A sum amounts to Rs. 7727.104 at 12 percent per annum, compounded annually, after 3 years. For this compound interest investment, what is the original principal amount in rupees that was invested?

Introduction / Context:
This question asks you to work backwards in a compound interest scenario. Instead of being given the principal and asked to find the amount, you are given the final amount after 3 years at a known compound rate and must find the original principal. Reverse calculations like this are very common in banking and finance aptitude questions and rely on inverting the compound interest growth factor.

Given Data / Assumptions:

• Final amount after 3 years, A = Rs. 7727.104.
• Annual rate of compound interest r = 12 percent.
• Time t = 3 years.
• Interest is compounded annually.
• Principal P is unknown and must be found.

Concept / Approach:
The compound interest amount formula is:
A = P * (1 + r/100)^t.
Here, r = 12 and t = 3, so the growth factor is (1.12)^3. To find P, we rearrange the formula to P = A / (1 + r/100)^t. Once we compute (1.12)^3 and divide the final amount by this factor, we obtain the principal. We then compare the result with the given options.

Step-by-Step Solution:
Step 1: Write the compound amount formula: A = P * (1 + r/100)^t. Step 2: Substitute the given values: 7727.104 = P * (1 + 12/100)^3 = P * (1.12)^3. Step 3: Compute (1.12)^2 = 1.2544, then multiply by 1.12 again: (1.12)^3 = 1.404928. Step 4: Rearrange to solve for P: P = 7727.104 / 1.404928. Step 5: Perform the division: P ≈ 5500. Step 6: Therefore, the original principal invested is Rs. 5500. Step 7: We match this to the nearest option, which is exactly Rs. 5500.

Verification / Alternative check:
To verify, start with P = 5500 and compute the amount after 3 years at 12 percent with annual compounding. After 1 year: A₁ = 5500 * 1.12 = 6160. After 2 years: A₂ = 6160 * 1.12 = 6900.8. After 3 years: A₃ = 6900.8 * 1.12 = 7727.104, which matches the given amount exactly. This confirms that the principal must be Rs. 5500 and validates our reversed calculation.

Why Other Options Are Wrong:
Rs. 5000, Rs. 5200, Rs. 5350, and Rs. 6000, when grown at 12 percent compound interest for 3 years, do not reach 7727.104. For example, 5000 * (1.12)^3 is significantly less than 7727.104, while 6000 * (1.12)^3 is noticeably higher. Only 5500 reproduces the exact final amount. Thus, the other options represent incorrect guesses of the principal or rounding errors in reversing the growth factor.

Common Pitfalls:
Some students wrongly use the simple interest formula and set 7727.104 = P * (1 + r * t / 100), which leads to the wrong principal because the problem clearly mentions compounding. Others approximate (1.12)^3 too crudely or treat 12 percent over 3 years as 36 percent simple growth, which again is inappropriate. Precisely computing or carefully estimating the compound factor is important to arrive at the correct answer.

Final Answer:
The original principal that grows to Rs. 7727.104 at 12 percent per annum compounded annually over 3 years is Rs. 5500.