1. **State the problem:** Lush Gardens Co. bought a truck for 50000. It paid 5000 as a down payment, so the financed amount is $50000 - 5000 = 45000$. 2. **Identify the loan details:** - Principal $P = 45000$ - Interest rate $r = 5.53\%$ compounded semi-annually - Monthly payment $PMT = 1500$ - We want to find the time $t$ (in years and months) to pay off the loan. 3. **Convert the interest rate to an effective monthly rate:** Since interest is compounded semi-annually, the nominal annual rate is 5.53%, compounded twice a year. The semi-annual interest rate is $\frac{5.53}{2} = 2.765\% = 0.02765$ per half year. The effective monthly interest rate $i$ is: $$i = \left(1 + 0.02765\right)^{\frac{1}{6}} - 1$$ Calculate: $$i = (1.02765)^{0.1667} - 1 \approx 0.00454$$ 4. **Use the amortization formula to find the number of months $n$:** The formula for the present value of an annuity is: $$P = PMT \times \frac{1 - (1 + i)^{-n}}{i}$$ Rearranged to solve for $n$: $$1 - (1 + i)^{-n} = \frac{P \times i}{PMT}$$ $$ (1 + i)^{-n} = 1 - \frac{P \times i}{PMT}$$ $$ -n \ln(1 + i) = \ln\left(1 - \frac{P \times i}{PMT}\right)$$ $$ n = -\frac{\ln\left(1 - \frac{P \times i}{PMT}\right)}{\ln(1 + i)}$$ 5. **Plug in the values:** $$1 - \frac{45000 \times 0.00454}{1500} = 1 - \frac{204.3}{1500} = 1 - 0.1362 = 0.8638$$ Calculate logarithms: $$n = -\frac{\ln(0.8638)}{\ln(1.00454)} = -\frac{-0.146}{0.00453} \approx 32.2 \text{ months}$$ 6. **Convert months to years and months:** $$32.2 \text{ months} = 2 \text{ years and } 8.2 \text{ months}$$ Round up to the next payment period: $$2 \text{ years and } 9 \text{ months}$$ **Final answer:** It will take approximately **2 years and 9 months** to settle the loan.