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Sagot :
Given the problem, we need to identify which of the provided expressions correctly represents the formula for calculating the monthly payment on a 30-year loan for \[tex]$190,000 at an annual interest rate of 11.4%, compounded monthly.
To start, let's outline the standard formula for calculating the monthly payment \( M \) of a fixed-rate mortgage loan:
\[ M = \frac{P \cdot r(1 + r)^n}{(1 + r)^n - 1} \]
where:
- \( P \) is the loan principal (the amount borrowed)
- \( r \) is the monthly interest rate
- \( n \) is the total number of payments (months)
The given parameters are:
- Principal \( P = \$[/tex]190,000 \)
- Annual interest rate [tex]\( \text{annual\_interest\_rate} = 11.4\% \)[/tex] (or 0.114)
- Loan period [tex]\( \text{years} = 30 \)[/tex]
### Step-by-step solution:
1. Convert the annual interest rate to a monthly interest rate:
[tex]\[ \text{monthly\_interest\_rate} = \frac{11.4\%}{12} = \frac{0.114}{12} = 0.0095 \][/tex]
2. Calculate the number of months (payments):
[tex]\[ \text{months} = 30 \times 12 = 360 \][/tex]
3. Identify the correct formula by comparing it with the standard formula:
- The monthly interest rate [tex]\( r = 0.0095 \)[/tex]
- The number of payments [tex]\( n = 360 \)[/tex]
- The principal [tex]\( P = \$190,000 \)[/tex]
Substituting these values into the standard formula, we get:
[tex]\[ M = \frac{190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} - 1} \][/tex]
By analyzing the given choices:
\- Option A: [tex]\(\frac{\$ 190,000 \cdot 0.0095(1-0.0095)^{360}}{(1-0.0095)^{360}-1}\)[/tex] contains errors involving incorrect interest rate adjustments.
\- Option B: [tex]\(\frac{\$ 190.000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{300}+1}\)[/tex] incorrectly adjusts the exponent on the denominator.
\- Option C: [tex]\(\frac{\$ 190.000 \cdot 0.0095(1-0.0095)^{360}}{(1-0.0095)^{300}+1}\)[/tex] also contains incorrect interest rate adjustments and exponents.
\- Option D: [tex]\(\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}\)[/tex] correctly uses the loan parameters and the formula structure.
Thus, the correct expression that can be used to calculate the monthly payment for the specified loan is:
[tex]\[ \boxed{\text{D. } \frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}} \][/tex]
- Annual interest rate [tex]\( \text{annual\_interest\_rate} = 11.4\% \)[/tex] (or 0.114)
- Loan period [tex]\( \text{years} = 30 \)[/tex]
### Step-by-step solution:
1. Convert the annual interest rate to a monthly interest rate:
[tex]\[ \text{monthly\_interest\_rate} = \frac{11.4\%}{12} = \frac{0.114}{12} = 0.0095 \][/tex]
2. Calculate the number of months (payments):
[tex]\[ \text{months} = 30 \times 12 = 360 \][/tex]
3. Identify the correct formula by comparing it with the standard formula:
- The monthly interest rate [tex]\( r = 0.0095 \)[/tex]
- The number of payments [tex]\( n = 360 \)[/tex]
- The principal [tex]\( P = \$190,000 \)[/tex]
Substituting these values into the standard formula, we get:
[tex]\[ M = \frac{190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{360} - 1} \][/tex]
By analyzing the given choices:
\- Option A: [tex]\(\frac{\$ 190,000 \cdot 0.0095(1-0.0095)^{360}}{(1-0.0095)^{360}-1}\)[/tex] contains errors involving incorrect interest rate adjustments.
\- Option B: [tex]\(\frac{\$ 190.000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{300}+1}\)[/tex] incorrectly adjusts the exponent on the denominator.
\- Option C: [tex]\(\frac{\$ 190.000 \cdot 0.0095(1-0.0095)^{360}}{(1-0.0095)^{300}+1}\)[/tex] also contains incorrect interest rate adjustments and exponents.
\- Option D: [tex]\(\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}\)[/tex] correctly uses the loan parameters and the formula structure.
Thus, the correct expression that can be used to calculate the monthly payment for the specified loan is:
[tex]\[ \boxed{\text{D. } \frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}} \][/tex]
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