₹X was invested in scheme A, which offers simple interest at 10 pcpa for 2 years. The amount received from scheme A was invested into scheme B, which offers compound interest (compounded annually) at 20 pcpa for 2 years. If the amount received from scheme B is ₹5356.80,what is the value of X?

Options

A. ₹ 3100
B. ₹ 2700
C. ₹ 2800
D. ₹ 3700

Answer

To find the value of ₹X that was initially invested in scheme A, we can follow these steps:

### Step 1: Calculate the Simple Interest from Scheme A
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(P\) = Principal amount (₹X)
- \(R\) = Rate of interest (10% per annum)
- \(T\) = Time (2 years)

Substituting the values:
\[ SI = \frac{X \times 10 \times 2}{100} = \frac{20X}{100} = \frac{X}{5} \]

### Step 2: Calculate the Total Amount Received from Scheme A
The total amount (A) received from scheme A after 2 years is:
\[ A = P + SI = X + \frac{X}{5} \]
To combine the terms:
\[ A = X + \frac{X}{5} = \frac{5X}{5} + \frac{X}{5} = \frac{6X}{5} \]

### Step 3: Invest the Amount in Scheme B
This amount \(\frac{6X}{5}\) is then invested in scheme B, which offers compound interest at 20% per annum for 2 years.

### Step 4: Calculate the Amount from Scheme B
The formula for Compound Interest (CI) is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \(P = \frac{6X}{5}\)
- \(R = 20\%\)
- \(T = 2\)

Substituting the values:
\[ A = \frac{6X}{5} \left(1 + \frac{20}{100}\right)^2 = \frac{6X}{5} \left(1 + 0.2\right)^2 = \frac{6X}{5} \left(1.2\right)^2 \]
Calculating \((1.2)^2\):
\[ (1.2)^2 = 1.44 \]
Thus,
\[ A = \frac{6X}{5} \times 1.44 = \frac{6 \times 1.44X}{5} = \frac{8.64X}{5} \]

### Step 5: Set the Amount Equal to ₹5356.80
We know from the problem that the amount received from scheme B is ₹5356.80, so:
\[ \frac{8.64X}{5} = 5356.80 \]

### Step 6: Solve for X
To find \(X\), we can multiply both sides by 5:
\[ 8.64X = 5356.80 \times 5 \]
Calculating the right side:
\[ 5356.80 \times 5 = 26784 \]
Now, we have:
\[ 8.64X = 26784 \]
Dividing both sides by 8.64:
\[ X = \frac{26784}{8.64} \]
Calculating the division:
\[ X = 3100 \]

### Final Answer
Thus, the value of \(X\) is ₹3100.

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