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Answer :
To determine which expression can be used to calculate the monthly payment for a 30-year loan of [tex]$190,000 at an interest rate of 11.4% compounded monthly, we start with the standard formula for calculating monthly loan payments on an annuity (such as a mortgage):
\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]
where:
- \( M \) is the monthly payment,
- \( P \) is the principal loan amount ($[/tex]190,000 in this case),
- [tex]\( r \)[/tex] is the monthly interest rate,
- [tex]\( n \)[/tex] is the total number of payments.
Step-by-step Calculation:
1. Determine the monthly interest rate:
- The annual interest rate is 11.4%.
- To find the monthly interest rate, we divide the annual rate by 12 (the number of months in a year):
[tex]\[ r = \frac{11.4\%}{12} = \frac{0.114}{12} = 0.0095 \][/tex]
2. Calculate the total number of payments:
- For a 30-year loan with monthly payments:
[tex]\[ n = 30 \times 12 = 360 \][/tex]
3. Apply the formula:
- Substitute the values into the formula:
- Numerator: [tex]\( 190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360} \)[/tex]
- Denominator: [tex]\( (1 + 0.0095)^{360} - 1 \)[/tex]
4. Check the expressions:
- Expression A:
[tex]\[ \frac{190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{860} - 1} \][/tex]
- The denominator is incorrect here because it has 860 instead of 360, which affects the calculation.
- Expression B:
[tex]\[ \frac{190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{300} + 1} \][/tex]
- The denominator is incorrect here because it has 300 instead of 360, and also the addition of 1 is not correct according to the formula.
- Expression C:
[tex]\[ \frac{190,000 \cdot 0.0095 \cdot (1 - 0.0095)^{360}}{(1 - 0.0095)^{860} - 1} \][/tex]
- The expression incorrectly uses subtraction instead of addition for the interest rate inside the expressions, making it invalid.
- Expression D:
[tex]\[ \frac{190,000 \cdot 0.0095 \cdot (1 - 0.0095)^{360}}{(1 - 0.0095)^{360} + 1} \][/tex]
- Similar to Expression C, this one also uses subtraction, which is incorrect.
Given the analysis, none of the expressions accurately matches the standard formula for calculating the monthly mortgage payment based on the loan details provided.
\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]
where:
- \( M \) is the monthly payment,
- \( P \) is the principal loan amount ($[/tex]190,000 in this case),
- [tex]\( r \)[/tex] is the monthly interest rate,
- [tex]\( n \)[/tex] is the total number of payments.
Step-by-step Calculation:
1. Determine the monthly interest rate:
- The annual interest rate is 11.4%.
- To find the monthly interest rate, we divide the annual rate by 12 (the number of months in a year):
[tex]\[ r = \frac{11.4\%}{12} = \frac{0.114}{12} = 0.0095 \][/tex]
2. Calculate the total number of payments:
- For a 30-year loan with monthly payments:
[tex]\[ n = 30 \times 12 = 360 \][/tex]
3. Apply the formula:
- Substitute the values into the formula:
- Numerator: [tex]\( 190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360} \)[/tex]
- Denominator: [tex]\( (1 + 0.0095)^{360} - 1 \)[/tex]
4. Check the expressions:
- Expression A:
[tex]\[ \frac{190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{860} - 1} \][/tex]
- The denominator is incorrect here because it has 860 instead of 360, which affects the calculation.
- Expression B:
[tex]\[ \frac{190,000 \cdot 0.0095 \cdot (1 + 0.0095)^{360}}{(1 + 0.0095)^{300} + 1} \][/tex]
- The denominator is incorrect here because it has 300 instead of 360, and also the addition of 1 is not correct according to the formula.
- Expression C:
[tex]\[ \frac{190,000 \cdot 0.0095 \cdot (1 - 0.0095)^{360}}{(1 - 0.0095)^{860} - 1} \][/tex]
- The expression incorrectly uses subtraction instead of addition for the interest rate inside the expressions, making it invalid.
- Expression D:
[tex]\[ \frac{190,000 \cdot 0.0095 \cdot (1 - 0.0095)^{360}}{(1 - 0.0095)^{360} + 1} \][/tex]
- Similar to Expression C, this one also uses subtraction, which is incorrect.
Given the analysis, none of the expressions accurately matches the standard formula for calculating the monthly mortgage payment based on the loan details provided.
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