1. **State the problem:** Lush Gardens Co. bought a truck for 66000. It paid 6600 as a down payment and financed the rest at 5.50% interest compounded semi-annually. Monthly payments of 2100 are made. We need to find how long it will take to pay off the loan. 2. **Calculate the loan principal:** $$\text{Principal} = 66000 - 6600 = 59400$$ 3. **Convert the interest rate to an effective monthly rate:** The nominal annual rate is 5.50% compounded semi-annually, so the semi-annual rate is $$\frac{5.50}{2} = 2.75\% = 0.0275$$ per 6 months. The effective annual rate (EAR) is: $$EAR = (1 + 0.0275)^2 - 1 = 1.0275^2 - 1 = 1.05650625 - 1 = 0.05650625$$ The effective monthly interest rate is: $$i = (1 + EAR)^{\frac{1}{12}} - 1 = (1.05650625)^{\frac{1}{12}} - 1 \approx 0.00458$$ 4. **Set up the loan amortization formula:** The loan is paid off with monthly payments $P = 2100$, principal $PV = 59400$, monthly interest rate $i = 0.00458$, and number of months $n$ unknown. The formula for the present value of an annuity is: $$PV = P \times \frac{1 - (1 + i)^{-n}}{i}$$ 5. **Solve for $n$:** Rearranged: $$\frac{PV \times i}{P} = 1 - (1 + i)^{-n}$$ $$ (1 + i)^{-n} = 1 - \frac{PV \times i}{P}$$ $$ -n \ln(1 + i) = \ln\left(1 - \frac{PV \times i}{P}\right)$$ $$ n = - \frac{\ln\left(1 - \frac{PV \times i}{P}\right)}{\ln(1 + i)}$$ Calculate: $$1 - \frac{59400 \times 0.00458}{2100} = 1 - \frac{271.452}{2100} = 1 - 0.12926 = 0.87074$$ $$n = - \frac{\ln(0.87074)}{\ln(1.00458)} = - \frac{-0.1385}{0.00457} \approx 30.3 \text{ months}$$ 6. **Convert months to years and months:** $$30.3 \text{ months} \approx 2 \text{ years and } 6 \text{ months}$$ Since payments are monthly, round up to the next whole month: 31 months. Final answer: **2 years and 7 months**.