A person invested Rs. 20000 in a bank which is offering 10% per annum simple interest. After two years he withdrew the money from the bank and deposited the total amount in another bank which gives an interest rate of r% p.a. compounded annually. After 2 years he received an amount of Rs. 2460 more than what he had invested in that bank. What is the value of r?

Options

A. 0.1
B. 0.15
C. 0.05
D. 0.12

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Calculate the Simple Interest (SI)
The formula for Simple Interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(P\) = Principal amount (initial investment)
- \(R\) = Rate of interest per annum
- \(T\) = Time in years

Given:
- \(P = 20000\)
- \(R = 10\%\)
- \(T = 2\)

Substituting the values into the formula:
\[ SI = \frac{20000 \times 10 \times 2}{100} = \frac{400000}{100} = 4000 \]

### Step 2: Calculate the Total Amount after 2 Years
The total amount \(A\) after 2 years is given by:
\[ A = P + SI \]
Substituting the values:
\[ A = 20000 + 4000 = 24000 \]

### Step 3: Set Up the Compound Interest (CI) Equation
The person then deposits this amount in another bank that offers compound interest. The formula for Compound Interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^t \]
Where:
- \(A\) = Total amount after time \(t\)
- \(P\) = Principal amount (new deposit)
- \(r\) = Rate of interest per annum
- \(t\) = Time in years

Here:
- \(A = 24000\)
- \(P = 24000\)
- \(t = 2\)

### Step 4: Calculate the Interest Earned
According to the problem, the amount received after 2 years is Rs. 2460 more than what he had invested in the second bank:
\[ A = 24000 + 2460 = 26460 \]

### Step 5: Substitute into the CI Formula
Now we substitute into the CI formula:
\[ 26460 = 24000 \left(1 + \frac{r}{100}\right)^2 \]

### Step 6: Solve for \(r\)
First, divide both sides by 24000:
\[ \frac{26460}{24000} = \left(1 + \frac{r}{100}\right)^2 \]
Calculating the left side:
\[ \frac{26460}{24000} = 1.1025 \]

Now, we take the square root of both sides:
\[ 1 + \frac{r}{100} = \sqrt{1.1025} \]

Calculating the square root:
\[ \sqrt{1.1025} = 1.05 \]

Now, subtract 1 from both sides:
\[ \frac{r}{100} = 1.05 - 1 = 0.05 \]

Finally, multiply by 100 to find \(r\):
\[ r = 0.05 \times 100 = 5\% \]

### Conclusion
The value of \(r\) is \(5\%\).

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