A bank offers a fixed deposit scheme with 8% interest in the first year and 9% interest in the second year (compounded annually). If ₹17,658 is received at the end of 2 years, what was the initial amount (principal, in rupees) invested?

Introduction:
This problem tests reverse calculation of principal when the final amount is given under year-wise compounding with different rates each year. The essential idea is that compounding multiplies the principal by (1 + rate/100) each year. With two different annual rates, the final amount equals P * (1 + 8/100) * (1 + 9/100). So the principal is found by dividing the final amount by this combined growth factor.

Given Data / Assumptions:

• Amount received after 2 years = ₹17,658
• Year 1 rate = 8% (annual compounding)
• Year 2 rate = 9% (annual compounding)
• Principal = P (unknown)
• Compounding rule: Amount after year = previous amount * (1 + r/100)

Concept / Approach:
Compute the two-year multiplication factor: (1.08) for the first year and (1.09) for the second year. Multiply them to get the net factor. Then P = final amount / net factor. This is the cleanest method because rates vary by year.

Step-by-Step Solution:
After year 1: amount = P * 1.08 After year 2: amount = (P * 1.08) * 1.09 Final amount = P * (1.08 * 1.09) 1.08 * 1.09 = 1.1772 17658 = P * 1.1772 P = 17658 / 1.1772 = 15000

Verification / Alternative check:
If P = 15000, after year 1 amount = 15000*1.08 = 16200. After year 2 amount = 16200*1.09 = 17658, matching the given amount exactly.

Why Other Options Are Wrong:
Any principal other than ₹15,000 would produce a final amount different from ₹17,658 because the growth factor 1.1772 is fixed by the rates and compounding pattern.

Common Pitfalls:
Adding rates (8%+9%) and treating it like simple interest, or forgetting that the second year’s interest is earned on the first year’s grown amount (compounding effect).

Final Answer:
The initial investment (principal) was ₹15,000.