Question
roy took out a 30 year loan for $155,000 at an 5.5%, compounded monthly. approximately what would be the total cost of his loan if he paid it off 4 years early?
A. $37,842.06
B. $316,825.20
C. $312,423.90
D. $274,581.84
A. $37,842.06
B. $316,825.20
C. $312,423.90
D. $274,581.84
Answers
GPT-4o mini
To determine the total cost of Roy's loan if he pays it off 4 years early, we first need to calculate his monthly payment for the original loan, and then find the total cost after 26 years of payments instead of the full 30 years.
### Step 1: Calculate the monthly payment
The Monthly Payment (M) for a fixed-rate mortgage can be calculated using the formula:
\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( P \) = loan principal ($155,000)
- \( r \) = monthly interest rate (annual interest rate / 12 months)
- \( n \) = total number of payments (loan term in months)
Given:
- Annual interest rate = 5.5% = 0.055
- Monthly interest rate \( r = \frac{0.055}{12} \approx 0.0045833 \)
- Total number of payments for a 30-year loan \( n = 30 \times 12 = 360 \)
Plugging in the values:
\[
M = 155,000 \frac{0.0045833(1 + 0.0045833)^{360}}{(1 + 0.0045833)^{360} - 1}
\]
Calculating \( (1 + 0.0045833)^{360} \):
\[
(1 + 0.0045833)^{360} \approx (1.0045833)^{360} \approx 5.095157
\]
Now we can calculate \( M \):
\[
M = 155,000 \frac{0.0045833 \cdot 5.095157}{5.095157 - 1}
\]
Calculating the values:
\[
M \approx 155,000 \frac{0.023377576}{4.095157} \approx 155,000 \cdot 0.005716576 \approx 886.73
\]
So, the monthly payment is approximately **$886.73**.
### Step 2: Calculate the total payments over 30 years
Total payments over 30 years are:
\[
\text{Total Payments} = M \times n = 886.73 \times 360 \approx 319,828.80
\]
### Step 3: Calculate total payments if paid off after 26 years
If Roy pays off the loan after 26 years:
\[
n' = 26 \times 12 = 312
\]
The total payments in this case would be:
\[
\text{Total Payments after 26 years} = M \times n' = 886.73 \times 312 \approx 276,724.76
\]
### Step 4: Approximate total cost of the loan
Since none of the provided options directly match the calculated amount of **$276,724.76**, we can round and compare it to the options provided:
- A. $37,842.06
- B. $316,825.20
- C. $312,423.90
- D. $274,581.84
### Conclusion
The closest option to our estimate of **$276,724.76** would be:
**D. $274,581.84**
Thus, the total cost of Roy's loan if he paid it off 4 years early is most accurately represented by **option D**.
### Step 1: Calculate the monthly payment
The Monthly Payment (M) for a fixed-rate mortgage can be calculated using the formula:
\[
M = P \frac{r(1 + r)^n}{(1 + r)^n - 1}
\]
Where:
- \( P \) = loan principal ($155,000)
- \( r \) = monthly interest rate (annual interest rate / 12 months)
- \( n \) = total number of payments (loan term in months)
Given:
- Annual interest rate = 5.5% = 0.055
- Monthly interest rate \( r = \frac{0.055}{12} \approx 0.0045833 \)
- Total number of payments for a 30-year loan \( n = 30 \times 12 = 360 \)
Plugging in the values:
\[
M = 155,000 \frac{0.0045833(1 + 0.0045833)^{360}}{(1 + 0.0045833)^{360} - 1}
\]
Calculating \( (1 + 0.0045833)^{360} \):
\[
(1 + 0.0045833)^{360} \approx (1.0045833)^{360} \approx 5.095157
\]
Now we can calculate \( M \):
\[
M = 155,000 \frac{0.0045833 \cdot 5.095157}{5.095157 - 1}
\]
Calculating the values:
\[
M \approx 155,000 \frac{0.023377576}{4.095157} \approx 155,000 \cdot 0.005716576 \approx 886.73
\]
So, the monthly payment is approximately **$886.73**.
### Step 2: Calculate the total payments over 30 years
Total payments over 30 years are:
\[
\text{Total Payments} = M \times n = 886.73 \times 360 \approx 319,828.80
\]
### Step 3: Calculate total payments if paid off after 26 years
If Roy pays off the loan after 26 years:
\[
n' = 26 \times 12 = 312
\]
The total payments in this case would be:
\[
\text{Total Payments after 26 years} = M \times n' = 886.73 \times 312 \approx 276,724.76
\]
### Step 4: Approximate total cost of the loan
Since none of the provided options directly match the calculated amount of **$276,724.76**, we can round and compare it to the options provided:
- A. $37,842.06
- B. $316,825.20
- C. $312,423.90
- D. $274,581.84
### Conclusion
The closest option to our estimate of **$276,724.76** would be:
**D. $274,581.84**
Thus, the total cost of Roy's loan if he paid it off 4 years early is most accurately represented by **option D**.