The interest earned by investing a sum of money in scheme A for two years is ₹450 more than the interest earned when the same sum is invested in scheme B for the same period. If scheme A offers compound interest (compounded annually) at `30%` pa and scheme B offers SI at the same rate of interest respectively, what was the sum invested in each scheme?

Options

A. ₹ 1800
B. ₹ 1600
C. ₹ 2200
D. ₹ 5000

Answer

To solve the problem, we need to find the principal amount invested in both schemes A and B based on the interest earned over two years. Let's break down the solution step by step.

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

### Step 2: Calculate Interest for Scheme A
Scheme A offers compound interest at a rate of 30% per annum. The formula for compound interest (CI) for 2 years is given by:
\[ \text{CI} = P \left(1 + \frac{r}{100}\right)^t - P \]
Where:
- \( r = 30 \)
- \( t = 2 \)

Substituting the values:
\[ \text{CI} = P \left(1 + \frac{30}{100}\right)^2 - P = P \left(1.3\right)^2 - P = P \left(1.69\right) - P = 0.69P \]

### Step 3: Calculate Interest for Scheme B
Scheme B offers simple interest (SI) at the same rate of 30% per annum. The formula for simple interest is:
\[ \text{SI} = \frac{P \cdot r \cdot t}{100} \]
Substituting the values:
\[ \text{SI} = \frac{P \cdot 30 \cdot 2}{100} = \frac{60P}{100} = 0.6P \]

### Step 4: Set Up the Equation
According to the problem, the interest earned from scheme A is ₹450 more than the interest earned from scheme B. Therefore, we can set up the following equation:
\[ 0.69P = 0.6P + 450 \]

### Step 5: Solve the Equation
Now, we will solve for \( P \):
\[ 0.69P - 0.6P = 450 \]
\[ 0.09P = 450 \]
\[ P = \frac{450}{0.09} = 5000 \]

### Conclusion
The principal amount invested in each scheme is ₹5000.