You can afford a $$ \$ 200 $$ per month car payment. You've found a 5 year loan at $$ 5 \% $$ interest. Hov big of a loan can you afford?

Question

UpStudy AI · Accepted Answer

We start with the formula for the present value of an annuity that represents the loan amount $$ L $$:

$$
L = P \cdot \frac{1 - \left(1 + r\right)^{-n}}{r}
$$

where  
$$ P $$ is the monthly payment,  
$$ r $$ is the monthly interest rate, and  
$$ n $$ is the total number of monthly payments.

1. The monthly payment is given as $$ P = 200 $$.

2. The annual interest rate is $$ 5\% $$ so the monthly rate is

$$
r = \frac{0.05}{12} \approx 0.00416667.
$$

3. The term of the loan is 5 years which gives

$$
n = 5 \times 12 = 60 \text{ months}.
$$

4. Substitute these values into the annuity formula:

$$
L = 200 \cdot \frac{1 - \left(1 + 0.00416667\right)^{-60}}{0.00416667}.
$$

5. Calculate $$ 1 + r $$:

$$
1 + 0.00416667 \approx 1.00416667.
$$

6. Raise this to the $$-60$$ power:

$$
\left(1.00416667\right)^{-60} \approx \frac{1}{\left(1.00416667\right)^{60}}.
$$

Calculating $$ \left(1.00416667\right)^{60} $$ approximately:

$$
\left(1.00416667\right)^{60} \approx 1.2834,
$$

so that

$$
\left(1.00416667\right)^{-60} \approx \frac{1}{1.2834} \approx 0.7782.
$$

7. Now evaluate the numerator:

$$
1 - 0.7782 \approx 0.2218.
$$

8. Divide by $$ r $$:

$$
\frac{0.2218}{0.00416667} \approx 53.23.
$$

9. Finally, multiply by the monthly payment $$ P $$:

$$
L \approx 200 \times 53.23 \approx 10,\!646.
$$

Thus, you can afford a loan of approximately $$ \$10,\!646 $$.
You can afford a loan of approximately \$10,646.

You can afford a \( \$ 200 \) per month car payment. You've found a 5 year loan at \( 5 \% \) interest. Hov big of a loan can you afford?