You can afford a $\$ 250$ per month car payment. You've found a $3$ year loan at $5 \%$ interest. How big of a loan can you afford? $\newline$ $\$\\\newline$ Enter an integer or decimal number [more.].

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Math Problems

Grade 6

Debit cards and credit cards

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Q. You can afford a

\$ 250

per month car payment. You've found a

3

year loan at

5 \%

interest. How big of a loan can you afford?

\newline

\$\\\newline

Enter an integer or decimal number [more.].

Calculate Monthly Interest Rate: To solve this problem, we need to use the formula for an installment loan payment, which is typically represented as $P = \frac{rPV}{1 - (1 + r)^{-n}}$ , where $P$ is the monthly payment, $r$ is the monthly interest rate, $V$ is the loan amount (which we are trying to find), and $n$ is the total number of payments. $\newline$ First, we need to convert the annual interest rate to a monthly interest rate by dividing by $12$ (since there are $12$ months in a year). $\newline$ Annual interest rate: $5\%$ $\newline$ Monthly interest rate: $\frac{5\%}{12}$
Determine Total Number of Payments: Now, let's calculate the monthly interest rate in decimal form. $\newline$ Monthly interest rate (decimal): $\frac{5}{100} \div 12 = 0.00416666667$ $\newline$ Next, we need to determine the total number of payments over the $3$ years. $\newline$ Total number of payments ( $n$ ): $3 \text{ years} \times 12 \text{ months/year} = 36 \text{ months}$
Plug Values into Formula: We can now plug the values into the formula to solve for $V$ , the loan amount. $P = \$250$ per month $r = 0.00416666667$ per month $n = 36$ monthsUsing the formula: $\$250 = \frac{(0.00416666667 \times V)}{1 - (1 + 0.00416666667)^{-36}}$
Isolate V: To isolate V, we need to multiply both sides of the equation by the denominator on the right side and then divide by the monthly interest rate. $\newline$ $\$250 \times (1 - (1 + 0.00416666667)^{-36}) = 0.00416666667 \times V$
Calculate Value of $(1 - (1 + 0.00416666667)^{-36})$ : Now we need to calculate the value of $(1 - (1 + 0.00416666667)^{-36})$ . $(1 + 0.00416666667)^{-36} = \frac{1}{(1 + 0.00416666667)^{36}}$ We calculate $(1 + 0.00416666667)^{36}$ using a calculator. $(1 + 0.00416666667)^{36} \approx 1.157625687$ Now, we take the reciprocal of this value. $\frac{1}{1.157625687} \approx 0.8639053255$ Then, we subtract this value from $1$ . $1 - 0.8639053255 \approx 0.1360946745$
Solve for V: We now have all the values needed to solve for V. $\newline$ $\$250 \times 0.1360946745 = 0.00416666667 \times V$ $\newline$ $\$34.023668625 = 0.00416666667 \times V$ $\newline$ To find V, we divide both sides by the monthly interest rate. $\newline$ $V = \frac{\$34.023668625}{0.00416666667}$ $\newline$ $V \approx \$8164.80$