A certain sum . At compound interest, becomes rupes 7.396 in 2 year and rupes 7,950. 70 in 3 years Find the rate of interest.

Answer

To find the rate of interest given the amounts at compound interest for 2 years and 3 years, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the amounts for the given years:**
- Amount after 2 years (A1) = ₹7396
- Amount after 3 years (A2) = ₹7950.70

2. **Use the formula for compound interest:**
The formula for the amount at compound interest is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
where:
- \(A\) = Amount after n years
- \(P\) = Principal amount (initial sum)
- \(r\) = Rate of interest per annum
- \(n\) = Number of years

3. **Set up the equations for the two amounts:**
- For 2 years:
\[ A_1 = P \left(1 + \frac{r}{100}\right)^2 \]
Substituting the value of \(A_1\):
\[ 7396 = P \left(1 + \frac{r}{100}\right)^2 \quad \text{(Equation 1)} \]

- For 3 years:
\[ A_2 = P \left(1 + \frac{r}{100}\right)^3 \]
Substituting the value of \(A_2\):
\[ 7950.70 = P \left(1 + \frac{r}{100}\right)^3 \quad \text{(Equation 2)} \]

4. **Divide Equation 2 by Equation 1:**
\[ \frac{7950.70}{7396} = \frac{P \left(1 + \frac{r}{100}\right)^3}{P \left(1 + \frac{r}{100}\right)^2} \]
This simplifies to:
\[ \frac{7950.70}{7396} = 1 + \frac{r}{100} \]

5. **Calculate the left side:**
\[ \frac{7950.70}{7396} \approx 1.075 \]

6. **Set the equation:**
\[ 1 + \frac{r}{100} = 1.075 \]

7. **Isolate \(r\):**
\[ \frac{r}{100} = 1.075 - 1 = 0.075 \]
\[ r = 0.075 \times 100 = 7.5 \]

8. **Conclusion:**
The rate of interest \(r\) is **7.5%**.