A certain sum is invested for T years. It amounts to ₹ 400 at 10 % per annum. But when invested at 4% per annum. it amounts to ₹200. Find the time(T)?

Options

A. 39 years
B. 41years
C. 45 years
D. 50 years

Explanation

To solve the problem, we need to find the time \( T \) for which a certain sum is invested, given two different interest rates and amounts. We will use the formula for simple interest and set up equations based on the information provided.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have two scenarios:
- When invested at 10% per annum, the amount becomes ₹400.
- When invested at 4% per annum, the amount becomes ₹200.
- We need to find the time \( T \) in years.

2. **Setting Up the Equations**:
- Let \( P \) be the principal amount (the initial sum invested).
- For the first scenario (10% per annum):
\[ \text{Amount} = P + \text{Interest} \]
\[ 400 = P + \left(\frac{P \times 10 \times T}{100}\right) \]
Rearranging gives:
\[ 400 = P + \frac{PT}{10} \]
\[ 400 = P \left(1 + \frac{T}{10}\right) \tag{1} \]

- For the second scenario (4% per annum):
\[ 200 = P + \left(\frac{P \times 4 \times T}{100}\right) \]
Rearranging gives:
\[ 200 = P + \frac{PT}{25} \]
\[ 200 = P \left(1 + \frac{T}{25}\right) \tag{2} \]

3. **Equating the Two Equations**:
- From equations (1) and (2), we can set them equal to each other:
\[ P \left(1 + \frac{T}{10}\right) = 400 \]
\[ P \left(1 + \frac{T}{25}\right) = 200 \]

4. **Eliminating \( P \)**:
- From equation (1):
\[ P = \frac{400}{1 + \frac{T}{10}} \tag{3} \]
- Substitute \( P \) from (3) into (2):
\[ \frac{400}{1 + \frac{T}{10}} \left(1 + \frac{T}{25}\right) = 200 \]

5. **Cross-Multiplying and Simplifying**:
- Cross-multiplying gives:
\[ 400 \left(1 + \frac{T}{25}\right) = 200 \left(1 + \frac{T}{10}\right) \]
- Expanding both sides:
\[ 400 + \frac{400T}{25} = 200 + \frac{200T}{10} \]
\[ 400 + 16T = 200 + 20T \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ 400 - 200 = 20T - 16T \]
\[ 200 = 4T \]
\[ T = \frac{200}{4} = 50 \]

7. **Final Answer**:
- The time \( T \) is **50 years**.