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A [tex]$\$[/tex] 525,000[tex]$ adjustable-rate mortgage is expected to have the following payments:

\begin{tabular}{|c|l|l|}
\hline
Year & Interest Rate & Monthly Payment \\
\hline
$[/tex]1-5[tex]$ & $[/tex]4 \%[tex]$ & $[/tex]\[tex]$ 2,506.43$[/tex] \\
\hline
[tex]$6-15$[/tex] & [tex]$6 \%$[/tex] & [tex]$\$[/tex] 3,059.46[tex]$ \\
\hline
$[/tex]16-25[tex]$ & $[/tex]8 \%[tex]$ & $[/tex]\[tex]$ 3,464.78$[/tex] \\
\hline
[tex]$26-30$[/tex] & [tex]$10 \%$[/tex] & [tex]$\$[/tex] 3,630.65[tex]$ \\
\hline
\end{tabular}

A fixed-rate mortgage in the same amount is offered with an interest rate of $[/tex]4.65 \%[tex]$. What is the difference in the total cost between the two mortgages, rounded to the nearest dollar?

Note: A spreadsheet was used to calculate the correct answer. Your answer may vary slightly depending on the technology used.

A. $[/tex]\[tex]$ 176,580$[/tex]
B. [tex]$\$[/tex] 878,626$

Sagot :

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.