350 The banks's interest rate is 6.9% /a compounded monthly. About how many years will it take Jane to repay student loan. She plans to pay 1500 up front and then make monthly payments of her loan? [4]

## Step 1: Calculate the remaining loan amount.<br />### Jane pays $1500 upfront, leaving $350 - $1500 = $-1150$. Since the loan is being repaid, we consider the absolute value of the remaining amount, which is $1150.<br /><br />## Step 2: Calculate the monthly interest rate.<br />### The annual interest rate is $6.9\% = 0.069$. The monthly interest rate is $\frac{0.069}{12} = 0.00575$.<br /><br />## Step 3: Determine the number of months to repay.<br />### Let $n$ be the number of months. We use the formula for the future value of an annuity: $FV = P \cdot \frac{(1+r)^n - 1}{r}$, where $FV$ is the future value (loan amount), $P$ is the monthly payment, $r$ is the monthly interest rate. In our case, $1150 = 350 \cdot \frac{(1+0.00575)^n - 1}{0.00575}$. Simplifying, we get $\frac{1150 \cdot 0.00575}{350} + 1 = (1.00575)^n$, which is $1.018857 = (1.00575)^n$. Taking the logarithm of both sides: $n = \frac{\log(1.018857)}{\log(1.00575)} \approx 3.27$.<br /><br />## Step 4: Convert months to years.<br />### Since there are 12 months in a year, it will take Jane approximately $\frac{3.27}{12} \approx 0.27$ years to repay the loan.