Ajitha borrowed Rs 90,000 at 10% per annum simple interest. On the same day, she lent the sum to her friend at the same rate but compounded annually. How much does she gain at the end of 3 years ?

Options

A. Rs 2,907
B. Rs 2,970
C. Rs 2,800
D. Rs 2,790

Answer

To solve the problem, we need to calculate the total amount Ajitha has to pay back after borrowing the money at simple interest and the total amount she receives from lending the same amount at compound interest. Finally, we will find the difference between these two amounts to determine her gain.

### Step-by-Step Solution:

1. **Calculate Simple Interest (SI) for the borrowed amount:**
- Formula for Simple Interest:
\[ SI = \frac{P \times R \times T}{100} \]
- Where:
- \( P = 90,000 \) (Principal)
- \( R = 10\% \) (Rate of interest)
- \( T = 3 \) years (Time)
- Substituting the values:
\[ SI = \frac{90,000 \times 10 \times 3}{100} = \frac{2,700,000}{100} = 27,000 \]

2. **Calculate the total amount to be paid back:**
- Total Amount (A) after 3 years:
\[ A = P + SI = 90,000 + 27,000 = 1,17,000 \]

3. **Calculate Compound Interest (CI) for the lent amount:**
- Formula for Compound Interest:
\[ A = P \left(1 + \frac{R}{100}\right)^T \]
- Substituting the values:
\[ A = 90,000 \left(1 + \frac{10}{100}\right)^3 = 90,000 \left(1.1\right)^3 \]
- Calculate \( (1.1)^3 \):
\[ (1.1)^3 = 1.331 \]
- Now calculate the total amount:
\[ A = 90,000 \times 1.331 = 1,19,790 \]

4. **Calculate the gain:**
- Gain = Amount received from lending - Amount paid back for borrowing:
\[ \text{Gain} = 1,19,790 - 1,17,000 = 2,790 \]

### Final Answer:
Ajitha's gain at the end of 3 years is **Rs 2,790**.