A person invested two equal amounts in two different schemes. In first scheme,amount is invested at `8%` per annum on SI for T years SI received is Rs2000 while in second scheme,amount is invested at `10%` per annum for 2 years at CI and the compound interest received is Rs1050. Find the value of T.

Options

A. 4 years
B. 8 years
C. 6 years
D. 5 years

Answer

To solve the problem step by step, we will break down the information given and apply the relevant formulas for both Simple Interest (SI) and Compound Interest (CI).

### Step 1: Define the Variables
Let the equal amount invested in both schemes be \( X \).

### Step 2: Calculate the Simple Interest for the First Scheme
In the first scheme:
- Rate = 8% per annum
- Simple Interest (SI) = Rs 2000
- Time = \( T \) years

Using the formula for Simple Interest:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Substituting the known values:
\[ 2000 = \frac{X \times 8 \times T}{100} \]

### Step 3: Rearranging the Equation for SI
Rearranging the equation to find \( T \):
\[ 2000 = \frac{8XT}{100} \]
Multiplying both sides by 100:
\[ 200000 = 8XT \]
Dividing both sides by \( 8X \):
\[ T = \frac{200000}{8X} = \frac{25000}{X} \]

### Step 4: Calculate the Compound Interest for the Second Scheme
In the second scheme:
- Rate = 10% per annum
- Time = 2 years
- Compound Interest (CI) = Rs 1050

Using the formula for Compound Interest:
\[ \text{CI} = P \left(1 + \frac{R}{100}\right)^T - P \]
Substituting the known values:
\[ 1050 = X \left(1 + \frac{10}{100}\right)^2 - X \]

### Step 5: Simplifying the CI Equation
Calculating \( \left(1 + \frac{10}{100}\right)^2 \):
\[ \left(1 + 0.1\right)^2 = 1.1^2 = 1.21 \]
Now substituting back into the equation:
\[ 1050 = X \cdot 1.21 - X \]
This simplifies to:
\[ 1050 = 1.21X - X \]
\[ 1050 = 0.21X \]

### Step 6: Solving for \( X \)
Dividing both sides by 0.21:
\[ X = \frac{1050}{0.21} = 5000 \]

### Step 7: Substitute \( X \) back to find \( T \)
Now that we have \( X = 5000 \), we can substitute this value back into the equation for \( T \):
\[ T = \frac{25000}{5000} = 5 \]

### Conclusion
The value of \( T \) is 5 years.