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Answer :
- Calculate the monthly interest rate: $\frac{0.114}{12} = 0.0095$.
- Calculate the total number of payments: $30 \times 12 = 360$.
- Apply the monthly payment formula: $M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$.
- Substitute the values to find the correct expression: $\boxed{\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}}$.
### Explanation
1. Understanding the Problem
We are asked to identify the correct formula for calculating the monthly payment of a loan. The principal loan amount is $190,000, the annual interest rate is 11.4%, and the loan term is 30 years. The interest is compounded monthly.
2. Calculating Monthly Interest Rate
First, we need to calculate the monthly interest rate. We divide the annual interest rate by 12:$$\frac{0.114}{12} = 0.0095$$So the monthly interest rate is 0.0095.
3. Calculating Total Number of Payments
Next, we calculate the total number of payments. Since the loan term is 30 years and payments are made monthly, the total number of payments is:$$30 \times 12 = 360$$So there are 360 payments in total.
4. Stating the Monthly Payment Formula
The standard formula for the monthly payment (M) of a loan is:$$M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$$where P is the principal loan amount, r is the monthly interest rate, and n is the number of payments.
5. Substituting Values into the Formula
Now, we substitute the given values into the formula:$$M = 190000 \cdot \frac{0.0095(1+0.0095)^{360}}{(1+0.0095)^{360} - 1}$$
6. Identifying the Correct Option
Comparing the derived formula with the given options, we can see that option C matches our result:$$\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}$$
7. Final Answer
Therefore, the correct expression to calculate the monthly payment for the loan is option C.
### Examples
Understanding loan payments is crucial in personal finance. For instance, when buying a house, a car, or even funding education, loans are often involved. Knowing how the monthly payment is calculated helps in budgeting and comparing different loan offers to make informed financial decisions. This formula allows you to see how much of each payment goes towards interest and principal, aiding in long-term financial planning.
- Calculate the total number of payments: $30 \times 12 = 360$.
- Apply the monthly payment formula: $M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$.
- Substitute the values to find the correct expression: $\boxed{\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}}$.
### Explanation
1. Understanding the Problem
We are asked to identify the correct formula for calculating the monthly payment of a loan. The principal loan amount is $190,000, the annual interest rate is 11.4%, and the loan term is 30 years. The interest is compounded monthly.
2. Calculating Monthly Interest Rate
First, we need to calculate the monthly interest rate. We divide the annual interest rate by 12:$$\frac{0.114}{12} = 0.0095$$So the monthly interest rate is 0.0095.
3. Calculating Total Number of Payments
Next, we calculate the total number of payments. Since the loan term is 30 years and payments are made monthly, the total number of payments is:$$30 \times 12 = 360$$So there are 360 payments in total.
4. Stating the Monthly Payment Formula
The standard formula for the monthly payment (M) of a loan is:$$M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1}$$where P is the principal loan amount, r is the monthly interest rate, and n is the number of payments.
5. Substituting Values into the Formula
Now, we substitute the given values into the formula:$$M = 190000 \cdot \frac{0.0095(1+0.0095)^{360}}{(1+0.0095)^{360} - 1}$$
6. Identifying the Correct Option
Comparing the derived formula with the given options, we can see that option C matches our result:$$\frac{\$ 190,000 \cdot 0.0095(1+0.0095)^{360}}{(1+0.0095)^{360}-1}$$
7. Final Answer
Therefore, the correct expression to calculate the monthly payment for the loan is option C.
### Examples
Understanding loan payments is crucial in personal finance. For instance, when buying a house, a car, or even funding education, loans are often involved. Knowing how the monthly payment is calculated helps in budgeting and comparing different loan offers to make informed financial decisions. This formula allows you to see how much of each payment goes towards interest and principal, aiding in long-term financial planning.
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