Question
Calculate the monthly payment for a $2,500 loan at 10% annual interest for two years.
Use the formula, M = Pm(1+m)^na / (1+m)^na-1 as needed.
(1 point0
$1,021.80
$2,043.60
$1,384.80
$2,769.60
Use the formula, M = Pm(1+m)^na / (1+m)^na-1 as needed.
(1 point0
$1,021.80
$2,043.60
$1,384.80
$2,769.60
Answers
GPT-4o mini
To calculate the monthly payment for a loan using the formula \( M = \frac{Pm(1+m)^{na}}{(1+m)^{na}-1} \), we need to define the variables:
- \( P \) = principal loan amount = $2,500
- \( m \) = monthly interest rate = annual interest rate / 12 = \( \frac{10\%}{12} = 0.10/12 \)
- \( na \) = total number of payments (months) = 2 years * 12 months/year = 24 months
First, we calculate the monthly interest rate:
\[
m = \frac{10}{100 \times 12} = \frac{0.10}{12} \approx 0.0083333
\]
Next, we calculate \( (1 + m)^{na} \):
\[
(1 + 0.0083333)^{24} \approx (1.0083333)^{24} \approx 1.220386 (using a calculator)
\]
Now, substitute \( P \), \( m \), and \( na \) into the payment formula:
\[
M = \frac{2500 \times 0.0083333 \times 1.220386}{1.220386 - 1}
\]
Calculate the numerator:
\[
2500 \times 0.0083333 \times 1.220386 \approx 25.5
\]
Now compute the denominator:
\[
1.220386 - 1 \approx 0.220386
\]
Now substitute these values back into the formula:
\[
M \approx \frac{25.5}{0.220386} \approx 115.38
\]
Thus, the monthly payment is approximately **$115.38**.
Since this calculation doesn’t match any of the options, let's recalculate the whole formula step-by-step to ensure accuracy:
1. Compute:
\(\ (1 + m)^{na} = (1.0083333)^{24} \approx 1.21964\)
2. Now we recalculate:
\[
M = \frac{2500 \times 0.0083333 \times 1.21964}{1.21964 - 1}
\]
\[
= \frac{2500 \times 0.0083333 \times 1.21964}{0.21964}
\]
\[
= \frac{25.5}{0.21964} \approx 116.74
\]
Our answer still does not match the options. Clearly, there may be a misunderstanding with the provided options.
Let's verify using another method like an online loan calculator, or if I missed something based on how they computed theirs, maybe they had a slightly different consideration. However, please confirm the problem details or context where the multiple-choice options were provided.
- \( P \) = principal loan amount = $2,500
- \( m \) = monthly interest rate = annual interest rate / 12 = \( \frac{10\%}{12} = 0.10/12 \)
- \( na \) = total number of payments (months) = 2 years * 12 months/year = 24 months
First, we calculate the monthly interest rate:
\[
m = \frac{10}{100 \times 12} = \frac{0.10}{12} \approx 0.0083333
\]
Next, we calculate \( (1 + m)^{na} \):
\[
(1 + 0.0083333)^{24} \approx (1.0083333)^{24} \approx 1.220386 (using a calculator)
\]
Now, substitute \( P \), \( m \), and \( na \) into the payment formula:
\[
M = \frac{2500 \times 0.0083333 \times 1.220386}{1.220386 - 1}
\]
Calculate the numerator:
\[
2500 \times 0.0083333 \times 1.220386 \approx 25.5
\]
Now compute the denominator:
\[
1.220386 - 1 \approx 0.220386
\]
Now substitute these values back into the formula:
\[
M \approx \frac{25.5}{0.220386} \approx 115.38
\]
Thus, the monthly payment is approximately **$115.38**.
Since this calculation doesn’t match any of the options, let's recalculate the whole formula step-by-step to ensure accuracy:
1. Compute:
\(\ (1 + m)^{na} = (1.0083333)^{24} \approx 1.21964\)
2. Now we recalculate:
\[
M = \frac{2500 \times 0.0083333 \times 1.21964}{1.21964 - 1}
\]
\[
= \frac{2500 \times 0.0083333 \times 1.21964}{0.21964}
\]
\[
= \frac{25.5}{0.21964} \approx 116.74
\]
Our answer still does not match the options. Clearly, there may be a misunderstanding with the provided options.
Let's verify using another method like an online loan calculator, or if I missed something based on how they computed theirs, maybe they had a slightly different consideration. However, please confirm the problem details or context where the multiple-choice options were provided.