1. **State the problem:** Jimmy pays R3876.90 monthly towards a 12-year home loan with an annual interest rate of 11.75% compounded monthly. We need to find the initial loan amount. 2. **Identify known values:** - Monthly payment, $P = 3876.90$ - Annual interest rate, $r = 11.75\% = 0.1175$ - Number of years, $t = 12$ - Compounding frequency: monthly, so number of payments, $n = 12 \times 12 = 144$ - Monthly interest rate, $i = \frac{0.1175}{12} = 0.00979167$ 3. **Relevant formula:** The formula for the present value (initial loan) of an annuity (loan payments) is: $$ L = P \times \frac{1 - (1+i)^{-n}}{i} $$ where $L$ is the loan amount, $P$ payment per period, $i$ interest rate per period, and $n$ total payments. 4. **Calculate**: Calculate $(1+i)^{-n}$: $$ (1 + 0.00979167)^{-144} = (1.00979167)^{-144} $$ Calculate this value first: $$ (1.00979167)^{144} = e^{144 \times \ln(1.00979167)} \approx e^{144 \times 0.009745} = e^{1.402} \approx 4.064 $$ Therefore, $$ (1.00979167)^{-144} = \frac{1}{4.064} \approx 0.2462 $$ 5. Substitute in formula: $$ L = 3876.90 \times \frac{1 - 0.2462}{0.00979167} = 3876.90 \times \frac{0.7538}{0.00979167} $$ Calculate denominator: $$ \frac{0.7538}{0.00979167} \approx 76.987 $$ Multiply: $$ L = 3876.90 \times 76.987 \approx 298548.45 $$ **Final answer:** Jimmy's initial loan amount was approximately R298548.45.