The compound interest earned on a certain sum of money at 15% per annum for 2 years is Rs. 20640. What is the original sum invested in rupees?

Introduction / Context:
This question focuses on reversing the compound interest process using the known value of the interest alone rather than the final amount. When compound interest and the rate and time are known, the candidate must correctly identify the growth factor and then isolate the principal using algebra. Such problems test the ability to move comfortably between principal, amount, and interest within the compound interest framework.

Given Data / Assumptions:

• Compound interest for 2 years = Rs. 20640.
• Rate of interest r = 15% per annum.
• Time period n = 2 years.
• Interest is compounded annually.
• We need to find the principal P.
• Relationship: Compound interest CI = A - P, and A = P * (1 + r) ^ n.

Concept / Approach:
First, we compute the overall growth factor for two years at 15% per year with annual compounding. That factor is (1.15) ^ 2. The amount A is then P multiplied by this factor. Since CI = A - P, we can express CI as P * ((1 + r) ^ n - 1). Here, CI is given, so we solve P = CI / ((1 + r) ^ n - 1). This formula directly converts the known compound interest into the principal by dividing out the net growth that applies to the principal over the given period.

Step-by-Step Solution:
Step 1: Convert the rate into decimal form: r = 15% = 0.15. Step 2: Compute the growth factor for 2 years: (1 + r) ^ n = (1.15) ^ 2. Step 3: Compute (1.15) ^ 2 = 1.3225. Step 4: For compound interest, CI = P * ((1 + r) ^ n - 1) = P * (1.3225 - 1) = P * 0.3225. Step 5: We are given CI = 20640. So 20640 = P * 0.3225. Step 6: Solve for the principal: P = 20640 / 0.3225. Step 7: Perform the division: 20640 / 0.3225 = 64000. Step 8: Hence, the original principal invested is Rs. 64000.

Verification / Alternative check:
To verify, we recompute the amount from P = 64000. The amount after 2 years is A = 64000 * (1.15) ^ 2 = 64000 * 1.3225 = 84640. The compound interest is then A - P = 84640 - 64000 = 20640, which matches the given interest perfectly. This confirms that our calculation for the principal is consistent and correct. Also, the value of 64000 roughly matches the scale one expects when the interest alone is over twenty thousand rupees at a high rate such as 15% per annum for two years.

Why Other Options Are Wrong:
If the principal were 60000, the compound interest would be 60000 * 0.3225 = 19350, which is smaller than the required 20640. A principal of 56000 would yield an even smaller interest of 56000 * 0.3225 = 18060. Similarly, a principal of 52000 would produce CI = 52000 * 0.3225 = 16770. A principal of 68000 would give CI = 68000 * 0.3225 = 21930, which is too large. Only 64000 generates the exact interest of 20640 at 15% per annum over two years.

Common Pitfalls:
Some students mistakenly use the simple interest formula when the question clearly states compound interest. Others may subtract 15% twice without compounding, which effectively converts the situation into a simple interest type calculation. Another common error is to forget that CI relates to the difference between the amount and principal and therefore to incorrectly divide the interest by the rate per year rather than by the combined growth factor minus one. Proper handling of percentage to decimal conversion and careful multiplication of 1.15 by itself are crucial to avoid arithmetic slips.

Final Answer:
The sum originally invested, which earns Rs. 20640 as compound interest in 2 years at 15% per annum, is Rs. 64000, corresponding to option A.