1. **State the problem:** We have three payments: $747$ due in 1 year, $843$ due in 4.5 years, and $1233$ due in 5 years. We want to find a single equivalent payment at 2.5 years, with an interest rate of 9.4% compounded annually. 2. **Formula used:** The equivalent single payment $P$ at time $t$ satisfies the equation $$P(1+i)^{t_0 - t} = ext{sum of all payments compounded or discounted to } t_0,$$ where $i=0.094$ is the interest rate, $t_0$ is the time of each payment, and $t=2.5$ years is the time of the equivalent payment. 3. **Calculate the present value of each payment at 2.5 years:** - For payment at 1 year: it is in the past relative to 2.5 years, so we compound it forward: $$747 \times (1+0.094)^{2.5 - 1} = 747 \times (1.094)^{1.5}$$ - For payment at 4.5 years: it is in the future relative to 2.5 years, so we discount it back: $$843 \times (1.094)^{2.5 - 4.5} = 843 \times (1.094)^{-2}$$ - For payment at 5 years: similarly discount back: $$1233 \times (1.094)^{2.5 - 5} = 1233 \times (1.094)^{-2.5}$$ 4. **Calculate each term:** - $(1.094)^{1.5} = 1.141857$ - $(1.094)^{-2} = \frac{1}{(1.094)^2} = \frac{1}{1.196836} = 0.835621$ - $(1.094)^{-2.5} = \frac{1}{(1.094)^{2.5}} = \frac{1}{1.288034} = 0.776646$ 5. **Calculate the equivalent payment:** $$P = 747 \times 1.141857 + 843 \times 0.835621 + 1233 \times 0.776646$$ $$P = 852.996 + 704.348 + 957.393 = 2514.737$$ 6. **Round the final answer:** $$P \approx 2514.74$$ **Final answer:** The equivalent single replacement payment at 2.5 years is $2514.74$.