7. Jane must repay her 19000 student loan. She plans to 350 The banks's interest rate is 6.9% /a compounded monthly. About how many years will it take Jane to repay her loan? [4]

## Step 1: Calculate the monthly interest rate.<br />### The annual interest rate is 6.9%, so the monthly interest rate is $\frac{6.9\%}{12} = 0.069/12 = 0.00575$.<br /><br />## Step 2: Use the loan amortization formula.<br />### The loan amortization formula is $P = \frac{r(PV)}{1-(1+r)^{-n}}$, where:<br />### * $P$ is the monthly payment ($\$350$)<br />### * $r$ is the monthly interest rate ($0.00575$)<br />### * $PV$ is the present value of the loan ($\$19000$)<br />### * $n$ is the number of months.<br />### We need to solve for $n$. The formula can be rearranged to: $n = -\frac{\ln(1 - \frac{r(PV)}{P})}{\ln(1+r)}$.<br /><br />## Step 3: Substitute the values and calculate $n$.<br />### Substituting the values, we get:<br />### $n = -\frac{\ln(1 - \frac{0.00575 \times 19000}{350})}{\ln(1+0.00575)} = -\frac{\ln(1 - 0.3117857)}{\ln(1.00575)} \approx -\frac{\ln(0.6882143)}{\ln(1.00575)} \approx \frac{0.3757}{0.005732} \approx 65.55$ months.<br /><br />## Step 4: Convert months to years.<br />### Divide the number of months by 12 to get the number of years: $\frac{65.55}{12} \approx 5.46$ years. Since the question asks for "about" how many years, we can round this to approximately 5.5 years.