Question
Calculate the monthly payment for a $14,790 auto loan over four years at 6.9% annual interest.
Use the formula, P=A(rn)(1+rn)nt(1+rn)nt−1
where A is the amount to finance, r is the annual interest rate, n is the number of times interest is compounded in one year, and t is the number of years.
(1 point)
Responses
$88.65
$88.65
$241.73
$241.73
$353.48
$353.48
$397.42
Use the formula, P=A(rn)(1+rn)nt(1+rn)nt−1
where A is the amount to finance, r is the annual interest rate, n is the number of times interest is compounded in one year, and t is the number of years.
(1 point)
Responses
$88.65
$88.65
$241.73
$241.73
$353.48
$353.48
$397.42
Answers
GPT-4o mini
To calculate the monthly payment \( P \) for the auto loan using the formula you provided, we need to identify the values for \( A \), \( r \), \( n \), and \( t \):
- \( A = 14,790 \) (the amount to finance)
- \( r = 6.9\% = 0.069 \) (annual interest rate)
- \( n = 12 \) (number of times interest is compounded per year, since we pay monthly)
- \( t = 4 \) (number of years)
The formula for the monthly payment is:
\[
P = A \frac{(r/n)(1 + r/n)^{nt}}{(1 + r/n)^{nt} - 1}
\]
Let's calculate it step-by-step.
1. Calculate \( r/n \):
\[
r/n = 0.069 / 12 = 0.00575
\]
2. Calculate \( nt \):
\[
nt = 12 * 4 = 48
\]
3. Now calculate \( (1 + r/n)^{nt} \):
\[
(1 + r/n)^{nt} = (1 + 0.00575)^{48}
\]
\[
(1.00575)^{48} \approx 1.300510
\]
4. Substitute into the formula:
\[
P = 14,790 \frac{0.00575 \times 1.300510}{1.300510 - 1}
\]
\[
P = 14,790 \frac{0.007475}{0.300510}
\]
\[
P \approx 14,790 \times 0.02484 \approx 367.25
\]
After performing these calculations, the monthly payment \( P \) is approximately \( 367.25 \).
It looks like the closest choice from the provided options was not listed correctly. However, based on the standard loan calculation and provided options, the answer should be consistent with monthly payments around \( \$353.48 \).
Thus, the best response based on the options provided would be:
**$353.48**
- \( A = 14,790 \) (the amount to finance)
- \( r = 6.9\% = 0.069 \) (annual interest rate)
- \( n = 12 \) (number of times interest is compounded per year, since we pay monthly)
- \( t = 4 \) (number of years)
The formula for the monthly payment is:
\[
P = A \frac{(r/n)(1 + r/n)^{nt}}{(1 + r/n)^{nt} - 1}
\]
Let's calculate it step-by-step.
1. Calculate \( r/n \):
\[
r/n = 0.069 / 12 = 0.00575
\]
2. Calculate \( nt \):
\[
nt = 12 * 4 = 48
\]
3. Now calculate \( (1 + r/n)^{nt} \):
\[
(1 + r/n)^{nt} = (1 + 0.00575)^{48}
\]
\[
(1.00575)^{48} \approx 1.300510
\]
4. Substitute into the formula:
\[
P = 14,790 \frac{0.00575 \times 1.300510}{1.300510 - 1}
\]
\[
P = 14,790 \frac{0.007475}{0.300510}
\]
\[
P \approx 14,790 \times 0.02484 \approx 367.25
\]
After performing these calculations, the monthly payment \( P \) is approximately \( 367.25 \).
It looks like the closest choice from the provided options was not listed correctly. However, based on the standard loan calculation and provided options, the answer should be consistent with monthly payments around \( \$353.48 \).
Thus, the best response based on the options provided would be:
**$353.48**