A certain sum of money is put at compound interest, compounded half-yearly. If the interest for two successive half-years are रु 650 and रु 760.50, find the rate of interest.

Explanation

To solve the problem of finding the rate of interest when a certain sum of money is put at compound interest compounded half-yearly, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
Let the principal amount (the sum of money invested) be \( P \) and the half-yearly rate of interest be \( r \% \).

2. **Identify the Interest for the First Half-Year**:
According to the problem, the interest for the first half-year is given as \( 650 \) rupees. This can be expressed as:
\[ \text{Interest for first half-year} = \frac{P \times r}{100} = 650 \]

3. **Identify the Interest for the Second Half-Year**:
The interest for the second half-year is given as \( 760.50 \) rupees. Since the interest is compounded, the principal for the second half-year becomes \( P + 650 \). Therefore, we can write:
\[ \text{Interest for second half-year} = \frac{(P + 650) \times r}{100} = 760.50 \]

4. **Set Up the Equations**:
We now have two equations:
\[ \frac{P \times r}{100} = 650 \quad \text{(1)} \]
\[ \frac{(P + 650) \times r}{100} = 760.50 \quad \text{(2)} \]

5. **Subtract the First Equation from the Second**:
To find the difference in interest, we can subtract equation (1) from equation (2):
\[ \frac{(P + 650) \times r}{100} - \frac{P \times r}{100} = 760.50 - 650 \]
Simplifying this gives:
\[ \frac{650 \times r}{100} = 110.50 \]

6. **Solve for \( r \)**:
Rearranging the equation:
\[ 650 \times r = 110.50 \times 100 \]
\[ 650 \times r = 11050 \]
\[ r = \frac{11050}{650} \]
\[ r = 17 \]

7. **Conclusion**:
The rate of interest per half-year is \( 17\% \). To find the annual rate, we multiply by 2:
\[ \text{Annual Rate} = 17 \times 2 = 34\% \]

### Final Answer:
The rate of interest is \( 17\% \) per half-year, or \( 34\% \) per annum.