a student has a job that leaves her with $300 per month in disposable income. she decides that she will use…

Explanation:

Step1: Calculate the monthly interest rate

The nominal annual interest rate is 9%, so the monthly interest rate $i=\frac{9%}{12}= 0.0075$.

Step2: Identify the number of periods

The loan is for 48 months, so $n = 48$.

Step3: Use the present - value of an ordinary annuity formula

The formula for the present - value of an ordinary annuity is $PV = A\times\frac{1-(1 + i)^{-n}}{i}$, where $A$ is the monthly payment, $i$ is the monthly interest rate, and $n$ is the number of periods. Here, $A = 300$, $i=0.0075$, and $n = 48$. Substitute the values into the formula: [ \begin{align*} PV&=300\times\frac{1-(1 + 0.0075)^{-48}}{0.0075}\ \end{align*} ] First, calculate $(1 + 0.0075)^{-48}$. Let $x=(1 + 0.0075)^{-48}=\frac{1}{(1.0075)^{48}}$. Using the formula for compound - interest $(a)^{n}$, where $a = 1.0075$ and $n = 48$, $(1.0075)^{48}\approx1.431408$. So $x=\frac{1}{1.431408}\approx0.6985$. Then, $1-(1 + 0.0075)^{-48}=1 - 0.6985 = 0.3015$. $\frac{1-(1 + 0.0075)^{-48}}{0.0075}=\frac{0.3015}{0.0075}=40.2$. $PV=300\times40.2 = 12060$.

Answer:

The maximum car purchase price she can afford with her loan is $$12060$.