1. **State the problem:** Gether borrows 27000 and pays 2832.18 interest at 2% annual interest compounded quarterly. We need to find the length of the loan in years. 2. **Formula:** Compound interest is calculated by $$A = P(1 + i)^n$$ where: - $A$ is the final amount - $P$ is the principal - $i$ is the interest rate per compounding period - $n$ is the number of compounding periods 3. **Identify values:** - $P = 27000$ - Interest paid = 2832.18, so final amount $A = 27000 + 2832.18 = 29832.18$ - Annual interest rate = 2% or 0.02 - Compounded quarterly means 4 compounding periods per year, so interest rate per period $i = \frac{0.02}{4} = 0.005$ 4. **Find $n$:** $$29832.18 = 27000(1 + 0.005)^n$$ Divide both sides by 27000: $$\frac{29832.18}{27000} = (1.005)^n$$ $$1.1053 = (1.005)^n$$ 5. **Solve for $n$ using logarithms:** $$\ln(1.1053) = n \ln(1.005)$$ $$n = \frac{\ln(1.1053)}{\ln(1.005)}$$ Calculate: $$n \approx \frac{0.1001}{0.004987} \approx 20.08$$ 6. **Convert $n$ to years:** Since $n$ is the number of quarters, years $= \frac{n}{4} = \frac{20.08}{4} = 5.02$ years **Final answer:** The length of the loan is approximately **5 years**.