Maths Paper \( 1(\mathrm{~A}) \) Question 8, PART 4 Adapted from DBE Nov 2016 Grade 11, P1, Q8.2 and 2015 Grade 11, P1, Q8 Maths formulas Convert a nominal interest rate of \( 11,3 \% \) p.a., compounded semi-anually, to an annual effective rate. INSTRUCTION: Round your answer to two decimal places. Answer: \( i_{\text {eff }}= \) \% p.a. Check answer 2 attempts remaining

To convert a nominal interest rate compounded semi-annually to an annual effective rate, we can use the following formula: \[ i_{\text{eff}} = \left(1 + \frac{r}{m}\right)^m - 1 \] where: - \( r \) is the nominal interest rate (as a decimal), - \( m \) is the number of compounding periods per year. In this case: - The nominal interest rate \( r = 11.3\% = 0.113 \), - The compounding frequency \( m = 2 \) (since it is compounded semi-annually). Now, we can substitute these values into the formula: \[ i_{\text{eff}} = \left(1 + \frac{0.113}{2}\right)^2 - 1 \] Calculating \( \frac{0.113}{2} \): \[ \frac{0.113}{2} = 0.0565 \] Now, substituting this back into the formula: \[ i_{\text{eff}} = \left(1 + 0.0565\right)^2 - 1 \] Calculating \( 1 + 0.0565 \): \[ 1 + 0.0565 = 1.0565 \] Now, we raise this to the power of 2: \[ (1.0565)^2 \approx 1.1135 \] Now, subtract 1: \[ i_{\text{eff}} \approx 1.1135 - 1 = 0.1135 \] To express this as a percentage, we multiply by 100: \[ i_{\text{eff}} \approx 0.1135 \times 100 = 11.35\% \] Rounding to two decimal places, we get: \[ i_{\text{eff}} \approx 11.35\% \] Thus, the annual effective rate is: \[ \boxed{11.35} \% \text{ p.a.} \]