The difference between the compound interest and the simple interest on a certain sum of money for 3 years at 10% per annum is Rs. 186. Assuming annual compounding for compound interest, what is the principal (sum of money)?

Introduction / Context:
This question involves comparing simple interest (SI) and compound interest (CI) over 3 years at the same rate. Over multiple years, CI grows faster than SI, and the difference between them depends on both the rate and the time. By using the known difference between CI and SI, we can work backward to find the principal sum invested.

Given Data / Assumptions:

Difference CI − SI over 3 years = Rs. 186
Rate of interest r = 10% per annum
Time t = 3 years
Interest is compounded annually for CI
Principal P is unknown and must be found

Concept / Approach:
For annual compounding at rate r, the amount after 3 years is:
A = P * (1 + r/100)^3. Thus:
CI = A − P = P * [(1 + r/100)^3 − 1]. Simple interest for 3 years is:
SI = P * r * t / 100 = P * 3r / 100. The difference is:
CI − SI = P * [(1 + r/100)^3 − 1 − 3r/100]. We substitute r = 10% and the known difference to solve for P.

Step-by-Step Solution:
Step 1: Compute (1 + r/100)^3 for r = 10. (1.10)^3 = 1.331. Step 2: Calculate the bracket term. (1.10)^3 − 1 − 3r/100 = 1.331 − 1 − 0.30 = 1.331 − 1.30 = 0.031. Step 3: Write the difference formula. CI − SI = P * 0.031. Given CI − SI = 186, we have 186 = 0.031P. Step 4: Solve for P. P = 186 / 0.031 = 6000.

Verification / Alternative check:
Compute SI and CI explicitly for P = Rs. 6000 at 10% for 3 years. SI = 6000 * 10 * 3 / 100 = 1800. Amount under CI: A = 6000 * 1.331 = 7986. CI = A − P = 7986 − 6000 = 1986. Difference CI − SI = 1986 − 1800 = 186, matching the given value and confirming that the principal is correct.

Why Other Options Are Wrong:
For P = 5500, the difference would be 5500 * 0.031 = 170.5. For 6500, it would be 6500 * 0.031 = 201.5. For 7200, it would be 7200 * 0.031 = 223.2. None of these match 186, so those principals cannot be correct.

Common Pitfalls:
Students often confuse the formulas for the 2-year and 3-year differences between CI and SI, applying the simpler 2-year shortcut P * r^2 / 100^2 incorrectly to a 3-year problem. Another common mistake is to miscalculate (1.10)^3 or the term 3r/100. Being careful with these computations is essential to arriving at the correct principal.

Final Answer:
The principal sum of money is Rs. 6000.