1. **Problem Statement:** Determine the number of years needed to pay off a loan of 55400 with monthly payments of 1132.14 at an annual interest rate of 17% compounded monthly. 2. **Identify known values:** - Principal, $P = 55400$ - Monthly payment, $PMT = 1132.14$ - Annual nominal interest rate, $r = 0.17$ - Number of compounding periods per year, $n = 12$ 3. **Calculate monthly interest rate:** $$i = \frac{r}{n} = \frac{0.17}{12} \approx 0.0141667$$ 4. **Use the loan amortization formula to find the total number of payments $N$:** $$PMT = P \times \frac{i(1+i)^N}{(1+i)^N - 1}$$ Rearranged to solve for $N$: $$N = \frac{\log(\frac{PMT}{PMT - P \times i})}{\log(1+i)}$$ 5. **Substitute the known values:** $$\frac{PMT}{PMT - P \times i} = \frac{1132.14}{1132.14 - 55400 \times 0.0141667} = \frac{1132.14}{1132.14 - 784.667} = \frac{1132.14}{347.473} \approx 3.257$$ 6. **Calculate $N$:** $$N = \frac{\log(3.257)}{\log(1.0141667)} = \frac{0.512}{0.00627} \approx 81.66 \text{ months}$$ 7. **Convert months to years:** $$\text{Years} = \frac{81.66}{12} \approx 6.805$$ **Final answer:** It will take approximately $6.81$ years to fully settle the loan with the given monthly payments and interest rate.