Difficulty: Medium
Correct Answer: 5%
Explanation:
Introduction / Context:
Here, the principal and the compound interest for a fixed period are given, and the rate of interest is unknown. This is a typical reverse compound interest problem. We must use the relationship between principal, rate, time, and compound interest to determine the annual interest rate at which Sonika invested her money.
Given Data / Assumptions:
Concept / Approach:
For compound interest compounded annually for 2 years, the amount A is:
A = P * (1 + r / 100)^2
The compound interest is:
CI = A - P = P * [(1 + r / 100)^2 - 1]
So:
CI / P = (1 + r / 100)^2 - 1
From this, we can solve for r by computing the ratio CI / P and then finding the square root.
Step-by-Step Solution:
Given P = 5800 and CI = 594.50.
Compute CI / P = 594.50 / 5800.
594.50 / 5800 = 0.1025.
So (1 + r / 100)^2 - 1 = 0.1025.
Therefore (1 + r / 100)^2 = 1.1025.
Take the square root: 1 + r / 100 = sqrt(1.1025) = 1.05.
So r / 100 = 1.05 - 1 = 0.05.
Therefore r = 5% per annum.
Verification / Alternative Check:
Check by forward calculation using r = 5%.
Amount factor for 2 years: (1.05)^2 = 1.1025.
Amount A = 5800 * 1.1025 = Rs 6394.50.
Compound interest = A - P = 6394.50 - 5800 = Rs 594.50, which matches the given figure.
Why Other Options Are Wrong:
6.5% and 6% would produce a higher effective factor than 1.1025 and hence a larger interest than Rs 594.50.
4.5% would give a lower factor and yield less interest than the given amount.
Common Pitfalls:
A common mistake is to assume simple interest and use CI = P * r * t / 100, which does not hold for compound interest.
Another error is incorrect division when computing CI / P or forgetting to take the square root to get 1 + r / 100.
Final Answer:
The correct rate of compound interest is 5% per annum.
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