Introduction / Context:
This question asks you to determine the time required for a given principal to earn a specified amount of compound interest at a fixed annual rate. Instead of directly asking for the final amount, the problem gives the interest earned and expects you to find the number of years for which the sum has been invested at compound interest.
Given Data / Assumptions:
- Principal P = Rs 2000.
- Compound interest CI = Rs 662.
- Rate of interest r = 10% per annum.
- Interest is compounded annually.
- Number of years n is unknown.
Concept / Approach:
The amount A after n years at compound interest is A = P * (1 + r/100)^n. Since CI = A - P, we have A = P + CI. Using this relation, we can express (1 + r/100)^n as A / P and then solve for n by recognizing powers of 1.1 or by checking integer values systematically.
Step-by-Step Solution:
Step 1: Compute the total amount after n years:
A = P + CI = 2000 + 662 = Rs 2662.
Step 2: Write the compound interest formula:
A = 2000 * (1.1)^n.
Step 3: Set up the equation:
2000 * (1.1)^n = 2662.
Step 4: Divide both sides by 2000:
(1.1)^n = 2662 / 2000 = 1.331.
Step 5: Recognize that 1.1^3 = 1.1 * 1.1 * 1.1 = 1.331.
Step 6: Therefore n = 3 years.
Verification / Alternative check:
Compute the amount for 3 years: A = 2000 * 1.1^3 = 2000 * 1.331 = 2662.
Compute CI explicitly: CI = 2662 - 2000 = 662, which matches the given interest value.
Why Other Options Are Wrong:
For 2 years, factor is 1.1^2 = 1.21, giving A = 2420, CI = 420, which is too small.
For 4 or 5 years, the amount would be much larger than 2662, leading to interest amounts much greater than 662.
Common Pitfalls:
Learners may try to divide the interest evenly by the number of years, which corresponds to simple interest, not compound interest. Others may attempt to solve for n using logarithms without noticing that this is a standard power of 1.1 and can be recognized directly.
Final Answer:
The sum will yield Rs 662 as compound interest in 3 years.
Discussion & Comments