A sum of `Rs. 20000` becomes `Rs. 32000` in 12 years, when invested in a scheme of simple interest. If the same sum is invested in a scheme of compound Interest with same yearly interest rate (compounded annually), what will be the amount (in Rs.) after 2 years?

Options

A. ` 21750`
B. `22050 `
C. ` 23250 `
D. `24650`

Answer

To solve the problem step by step, we will first determine the rate of interest from the simple interest scenario and then use that rate to calculate the compound interest for the given period.

### Step 1: Calculate the Simple Interest
The formula for simple interest (SI) is:
\[ SI = \text{Principal} \times \frac{\text{Rate}}{100} \times \text{Time} \]
Here, the principal (P) is Rs. 20,000, and the amount (A) after 12 years is Rs. 32,000.

First, we find the total simple interest earned over 12 years:
\[ SI = A - P = 32000 - 20000 = 12000 \]

### Step 2: Find the Rate of Interest
Now, we can use the simple interest formula to find the rate (R):
\[ 12000 = 20000 \times \frac{R}{100} \times 12 \]
Rearranging the equation to solve for R:
\[ R = \frac{12000 \times 100}{20000 \times 12} \]
Calculating R:
\[ R = \frac{1200000}{240000} = 5\% \]

### Step 3: Calculate the Amount with Compound Interest
Now that we have the rate of interest (5%), we can calculate the amount after 2 years using the compound interest formula:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where:
- \(P = 20000\)
- \(R = 5\)
- \(n = 2\)

Substituting the values:
\[ A = 20000 \left(1 + \frac{5}{100}\right)^2 \]
\[ A = 20000 \left(1 + 0.05\right)^2 \]
\[ A = 20000 \left(1.05\right)^2 \]
Calculating \(1.05^2\):
\[ 1.05^2 = 1.1025 \]
Now substitute back:
\[ A = 20000 \times 1.1025 = 22050 \]

### Final Answer
The amount after 2 years when invested in a scheme of compound interest is **Rs. 22050**.

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