Introduction / Context:
Here you are asked to calculate the compound interest on a fixed principal over multiple years at a given annual rate. This is a standard compound interest problem, and it checks whether you know how to use the formula for the amount and then separate the interest portion from the total amount.
Given Data / Assumptions:
- Principal P = Rs 4000.
- Rate of interest r = 10% per annum.
- Time period n = 4 years.
- Interest is compounded annually.
- We must find the compound interest earned in 4 years.
Concept / Approach:
The amount A after n years at compound interest is given by A = P * (1 + r/100)^n. Once we find the total amount, we subtract the principal to obtain the compound interest. This separates the profit (interest) from the initial investment.
Step-by-Step Solution:
Step 1: Use the formula A = P * (1 + r/100)^n.
Step 2: Substitute the values: P = 4000, r = 10, n = 4.
Step 3: Compute the factor: 1 + r/100 = 1 + 10/100 = 1.10.
Step 4: Raise this factor to the 4th power: 1.10^4 = 1.1 * 1.1 * 1.1 * 1.1 = 1.4641.
Step 5: Calculate the final amount: A = 4000 * 1.4641 = Rs 5856.40.
Step 6: Find the compound interest: CI = A - P = 5856.40 - 4000 = Rs 1856.40.
Verification / Alternative check:
You can verify by computing interest year by year: after 1st year interest = 400, new amount 4400; 2nd year interest = 440, amount 4840; 3rd year interest = 484, amount 5324; 4th year interest = 532.40, amount 5856.40. Adding yearly interests 400 + 440 + 484 + 532.40 = 1856.40, which matches the earlier calculation.
Why Other Options Are Wrong:
Rs. 1600 corresponds to simple interest for 4 years at 10%, not compound interest.
Rs. 1856 is only an approximation and ignores the fractional rupee, while the exact compound interest is Rs. 1856.40.
Rs. 1756.60 does not match any consistent interest calculation at 10% for 4 years.
Common Pitfalls:
Students often wrongly use the simple interest formula P * r * n / 100 instead of the compound interest formula. Another common mistake is to round intermediate results too early, which can lead to a slightly incorrect final amount.
Final Answer:
The compound interest earned in 4 years is Rs. 1856.40.
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