1. **Problem Statement:** If the rate of interest is $r\%$ per annum compounded half yearly, find the effective rate of interest. 2. **Formula Used:** The effective rate of interest (ERI) when interest is compounded more than once a year is given by: $$\text{ERI} = \left(1 + \frac{i}{n}\right)^n - 1$$ where $i$ is the nominal annual interest rate (in decimal), and $n$ is the number of compounding periods per year. 3. **Explanation:** - Here, the nominal rate $r\%$ means $i = \frac{r}{100}$. - Since interest is compounded half yearly, $n = 2$. 4. **Calculation:** Substitute $i = \frac{r}{100}$ and $n = 2$ into the formula: $$\text{ERI} = \left(1 + \frac{\frac{r}{100}}{2}\right)^2 - 1 = \left(1 + \frac{r}{200}\right)^2 - 1$$ 5. **Simplify:** $$\text{ERI} = \left(1 + \frac{r}{200}\right)^2 - 1 = 1 + 2 \times \frac{r}{200} + \left(\frac{r}{200}\right)^2 - 1 = \frac{r}{100} + \frac{r^2}{40000}$$ 6. **Final Answer:** The effective rate of interest as a decimal is: $$\text{ERI} = \frac{r}{100} + \frac{r^2}{40000}$$ or as a percentage: $$\text{ERI} = r + \frac{r^2}{400} \%$$ This means the effective rate is slightly higher than the nominal rate due to compounding twice a year.