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?

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We start with the standard loan payment formula for an annuity:

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

where

- $$P$$ is the monthly payment,
- $$L$$ is the loan amount,
- $$r$$ is the monthly interest rate,
- $$n$$ is the total number of payments.

Given that $$P = 200$$ dollars, the annual interest rate is $$5\%$$, and the loan term is $$5$$ years, we first convert the annual rate to a monthly rate and determine $$n$$:

$$
r = \frac{0.05}{12} \approx 0.004167
$$
$$
n = 5 \times 12 = 60
$$

To solve for the loan amount $$L$$, we rearrange the formula:

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

Substitute the known values:

$$
L = 200 \cdot \frac{1 - \left(1.004167\right)^{-60}}{0.004167}
$$

Now, compute the term $$\left(1.004167\right)^{-60}$$. First, calculate $$\left(1.004167\right)^{60}$$:

$$
\left(1.004167\right)^{60} \approx 1.28349
$$

Thus,

$$
\left(1.004167\right)^{-60} \approx \frac{1}{1.28349} \approx 0.7788
$$

Then,

$$
1 - \left(1.004167\right)^{-60} \approx 1 - 0.7788 = 0.2212
$$

Now, substitute back into the formula for $$L$$:

$$
L \approx 200 \cdot \frac{0.2212}{0.004167}
$$

Compute the division:

$$
\frac{0.2212}{0.004167} \approx 53.09
$$

Finally, multiply by the monthly payment:

$$
L \approx 200 \times 53.09 \approx 10,\!618
$$

Thus, you can afford a loan of approximately $$\$10,\!618$$.
You can afford a loan of approximately \$10,618.

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?

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