Which interest rate would you choose if you want to borrow money \( 19 \% \) p.a. compounded monthly or \( 21 \% \) p.a. effective? Justify your answer with calculations.

To compare the interest rates of \(19\%\) p.a. compounded monthly and \(21\%\) p.a. effective, we need to calculate the effective annual interest rate for both options. 1. **Interest Rate of \(19\%\) p.a. Compounded Monthly:** - The formula to calculate the effective annual interest rate for compound interest is: \[ \text{Effective Annual Interest Rate} = \left(1 + \frac{r}{n}\right)^n - 1 \] where: - \(r\) is the annual interest rate (in decimal form) - \(n\) is the number of compounding periods per year 2. **Interest Rate of \(21\%\) p.a. Effective:** - The effective annual interest rate is given directly as \(21\%\) p.a. Let's calculate the effective annual interest rate for both options and compare them to determine which interest rate is better. Calculate the value by following steps: - step0: Calculate: \(\left(1+\frac{0.19}{12}\right)^{12}-1\) - step1: Divide the terms: \(\left(1+\frac{19}{1200}\right)^{12}-1\) - step2: Add the terms: \(\left(\frac{1219}{1200}\right)^{12}-1\) - step3: Rewrite the expression: \(\frac{1219^{12}}{1200^{12}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{1219^{12}}{1200^{12}}-\frac{1200^{12}}{1200^{12}}\) - step5: Transform the expression: \(\frac{1219^{12}-1200^{12}}{1200^{12}}\) The effective annual interest rate for a \(19\%\) p.a. compounded monthly is approximately \(20.7451\%\). Now, let's compare this with the \(21\%\) p.a. effective interest rate. Numbers by following steps: - step0: Solution: \(21\) The effective annual interest rate for a \(21\%\) p.a. effective is \(21\%\). Comparing the two interest rates: - \(19\%\) p.a. compounded monthly is approximately \(20.7451\%\) - \(21\%\) p.a. effective is \(21\%\) Therefore, the interest rate of \(19\%\) p.a. compounded monthly is better than the \(21\%\) p.a. effective interest rate.