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
(1 point)
Responses
$85.15
$85.15
$115.40
$115.40
$125.72
$125.72
$138.48
Use the formula, M=Pm(1+m)na(1+m)na−1
(1 point)
Responses
$85.15
$85.15
$115.40
$115.40
$125.72
$125.72
$138.48
Answers
GPT-4o mini
To calculate the monthly payment \( M \) using the formula you provided, we first need to define the variables:
- \( P \) = principal amount (loan amount) = $2,500
- \( r \) = annual interest rate = 10% = 0.10
- \( n \) = number of payments per year = 12 (monthly payments)
- \( t \) = number of years = 2
1. **Calculate the monthly interest rate \( m \)**:
\[
m = \frac{r}{n} = \frac{0.10}{12} = 0.008333
\]
2. **Calculate the total number of payments \( N \)**:
\[
N = n \cdot t = 12 \cdot 2 = 24
\]
3. **Use the formula**:
\[
M = P \cdot \frac{m(1+m)^{N}}{(1+m)^{N}-1}
\]
Plugging in the values:
\[
M = 2500 \cdot \frac{0.008333(1 + 0.008333)^{24}}{(1 + 0.008333)^{24} - 1}
\]
4. **Calculate \( (1 + m)^{N} \)**:
\[
(1 + 0.008333)^{24} \approx (1.008333)^{24} \approx 1.22039
\]
5. **Now calculate the monthly payment \( M \)**:
\[
M \approx 2500 \cdot \frac{0.008333 \cdot 1.22039}{1.22039 - 1}
\]
\[
M \approx 2500 \cdot \frac{0.01018325}{0.22039}
\]
\[
M \approx 2500 \cdot 0.046223 = 115.56
\]
Thus, the calculated monthly payment of approximately **$115.56** is not an exact match for one of your options. However, the closest match based on rounding could be **$115.40**. The correct answer from the options provided would be:
**$115.40**
- \( P \) = principal amount (loan amount) = $2,500
- \( r \) = annual interest rate = 10% = 0.10
- \( n \) = number of payments per year = 12 (monthly payments)
- \( t \) = number of years = 2
1. **Calculate the monthly interest rate \( m \)**:
\[
m = \frac{r}{n} = \frac{0.10}{12} = 0.008333
\]
2. **Calculate the total number of payments \( N \)**:
\[
N = n \cdot t = 12 \cdot 2 = 24
\]
3. **Use the formula**:
\[
M = P \cdot \frac{m(1+m)^{N}}{(1+m)^{N}-1}
\]
Plugging in the values:
\[
M = 2500 \cdot \frac{0.008333(1 + 0.008333)^{24}}{(1 + 0.008333)^{24} - 1}
\]
4. **Calculate \( (1 + m)^{N} \)**:
\[
(1 + 0.008333)^{24} \approx (1.008333)^{24} \approx 1.22039
\]
5. **Now calculate the monthly payment \( M \)**:
\[
M \approx 2500 \cdot \frac{0.008333 \cdot 1.22039}{1.22039 - 1}
\]
\[
M \approx 2500 \cdot \frac{0.01018325}{0.22039}
\]
\[
M \approx 2500 \cdot 0.046223 = 115.56
\]
Thus, the calculated monthly payment of approximately **$115.56** is not an exact match for one of your options. However, the closest match based on rounding could be **$115.40**. The correct answer from the options provided would be:
**$115.40**