To solve this problem, we need to calculate the total payments for both the adjustable-rate mortgage and the fixed-rate mortgage, and then find the difference between the two. Here's a step-by-step solution:
### Adjustable-Rate Mortgage Calculation
1. Payments from Year 1 to Year 5:
- Monthly Payment: \[tex]$2,506.43
- Number of Years: 5
- Total Payments for 5 Years:
\[
\$[/tex]2,506.43 \times 12 \times 5 = \[tex]$150,385.80
\]
2. Payments from Year 6 to Year 15:
- Monthly Payment: \$[/tex]3,059.46
- Number of Years: 10
- Total Payments for 10 Years:
[tex]\[
\$3,059.46 \times 12 \times 10 = \$367,135.20
\][/tex]
3. Payments from Year 16 to Year 25:
- Monthly Payment: \[tex]$3,464.78
- Number of Years: 10
- Total Payments for 10 Years:
\[
\$[/tex]3,464.78 \times 12 \times 10 = \[tex]$415,773.60
\]
4. Payments from Year 26 to Year 30:
- Monthly Payment: \$[/tex]3,630.65
- Number of Years: 5
- Total Payments for 5 Years:
[tex]\[
\$3,630.65 \times 12 \times 5 = \$181,831.00
\][/tex]
5. Total Payments for Adjustable-Rate Mortgage:
- Sum of all the payments:
[tex]\[
\$150,385.80 + \$367,135.20 + \$415,773.60 + \$181,831.00 = \$1,151,125.60
\][/tex]
### Fixed-Rate Mortgage Calculation
The total cost for the fixed-rate mortgage is provided as \[tex]$878,626.
### Difference in Total Costs
Now we calculate the difference in the total payments between the adjustable-rate mortgage and the fixed-rate mortgage:
\[
\$[/tex]1,151,133.60 - \[tex]$878,626 = \$[/tex]272,507.60
\]
### Conclusion
The difference in the total cost between the adjustable-rate mortgage and the fixed-rate mortgage is approximately \$272,508 when rounded to the nearest dollar.
