The compound interest earned in 2 years at 15% per annum on a certain sum is Rs. 20,640. What is the principal sum invested (in rupees)?

Introduction / Context:
Here, the total compound interest over a two-year period is given, rather than the final amount. You are asked to work backward to obtain the original principal. This requires understanding the relationship between amount, principal, and compound interest at a known rate and time period.

Given Data / Assumptions:

• Compound interest CI for 2 years = Rs. 20,640
• Rate r = 15% per annum
• Time T = 2 years
• Compounding is annual
• Principal P is unknown

Concept / Approach:
For compound interest at rate r for 2 years, the amount A is:
A = P * (1 + r / 100)^2and the compound interest is:
CI = A - PWe can write:
CI = P * [(1 + r / 100)^2 - 1]Given CI, we solve this equation for P.

Step-by-Step Solution:
Step 1: Compute the 2-year factor at 15%.1 + r / 100 = 1 + 15 / 100 = 1.15(1.15)^2 = 1.3225Step 2: Express CI in terms of P.CI = P * (1.3225 - 1) = P * 0.3225Step 3: Use the given CI to solve for P.20640 = P * 0.3225P = 20640 / 0.3225P = Rs. 64,000

Verification / Alternative check:
Verify by directly computing amount and interest from P = 64,000. Amount after 2 years:
A = 64000 * (1.15)^2 = 64000 * 1.3225 = Rs. 84,640CI = A - P = 84640 - 64000 = Rs. 20,640This matches the given compound interest, confirming the calculation.

Why Other Options Are Wrong:
Rs. 60,000, Rs. 56,000, Rs. 52,000, Rs. 48,000: Substituting any of these values into the formula A = P * 1.3225 produces compound interest different from 20,640. They therefore cannot be the correct principal.

Common Pitfalls:
Some students mistakenly treat 20,640 as the amount or use simple interest formulas. Others fail to compute the squared factor correctly or forget to subtract 1 when forming the CI expression P * [(1 + r / 100)^2 - 1]. Careful algebra and attention to the formula structure prevent such errors.

Final Answer:
The principal sum invested was Rs. 64,000.