1. **Problem statement:** You want to take a mortgage of 100000 FRW with an annual interest rate of 9%, making monthly payments of 800 FRW. We need to find how long it will take to pay off the mortgage and the total interest paid. 2. **Formula used:** The mortgage payment formula for monthly payments is: $$P = \frac{rPV}{1 - (1+r)^{-n}}$$ where: - $P$ is the monthly payment, - $r$ is the monthly interest rate, - $PV$ is the loan principal, - $n$ is the number of months. 3. **Given values:** - $PV = 100000$ - Annual interest rate = 9%, so monthly interest rate $r = \frac{9\%}{12} = 0.0075$ - Monthly payment $P = 800$ 4. **Find $n$:** Rearranging the formula to solve for $n$: $$800 = \frac{0.0075 \times 100000}{1 - (1+0.0075)^{-n}}$$ Calculate numerator: $$0.0075 \times 100000 = 750$$ So: $$800 = \frac{750}{1 - (1.0075)^{-n}}$$ Invert both sides: $$\frac{1}{800} = \frac{1 - (1.0075)^{-n}}{750}$$ Multiply both sides by 750: $$\frac{750}{800} = 1 - (1.0075)^{-n}$$ Simplify left side: $$0.9375 = 1 - (1.0075)^{-n}$$ Rearranged: $$(1.0075)^{-n} = 1 - 0.9375 = 0.0625$$ Take natural logarithm on both sides: $$-n \ln(1.0075) = \ln(0.0625)$$ Calculate logarithms: $$\ln(1.0075) \approx 0.007472$$ $$\ln(0.0625) = \ln\left(\frac{1}{16}\right) = -\ln(16) \approx -2.7726$$ Solve for $n$: $$-n \times 0.007472 = -2.7726 \implies n = \frac{2.7726}{0.007472} \approx 371.0$$ 5. **Interpretation:** It will take approximately 371 months to pay off the mortgage. 6. **Calculate total amount paid:** $$\text{Total paid} = 800 \times 371 = 296800$$ 7. **Calculate total interest paid:** $$\text{Interest} = \text{Total paid} - \text{Principal} = 296800 - 100000 = 196800$$ **Final answers:** - Time to pay off mortgage: approximately 371 months (about 30.9 years). - Total interest paid: 196800 FRW.