The amount-invested in scheme A at `20% `per annum compound interest for two years is 1.5 times that invested in scheme B at `10%` per annum for the same period. The interest is compounded annually in both the schemes. The compound interest received from scheme A is Rs 2025 more than that from scheme B. Find tlie amount invested in scheme A

Options

A. Rs 6750
B. Rs 4500
C. Rs 4400
D. Rs 6600

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Define the Variables
Let the amount invested in scheme B be \( x \). Then, the amount invested in scheme A will be \( 1.5x \) since it is given that the amount in scheme A is 1.5 times that in scheme B.

### Step 2: Identify the Interest Rates and Time Periods
The interest rate for scheme A is \( 20\% \) per annum and for scheme B is \( 10\% \) per annum. The time period for both schemes is \( 2 \) years.

### Step 3: Calculate the Compound Interest for Scheme A
The formula for compound interest is given by:
\[ \text{Compound Interest} = P \left(1 + \frac{r}{100}\right)^t - P \]
For scheme A:
- Principal \( P = 1.5x \)
- Rate \( r = 20\% \)
- Time \( t = 2 \)

So, the compound interest for scheme A is:
\[ CI_A = 1.5x \left(1 + \frac{20}{100}\right)^2 - 1.5x \]
Calculating:
\[ CI_A = 1.5x \left(1 + 0.2\right)^2 - 1.5x = 1.5x \left(1.2\right)^2 - 1.5x = 1.5x \cdot 1.44 - 1.5x = 2.16x - 1.5x = 0.66x \]

### Step 4: Calculate the Compound Interest for Scheme B
For scheme B:
- Principal \( P = x \)
- Rate \( r = 10\% \)
- Time \( t = 2 \)

So, the compound interest for scheme B is:
\[ CI_B = x \left(1 + \frac{10}{100}\right)^2 - x \]
Calculating:
\[ CI_B = x \left(1 + 0.1\right)^2 - x = x \left(1.1\right)^2 - x = x \cdot 1.21 - x = 1.21x - x = 0.21x \]

### Step 5: Set Up the Equation Based on the Given Information
According to the problem, the compound interest from scheme A is Rs 2025 more than that from scheme B:
\[ CI_A = CI_B + 2025 \]
Substituting the values we calculated:
\[ 0.66x = 0.21x + 2025 \]

### Step 6: Solve for \( x \)
Rearranging the equation:
\[ 0.66x - 0.21x = 2025 \]
\[ 0.45x = 2025 \]
Now, divide both sides by \( 0.45 \):
\[ x = \frac{2025}{0.45} = 4500 \]

### Step 7: Find the Amount Invested in Scheme A
Now, we can find the amount invested in scheme A:
\[ \text{Amount in Scheme A} = 1.5x = 1.5 \times 4500 = 6750 \]

### Final Answer
The amount invested in scheme A is **Rs 6750**.

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