1. **Problem statement:** We want to find how long it will take to save 16500 by making monthly deposits of 90 into an account with 7.1% annual interest compounded quarterly. 2. **Formula used:** The future value of an annuity with periodic deposits is given by $$FV = P \times \frac{(1 + r)^n - 1}{r}$$ where $P$ is the deposit per period, $r$ is the interest rate per period, and $n$ is the number of periods. 3. **Important rules:** - Interest is compounded quarterly, so the quarterly interest rate is $\frac{7.1}{100 \times 4} = 0.01775$. - Deposits are monthly, so we must adjust the formula to account for monthly deposits with quarterly compounding. 4. **Adjusting for monthly deposits with quarterly compounding:** - Quarterly interest rate $i = 0.01775$. - Monthly interest rate $j$ is not simply $i/3$ because compounding is quarterly. - Instead, the effective monthly interest rate is $j = (1 + i)^{1/3} - 1$. Calculate $j$: $$j = (1 + 0.01775)^{\frac{1}{3}} - 1 = 1.01775^{0.3333} - 1 \approx 0.0059$$ 5. **Using the annuity formula with monthly deposits and monthly interest rate $j$:** $$16500 = 90 \times \frac{(1 + 0.0059)^n - 1}{0.0059}$$ 6. **Solve for $n$:** $$\frac{16500 \times 0.0059}{90} = (1.0059)^n - 1$$ $$1.0817 = (1.0059)^n - 1$$ $$2.0817 = (1.0059)^n$$ 7. Take natural logarithm on both sides: $$\ln(2.0817) = n \times \ln(1.0059)$$ $$n = \frac{\ln(2.0817)}{\ln(1.0059)} \approx \frac{0.733}{0.00588} \approx 124.66$$ 8. **Convert $n$ months to years and months:** $$\text{years} = \lfloor \frac{124.66}{12} \rfloor = 10$$ $$\text{months} = 124.66 - 10 \times 12 = 4.66 \approx 5$$ **Final answer:** It will take 10 years and 5 months to save 16500 with monthly deposits of 90 at 7.1% interest compounded quarterly.