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.
Answers below:
$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.
Answers below:
$1,021.80
$2,043.60
$1,384.80
$2,769.60
Answers
GPT-4o mini
To calculate the monthly payment (M) for a loan using the formula \( M = \frac{P \cdot m(1 + m)^{na}}{(1 + m)^{na} - 1} \), we first need to define the variables:
- **P** = loan amount = $2,500
- **r** = annual interest rate = 10% = 0.10
- **n** = number of payments per year = 12 (monthly payments)
- **a** = number of years = 2
Now calculate the monthly interest rate \( m \):
\[
m = \frac{r}{n} = \frac{0.10}{12} = \frac{0.10}{12} = 0.0083333\ldots
\]
Next, calculate the total number of payments \( na \):
\[
na = n \times a = 12 \times 2 = 24
\]
Now substitute \( P \), \( m \), and \( na \) into the formula:
\[
M = \frac{2500 \cdot 0.0083333 \cdot (1 + 0.0083333)^{24}}{(1 + 0.0083333)^{24} - 1}
\]
Calculate \( (1 + m)^{na} \):
\[
(1 + 0.0083333)^{24} \approx (1.0083333)^{24} \approx 1.220396
\]
Now plug this value back into the formula:
\[
M = \frac{2500 \cdot 0.0083333 \cdot 1.220396}{1.220396 - 1}
\]
Calculate the numerator:
\[
2500 \cdot 0.0083333 \cdot 1.220396 \approx 25.004151
\]
Calculate the denominator:
\[
1.220396 - 1 = 0.220396
\]
Now divide to get \( M \):
\[
M = \frac{25.004151}{0.220396} \approx 113.36
\]
Thus, the monthly payment is approximately **$113.36**, which is not one of the provided options.
Let's recalculate step by step to ensure accuracy:
1. Calculate \( (1 + 0.0083333)^{24} \):
\[
(1.0083333)^{24} \approx 1.220396
\]
2. Calculate \( 2500 \cdot 0.0083333 \):
\[
2500 \cdot 0.0083333 = 20.8333
\]
3. The numerator becomes:
\[
20.8333 \cdot 1.220396 = 25.392005
\]
4. The denominator:
\[
1.220396 - 1 = 0.220396
\]
5. Final calculation:
\[
M = \frac{25.392005}{0.220396} \approx 115.10
\]
If calculations do not align with your provided answers, consider ensuring parameters defined correctly (loan amount, interest, payment period).
For clarity, comparing methodically with an online loan calculator or financial software for correctness is always beneficial.
Given $2,500 loan over two years at 10% may ideally amount closer to $1,384.80 or similar relative to loan durations, payments give error identically.
This suggests either formula step mistake assumed with multiples originally you provided thus confirming if estimate options wanted correctly processed here.
- **P** = loan amount = $2,500
- **r** = annual interest rate = 10% = 0.10
- **n** = number of payments per year = 12 (monthly payments)
- **a** = number of years = 2
Now calculate the monthly interest rate \( m \):
\[
m = \frac{r}{n} = \frac{0.10}{12} = \frac{0.10}{12} = 0.0083333\ldots
\]
Next, calculate the total number of payments \( na \):
\[
na = n \times a = 12 \times 2 = 24
\]
Now substitute \( P \), \( m \), and \( na \) into the formula:
\[
M = \frac{2500 \cdot 0.0083333 \cdot (1 + 0.0083333)^{24}}{(1 + 0.0083333)^{24} - 1}
\]
Calculate \( (1 + m)^{na} \):
\[
(1 + 0.0083333)^{24} \approx (1.0083333)^{24} \approx 1.220396
\]
Now plug this value back into the formula:
\[
M = \frac{2500 \cdot 0.0083333 \cdot 1.220396}{1.220396 - 1}
\]
Calculate the numerator:
\[
2500 \cdot 0.0083333 \cdot 1.220396 \approx 25.004151
\]
Calculate the denominator:
\[
1.220396 - 1 = 0.220396
\]
Now divide to get \( M \):
\[
M = \frac{25.004151}{0.220396} \approx 113.36
\]
Thus, the monthly payment is approximately **$113.36**, which is not one of the provided options.
Let's recalculate step by step to ensure accuracy:
1. Calculate \( (1 + 0.0083333)^{24} \):
\[
(1.0083333)^{24} \approx 1.220396
\]
2. Calculate \( 2500 \cdot 0.0083333 \):
\[
2500 \cdot 0.0083333 = 20.8333
\]
3. The numerator becomes:
\[
20.8333 \cdot 1.220396 = 25.392005
\]
4. The denominator:
\[
1.220396 - 1 = 0.220396
\]
5. Final calculation:
\[
M = \frac{25.392005}{0.220396} \approx 115.10
\]
If calculations do not align with your provided answers, consider ensuring parameters defined correctly (loan amount, interest, payment period).
For clarity, comparing methodically with an online loan calculator or financial software for correctness is always beneficial.
Given $2,500 loan over two years at 10% may ideally amount closer to $1,384.80 or similar relative to loan durations, payments give error identically.
This suggests either formula step mistake assumed with multiples originally you provided thus confirming if estimate options wanted correctly processed here.