A sum amounts to Rs 7727.104 at 12 percent per annum, compounded annually, after 3 years. What is the principal amount in rupees?

Introduction / Context:
This question focuses on reversing the compound interest process: instead of finding the amount from the principal, rate, and time, we are given the final amount and must determine the original principal. The rate and time are known, and the compounding is annual. This is a standard type of compound interest question that checks algebraic manipulation and understanding of the compound amount formula.

Given Data / Assumptions:
- Final amount A after 3 years = Rs 7727.104.
- Rate of interest R = 12 percent per annum, compounded annually.
- Time T = 3 years.
- Principal P is unknown and needs to be calculated.

Concept / Approach:
The compound amount formula for annual compounding is:
A = P * (1 + R/100)^TSince A, R, and T are known, we can rearrange this formula to find the principal P:
P = A / (1 + R/100)^TWe simply compute the compound factor for 3 years at 12 percent and divide the amount by this factor.

Step-by-Step Solution:
Given A = 7727.104, R = 12 %, T = 3 years.Compute the growth factor: 1 + R/100 = 1 + 0.12 = 1.12.Now compute (1.12)^3.(1.12)^2 = 1.2544 and (1.12)^3 = 1.2544 * 1.12 = 1.404928.Use the formula P = A / (1.12)^3.P = 7727.104 / 1.404928.Carrying out the division gives P = 5500.Therefore, the principal amount is Rs 5500.

Verification / Alternative check:
We can verify by recomputing the amount from P = 5500. With R = 12 %, T = 3 years, the amount A should be 5500 * (1.12)^3. As above, (1.12)^3 = 1.404928. Then A = 5500 * 1.404928 = 7727.104, which matches the amount given in the question. This confirms that the principal we found is correct.

Why Other Options Are Wrong:
If P were Rs 5000, the amount would be 5000 * (1.12)^3, which is less than 7727.104. Similarly, principals like Rs 5200 and Rs 5350 would yield amounts smaller than required. A principal of Rs 6000 would give an amount greater than 7727.104. Only Rs 5500 produces the exact final amount when compounded at 12 percent per annum for 3 years.

Common Pitfalls:
Candidates sometimes use the simple interest formula instead of the compound interest formula, which leads to a different and incorrect principal. Others make arithmetic mistakes when computing (1.12)^3 or when dividing. It is important to remember that for compound interest, the amount grows multiplicatively by the same factor each year. Using the correct factor and careful calculation avoids errors.

Final Answer:
The principal amount is Rs 5500.