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If you can afford a [tex]$425 per quarter loan payment, what is the largest size loan you can take if the 13-year loan will earn $[/tex]3.3\%[tex]$ interest compounded quarterly?

Identify your variables:
\[ \text{PMT} = \boxed{} \]
\[ r = \boxed{} \]
\[ n = \boxed{} \]
\[ t = \boxed{} \]

Which is the correct formula for solving this problem?
\[ I = P r t \]
\[ A = P e^{r t} \]
\[ A = P \left(1 + \frac{r}{n}\right)^{n t} \]
\[ P_0 = \text{PMT} \left(\frac{1 - \left(1 + \frac{r}{n}\right)^{-n t}}{\frac{r}{n}}\right) \]
\[ \text{PMT} = P_0 \left(\frac{\frac{r}{n}}{1 - \left(1 + \frac{r}{n}\right)^{-n t}}\right) \]
\[ \text{PMT} = A \left(\frac{\frac{r}{n}}{\left(1 + \frac{r}{n}\right)^{n t} - 1}\right) \]
\[ x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a} \]
\[ A = \text{PMT} \left(\frac{\left(1 + \frac{r}{n}\right)^{n t} - 1}{\frac{r}{n}}\right) \]

Solve for the size of the loan: $[/tex] \boxed{} $

Sagot :

GinnyAnsweridn
To find the largest size loan you can afford with a quarterly payment of [tex]$425 for a 13-year loan at an annual interest rate of 3.3%, compounded quarterly, we'll use the following present value annuity formula: \[ \text{PMT} = P_o \left( \frac{\frac{r}{n}}{1 - \left(1 + \frac{r}{n}\right)^{-nt}} \right) \] Here, our variables are: - \(\text{PMT} = 425\) (quarterly loan payment) - \(r = 0.033\) (annual interest rate) - \(n = 4\) (compounding periods per year, i.e., quarterly) - \(t = 13\) (loan duration in years) First, identify the correct formula for the present value of an ordinary annuity: \[ P_o = \text{PMT} \left( \frac{1 - \left(1 + \frac{r}{n}\right)^{-nt}}{\frac{r}{n}} \right) \] Next, let's solve step-by-step: 1. Calculate the rate per period: \[ \text{rate per period} = \frac{r}{n} = \frac{0.033}{4} = 0.00825 \] 2. Calculate the total number of periods: \[ \text{total periods} = n \times t = 4 \times 13 = 52 \] 3. Calculate the discount factor: \[ \text{discount factor} = \left(1 + \frac{r}{n}\right)^{-nt} = \left(1 + 0.00825\right)^{-52} \approx 0.6523069483825145 \] 4. Calculate the present value (largest size loan): \[ P_o = \text{PMT} \left( \frac{1 - \left(1 + \frac{r}{n}\right)^{-nt}}{\frac{r}{n}} \right) \] Substituting the values: \[ P_o = 425 \left( \frac{1 - 0.6523069483825145}{0.00825} \right) \approx 17911.46 \] Therefore, the largest size loan you can take is approximately $[/tex]\[tex]$17911.46$[/tex].
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