Introduction / Context:
This problem compares simple interest and compound interest on the same principal, at the same rate, over the same time. The key is that compound interest generates interest on interest, while simple interest does not. The question gives the numerical difference between the two types of interest over 3 years and asks you to work backwards to find the principal amount.
Given Data / Assumptions:
- Difference between compound interest and simple interest for 3 years = Rs 186.
- Rate of interest r = 10% per annum.
- Time period n = 3 years.
- Principal P is unknown and needs to be determined.
- Interest is compounded annually for the compound interest calculation.
Concept / Approach:
For 3 years, the difference between compound interest (CI) and simple interest (SI) on principal P at rate r% is given by a known formula: Difference = P * r^2 * (n - 1) / 100^2 when time n is expressed in years and interest is compounded annually. For n = 3, this simplifies to P * r^2 * 2 / 100^2, or more directly, we can compute CI and SI expressions and subtract.
Step-by-Step Solution:
Step 1: Simple interest for 3 years: SI = P * r * n / 100 = P * 10 * 3 / 100 = 0.3P.
Step 2: Compound interest for 3 years: CI = P * (1 + r/100)^3 - P = P * (1.1^3 - 1).
Step 3: Compute 1.1^3 = 1.331, so CI = P * (1.331 - 1) = 0.331P.
Step 4: Difference CI - SI = 0.331P - 0.3P = 0.031P.
Step 5: Given that this difference equals Rs 186, write 0.031P = 186.
Step 6: Solve for P: P = 186 / 0.031 = 6000.
Verification / Alternative check:
Compute SI with P = 6000: SI = 6000 * 10 * 3 / 100 = 1800.
Compute CI: A = 6000 * (1.1)^3 = 6000 * 1.331 = 7986, so CI = 7986 - 6000 = 1986.
Difference = CI - SI = 1986 - 1800 = 186, which matches the given value.
Why Other Options Are Wrong:
Rs. 5500, Rs. 6500, and Rs. 7200 do not satisfy the equation 0.031P = 186, and they produce differences between CI and SI that are not equal to Rs 186.
Common Pitfalls:
Students sometimes confuse the formula for CI and SI or attempt to compute interest year by year without keeping track of the extra interest on interest, which leads to incorrect differences. Another error is forgetting to convert the rate correctly into decimal form when performing calculations.
Final Answer:
The required principal is Rs. 6000.