You can afford a $\$ 300$ per month car payment. You've found a $5$ year loan at $6 \%$ interest. How big of a loan can you afford? $\newline$ $\$$

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Q. You can afford a

\$ 300

per month car payment. You've found a

5

year loan at

6 \%

interest. How big of a loan can you afford?

\newline

\$

Understand the problem: Understand the problem. $\newline$ We need to determine the size of the loan that can be afforded with a monthly payment of $\$300$ over a period of $5$ years at an annual interest rate of $6\%$ .
Convert annual interest rate: Convert the annual interest rate to a monthly interest rate. $\newline$ The annual interest rate is $6\%$ , so the monthly interest rate is $6\%$ divided by $12$ months. $\newline$ Monthly interest rate = $\frac{6\%}{12} = 0.5\%$ per month
Convert monthly interest rate: Convert the monthly interest rate from a percentage to a decimal. $0.5\%$ as a decimal is $0.005$ .
Calculate number of payments: Calculate the number of monthly payments over the life of the loan. $\newline$ Since the loan is for $5$ years and there are $12$ months in a year, the total number of payments is $5$ years $\times$ $12$ months/year. $\newline$ Total number of payments = $5 \times 12 = 60$ payments
Use present value formula: Use the formula for the present value of an annuity to calculate the loan amount. $\newline$ The formula is $P = (\text{PMT} \times (1 - (1 + r)^{-n})) / r$ , where: $\newline$ $P$ is the loan amount (present value), $\newline$ $\text{PMT}$ is the monthly payment ( $\$300$ ), $\newline$ $r$ is the monthly interest rate ( $0.005$ ), $\newline$ $n$ is the total number of payments ( $60$ ).
Calculate loan amount: Plug the values into the formula and calculate the loan amount. $\newline$ $P = (\$(300) \times (1 - (1 + 0.005)^{-60})) / 0.005$
Calculate value inside parentheses: Calculate the value inside the parentheses. $\newline$ $(1 + 0.005)^{-60} = (1.005)^{-60}$
Evaluate the exponent: Evaluate the exponent. $\newline$ $(1.005)^{-60} \approx 0.747258$
Continue calculation: Continue with the calculation. $\newline$ $P = (\$(300) \times (1 - 0.747258)) / 0.005$
Subtract from $1$ : Subtract $0.747258$ from $1$ . $\newline$ $1 - 0.747258 = 0.252742$
Multiply by monthly payment: Multiply the result by the monthly payment. $\$300 \times 0.252742 \approx \$75.8226$
Divide by monthly interest rate: Divide by the monthly interest rate. $\$75.8226 / 0.005 \approx \$15164.52$