1. **State the problem:** You want to find the maximum price to pay for a house today, given monthly rental income, a future sale price, and a desired monthly return rate. 2. **Given data:** - Monthly rental income: $1800 (received at the beginning of each month) - Sale price at the end of 47 months: $300000 - Desired monthly return rate: 0.4% or 0.004 as a decimal - Number of months: 47 3. **Formula used:** This is a present value problem with an annuity due (payments at the beginning of each period) plus a lump sum at the end. The present value of an annuity due is: $$PV_{annuity} = P \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)$$ where $P$ is the payment, $r$ is the monthly interest rate, and $n$ is the number of payments. The present value of the lump sum (sale price) is: $$PV_{lump} = \frac{F}{(1 + r)^n}$$ where $F$ is the future value. 4. **Calculate present value of rental income:** $$PV_{annuity} = 1800 \times \frac{1 - (1 + 0.004)^{-47}}{0.004} \times (1 + 0.004)$$ Calculate the term inside the fraction: $$1 + 0.004 = 1.004$$ $$1.004^{-47} = \frac{1}{1.004^{47}}$$ Calculate $1.004^{47}$: $$1.004^{47} \approx e^{47 \times \ln(1.004)} \approx e^{47 \times 0.003992} \approx e^{0.1876} \approx 1.2065$$ So: $$1.004^{-47} = \frac{1}{1.2065} \approx 0.8293$$ Now: $$1 - 0.8293 = 0.1707$$ Divide by $r$: $$\frac{0.1707}{0.004} = 42.675$$ Multiply by $(1 + r)$: $$42.675 \times 1.004 = 42.845$$ Multiply by $P$: $$1800 \times 42.845 = 77121$$ 5. **Calculate present value of sale price:** $$PV_{lump} = \frac{300000}{1.004^{47}} = \frac{300000}{1.2065} \approx 248700$$ 6. **Total present value (maximum price to pay):** $$PV = PV_{annuity} + PV_{lump} = 77121 + 248700 = 325821$$ 7. **Final answer:** The most you should pay for the house today is approximately **325821** dollars.