ILse bags a iv marked with a lash price of \( R 4600 \) on Ciedit. She signs a hire parchase agreement which requies her to pay interest at a rate of \( 19 \% \) per anam. She detemines that she will pay \( R 6785 \) in total for bhe tv in equal monthly instalments over \( n \) gear 7 Calalate the value of \( n \) write your answer in gears and months show all calalagions 2 Calculate the monthly instalments to be paid

To find the value of \( n \) (the number of years), we first need to determine the total interest paid and how that translates into monthly payments. 1. **Calculate the total interest paid**: Total cost paid = \( R 6785 \) Price of the TV = \( R 4600 \) Interest paid = Total cost - Price = \( R 6785 - R 4600 = R 2185 \) 2. **Calculate the total number of monthly instalments and monthly payment**: The formula for total payment on a hire purchase is: Total payments = Monthly payment × Number of months With the hire purchase agreement spanning several years, we can rewrite the total payment in relation to \( n \): \( R 6785 = R m \times (12n) \) Where \( m \) is the monthly installment. Our goal is to find \( n \) considering an interest rate of \( 19\% \) per annum. To find \( n \) and \( m \), we will assume a different strategy to solve this specifically for monthly installments. We can use the formula: \[ A = P \left(1 + \frac{r}{100}\right)^t \] Where: - \( A \) = Total amount paid, which is \$ 6785 - \( P \) = Principal amount, which is \$ 4600 - \( r \) = Interest rate per annum, which is \( 19 \% \) or \( 0.19 \) in decimal - \( t \) = Time in years Now, rearranging for \( t \): \[ 6785 = 4600 \left(1 + \frac{19}{100}\right)^{t} \] \[ 1.475 = \left(1.19\right)^{t} \] To find \( t \): Taking the natural logarithm of both sides results in: \[ \ln(1.475) = t \cdot \ln(1.19) \] \[ t \approx \frac{\ln(1.475)}{\ln(1.19)} \] Calculating these values gives us: \[ t \approx 2.744 \, \text{years} \] This translates to: - \( 2 \text{ years} \) and \( 8-9 \text{ months} \) as \( 0.744 \times 12 = 8.93 \). 3. **Monthly instalment calculation**: To find the monthly payment \( m \): \[ \text{Number of months} = 2 \times 12 + 9 \approx 33 \] Then, \[ m = \frac{6785}{33} \approx R 205.06 \] Thus, the values are: - **Years and Months:** 2 years and 9 months - **Monthly Instalment:** \( R 205.06 \) per month.