Rumi invested a certain sum for 2 years in scheme X at `20%` pa compound interest (compounded- annually). He also invested an equal sum in scheme Y at `15%` pa simple interest for 2 years. If the difference between the interests earned from in schemes X and Y is ₹350, what was the sum?

Options

A. ₹2000
B. ₹1200
C. ₹2500
D. ₹4000

Answer

To solve the problem step by step, we will calculate the compound interest from Scheme X and the simple interest from Scheme Y, and then find the sum based on the given difference.

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

### Step 2: Calculate the compound interest for Scheme X
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
Where:
- \( A \) = Amount after time \( T \)
- \( P \) = Principal
- \( R \) = Rate of interest
- \( T \) = Time in years

For Scheme X:
- \( R = 20\% \)
- \( T = 2 \)

Substituting the values:
\[ A_X = P \left(1 + \frac{20}{100}\right)^2 = P \left(1 + 0.2\right)^2 = P \left(1.2\right)^2 = P \times 1.44 \]
The compound interest earned from Scheme X is:
\[ CI_X = A_X - P = 1.44P - P = 0.44P \]

### Step 3: Calculate the simple interest for Scheme Y
The formula for simple interest is:
\[ SI = \frac{P \times R \times T}{100} \]

For Scheme Y:
- \( R = 15\% \)
- \( T = 2 \)

Substituting the values:
\[ SI_Y = \frac{P \times 15 \times 2}{100} = \frac{30P}{100} = 0.3P \]

### Step 4: Set up the equation based on the difference in interests
According to the problem, the difference between the compound interest from Scheme X and the simple interest from Scheme Y is ₹350:
\[ CI_X - SI_Y = 350 \]
Substituting the values we calculated:
\[ 0.44P - 0.3P = 350 \]
This simplifies to:
\[ 0.14P = 350 \]

### Step 5: Solve for \( P \)
To find \( P \), divide both sides by 0.14:
\[ P = \frac{350}{0.14} = 2500 \]

### Conclusion
The sum that Rumi invested in each scheme is ₹2500.

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