You can afford a $\$ 250$ per month car payment. Youve found a $3$ year loan at $5 \%$ interest. How big of a loan can you afford?

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Q. You can afford a

\$ 250

per month car payment. Youve found a

3

year loan at

5 \%

interest. How big of a loan can you afford?

Loan Calculation Overview: First, we need to understand that the loan will be paid off in monthly installments over $3$ years, which is $36$ months. We also know that the monthly payment is $\$250$ and the annual interest rate is $5\%$ . To find the maximum loan amount, we will use the formula for an installment loan which takes into account the interest rate.
Monthly Payment Formula: The formula for the monthly payment $M$ for a loan with principal amount $P$ , monthly interest rate $r$ , and number of payments $n$ is: $\newline$ $M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}$ $\newline$ We need to rearrange this formula to solve for $P$ , the principal amount, which is the maximum loan amount we can afford.
Convert Annual to Monthly Rate: First, we convert the annual interest rate to a monthly interest rate by dividing by $12$ , since there are $12$ months in a year. $\newline$ Monthly interest rate $r$ = Annual interest rate / $12$ $\newline$ $r = \frac{5\%}{12}$ $\newline$ $r = \frac{0.05}{12}$ $\newline$ $r = 0.0041666667$
Calculate Number of Payments: Next, we calculate the number of payments $n$ , which is the number of months in $3$ years. $n = 3 \text{ years} \times 12 \text{ months/year}$ $n = 36 \text{ months}$
Solve for Principal Amount: Now we can rearrange the formula to solve for $P$ : $P = \frac{M}{r(1+r)^n} \times [(1+r)^n - 1]$ We will plug in the values for $M$ , $r$ , and $n$ to find $P$ .
Substitute Values into Formula: Substitute the values into the rearranged formula: $\newline$ $P = \frac{250}{[0.0041666667(1+0.0041666667)^{36}]} \times [(1+0.0041666667)^{36} - 1]$ $\newline$ Now we need to calculate the values inside the brackets and then divide $250$ by that result to find $P$ .
Calculate $(1+r)^n$ : First, calculate $(1+r)^n$ : $(1+0.0041666667)^{36}$ This requires using a calculator to raise $(1+0.0041666667)$ to the power of $36$ .
Calculate Denominator: After calculating the above expression, we get: $(1+0.0041666667)^{36} \approx 1.1940523$
Divide Monthly Payment: Now we calculate the entire denominator of the formula: $\newline$ $\frac{0.0041666667 \times 1.1940523}{1.1940523 - 1}$ $\newline$ This simplifies to: $\newline$ $\frac{0.0049753}{0.1940523}$
Find Maximum Loan Amount: After calculating the denominator, we get: $\frac{0.0049753}{0.1940523} \approx 0.025636$
Find Maximum Loan Amount: After calculating the denominator, we get: $\newline$ $0.0049753 / 0.1940523 \approx 0.025636$ Finally, we divide the monthly payment by this result to find the principal amount ( $P$ ): $\newline$ $P = \frac{250}{0.025636}$
Find Maximum Loan Amount: After calculating the denominator, we get: $\newline$ $0.0049753 / 0.1940523 \approx 0.025636$ Finally, we divide the monthly payment by this result to find the principal amount ( $P$ ): $\newline$ $P = \frac{250}{0.025636}$ After performing the division, we find the maximum loan amount ( $P$ ) that can be afforded: $\newline$ $P \approx 9753.47$