Ram invests a certain sum in Scheme A offering simple interest @`5%` pa for 4 years. He further invests the amount obtained from Scheme A into Scheme B offering compound interest @`10%` pa (compounded annually) . for 2 years. If the interest obtained from Scheme B was ₹378, then what was the sum invested in Scheme A? (in₹)

Options

A. 1200
B. 1000
C. 1600
D. 1500

Answer

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

### Step 1: Define the initial investment
Let the sum invested in Scheme A be \( x \) (in ₹).

### Step 2: Calculate the amount obtained from Scheme A
Scheme A offers simple interest at a rate of 5% per annum for 4 years. The formula for calculating simple interest is:

\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount (initial investment)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

For Scheme A:
- \( P = x \)
- \( R = 5\% \)
- \( T = 4 \)

Calculating the simple interest:

\[ \text{Simple Interest} = \frac{x \times 5 \times 4}{100} = \frac{20x}{100} = \frac{x}{5} \]

Now, the total amount obtained from Scheme A after 4 years is:

\[ \text{Total Amount from Scheme A} = P + \text{Simple Interest} = x + \frac{x}{5} = \frac{5x}{5} + \frac{x}{5} = \frac{6x}{5} \]

### Step 3: Invest the amount in Scheme B
The amount obtained from Scheme A, \( \frac{6x}{5} \), is then invested in Scheme B, which offers compound interest at a rate of 10% per annum for 2 years.

### Step 4: Calculate the compound interest from Scheme B
The formula for compound interest is:

\[ A = P \left(1 + \frac{R}{100}\right)^T \]

Where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

For Scheme B:
- \( P = \frac{6x}{5} \)
- \( R = 10\% \)
- \( T = 2 \)

Calculating the amount after 2 years:

\[ A = \frac{6x}{5} \left(1 + \frac{10}{100}\right)^2 = \frac{6x}{5} \left(1 + 0.1\right)^2 = \frac{6x}{5} \left(1.1\right)^2 = \frac{6x}{5} \times 1.21 = \frac{6.726x}{5} \]

### Step 5: Calculate the compound interest earned
The compound interest (CI) earned from Scheme B is given as ₹378. The CI can also be calculated as:

\[ \text{CI} = A - P \]

Where \( A \) is the amount after 2 years and \( P \) is the principal amount. Therefore:

\[ \text{CI} = \frac{6.726x}{5} - \frac{6x}{5} = \frac{6.726x - 6x}{5} = \frac{0.726x}{5} \]

Setting this equal to ₹378:

\[ \frac{0.726x}{5} = 378 \]

### Step 6: Solve for \( x \)
Multiplying both sides by 5:

\[ 0.726x = 1890 \]

Now, divide both sides by 0.726:

\[ x = \frac{1890}{0.726} \approx 2600 \]

### Conclusion
The sum invested in Scheme A is approximately ₹2600.