A man invested a certain sum in scheme A at 20% p.a. for 3 years and earned ₹4800 as simple interest . He increased his sum by ₹ 'x' and invested in another scheme B at 20% p.a C.I for 2 year and received ₹4400 as compound interest. Find the value of 'x'?

Options

A. ₹1,750
B. ₹2000
C. ₹1250
D. None of these

Explanation

To solve the problem, we need to find the value of 'x' based on the information provided about the investments in scheme A and scheme B.

### Step 1: Calculate the Principal Amount for Scheme A
We know that the formula for Simple Interest (SI) is given by:

\[ SI = \frac{P \times R \times T}{100} \]

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

From the question, we have:
- \(SI = 4800\)
- \(R = 20\%\)
- \(T = 3\) years

Substituting these values into the formula, we get:

\[ 4800 = \frac{P \times 20 \times 3}{100} \]

### Step 2: Simplify the Equation
Now, simplify the equation:

\[ 4800 = \frac{60P}{100} \]

This simplifies to:

\[ 4800 = 0.6P \]

### Step 3: Solve for P
Now, to find \(P\), we can rearrange the equation:

\[ P = \frac{4800}{0.6} \]

Calculating this gives:

\[ P = 8000 \]

So, the principal amount invested in scheme A is ₹8000.

### Step 4: Calculate the Total Amount Invested in Scheme B
The man increases his sum by ₹x and invests it in scheme B. Therefore, the total amount invested in scheme B is:

\[ 8000 + x \]

### Step 5: Calculate the Compound Interest for Scheme B
The formula for Compound Interest (CI) is:

\[ CI = A - P \]

Where \(A\) is the total amount after interest, and \(P\) is the principal amount.

The formula for the amount \(A\) in compound interest is:

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

Substituting the values for scheme B:

- \(P = 8000 + x\)
- \(R = 20\%\)
- \(T = 2\) years

So we have:

\[ A = (8000 + x) \left(1 + \frac{20}{100}\right)^2 \]

This simplifies to:

\[ A = (8000 + x) \left(1.2\right)^2 \]

Calculating \(1.2^2\):

\[ 1.2^2 = 1.44 \]

Thus,

\[ A = (8000 + x) \times 1.44 \]

### Step 6: Calculate the Compound Interest
Now we know that the compound interest earned is ₹4400, so we can set up the equation:

\[ CI = A - P = 4400 \]

Substituting for \(A\):

\[ 4400 = (8000 + x) \times 1.44 - (8000 + x) \]

### Step 7: Simplify the Equation
Now, we simplify the equation:

\[ 4400 = (8000 + x)(1.44 - 1) \]

This simplifies to:

\[ 4400 = (8000 + x) \times 0.44 \]

### Step 8: Solve for x
Now, rearranging gives:

\[ 8000 + x = \frac{4400}{0.44} \]

Calculating the right side:

\[ 8000 + x = 10000 \]

Now, solving for \(x\):

\[ x = 10000 - 8000 \]

Thus:

\[ x = 2000 \]

### Final Answer
The value of \(x\) is ₹2000.

---