Question

A man lends some money to his friend at 5% per annum of interest rate. After 2 years, the difference between the Simple and the compound interest on money is Rs. 50. What will be the value of the amount at the end of 3 years if compounded annually?

• A. 21325.6
• B. 24512.5
• C. 22252.7
• D. 23152.5

Answer 1

The Correct Option is D

Let's solve the problem step by step.
Given:
- Interest rate \( r = 5\% \) per annum
- The difference between Simple Interest (SI) and Compound Interest (CI) after 2 years is Rs. 50
First, let's denote the principal amount as \( P \).
Simple Interest (SI) for 2 years:
\[ SI = P \times \frac{r}{100} \times t \]
For 2 years, \( t = 2 \):
\[ SI = P \times \frac{5}{100} \times 2 = 0.1P \]
Compound Interest (CI) for 2 years:
\[ CI = P \left(1 + \frac{r}{100}\right)^t - P \]
For 2 years, \( t = 2 \):
\[ CI = P \left(1 + \frac{5}{100}\right)^2 - P = P \left(1.05^2\right) - P = P \left(1.1025\right) - P = 0.1025P \]
Given the difference between CI and SI after 2 years is Rs. 50:
\[ CI - SI = 0.1025P - 0.1P = 0.0025P = 50 \]
\[ P = \frac{50}{0.0025} = 20000 \]
Now, we need to find the amount at the end of 3 years when compounded annually:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
For 3 years, \( t = 3 \):
\[ A = 20000 \left(1 + \frac{5}{100}\right)^3 = 20000 \left(1.05\right)^3 \]
Calculating \( 1.05^3 \):
\[ 1.05^3 = 1.157625 \]
\[ A = 20000 \times 1.157625 = 23152.5 \]
So, the value of the amount at the end of 3 years if compounded annually is:
Answer: D (23152.5)

Answer 2

Instead of solving for the principal via the algebraic difference-of-interest formula, use the shortcut fact that for a rate \(r\) over 2 years, the CI-SI difference equals \(P r^{2}\) (with \(r\) as a decimal).

\[P \times (0.05)^{2} = 50 \implies P \times 0.0025 = 50 \implies P = \frac{50}{0.0025} = 20000\]

Now the amount after 3 years at 5% compounded annually:
\[A = 20000(1.05)^{3}\]
Compute \((1.05)^3\) by first squaring: \(1.05^{2}=1.1025\), then multiplying by 1.05 again: \(1.1025\times1.05=1.157625\).
\[A = 20000\times1.157625 = 23152.5\]

1. Option A (21325.6): Does not match.
2. Option B (24512.5): Does not match.
3. Option C (22252.7): Does not match.
4. Option D (23152.5): Matches exactly.

The amount at the end of 3 years, compounded annually, is Rs. 23152.5.

Hence, the correct answer is Option D: 23152.5.