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

Question

UpStudy AI · Accepted Answer

We start with the formula for the monthly payment $$ M $$ on a loan amount $$ P $$ with a monthly interest rate $$ r $$ over $$ n $$ payments:

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

In this problem we are given:

- Monthly payment: $$ M = 200 $$
- Annual interest rate: $$ 5\% = 0.05 $$
- Monthly interest rate:

$$
r = \frac{0.05}{12}
$$

- Loan term: $$ 5 $$ years, so the number of monthly payments is:

$$
n = 5 \times 12 = 60
$$

We need to solve for $$ P $$. Rearranging the formula for $$ P $$:

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

Substitute the given values:

$$
P = 200 \cdot \frac{1 - \left(1+\frac{0.05}{12}\right)^{-60}}{\frac{0.05}{12}}
$$

Let’s break down the computation step by step.

1. Calculate the monthly interest rate:

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

2. Compute $$ 1 + r $$:

$$
1 + r \approx 1.00416667
$$

3. Calculate $$ \left(1 + r\right)^{-60} $$:

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

We first compute $$ (1.00416667)^{60} $$. Using the exponential approximation:

$$
(1.00416667)^{60} \approx e^{60 \cdot \ln(1.00416667)} \approx e^{60 \cdot 0.00415888} \approx e^{0.249533} \approx 1.283
$$

Thus,

$$
\left(1.00416667\right)^{-60} \approx \frac{1}{1.283} \approx 0.779
$$

4. Compute the numerator:

$$
1 - 0.779 \approx 0.221
$$

5. Divide by $$ r $$:

$$
\frac{0.221}{0.00416667} \approx 53.04
$$

6. Multiply by the monthly payment:

$$
P \approx 200 \times 53.04 \approx 10\,608
$$

Thus, you can afford a loan of approximately $$ \$10\,608 $$.
You can afford a loan of approximately \$10,608.

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