.08 and 1.09. Only 15000, when grown by 8% and then 9%, gives exactly Rs. 17658.

Common Pitfalls:
A common mistake is to treat the two-year interest as simple and just apply an average rate, which is incorrect because the second year's interest is calculated on a larger base. Another error is to incorrectly multiply the growth factors or forget to convert percentages into decimals. Writing the expression A = P * 1.08 * 1.09 clearly and solving systematically helps avoid such confusion.

Final Answer:
The amount initially invested in the fixed deposit is Rs. 15000.