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Find the mortgage balance after the first three payments on a 30-year [tex]$100,000$[/tex] mortgage that was financed at an APR of [tex]$4.25\%$[/tex] and has a monthly payment of [tex]$491.94$[/tex].

\begin{tabular}{|l|l|l|l|}
\hline
\textbf{Payment Number} & \textbf{Interest Payment} & \textbf{Principal Payment} & \textbf{Mortgage Balance} \\
\hline
1 & [tex]$354.17$[/tex] & [tex]$137.77$[/tex] & Balance 1 \\
\hline
2 & [tex]$353.68$[/tex] & [tex]$138.26$[/tex] & Balance 2 \\
\hline
3 & [tex]$353.19$[/tex] & [tex]$138.75$[/tex] & Balance 3 \\
\hline
\end{tabular}


Sagot :

Certainly! Let's go through the detailed steps to find the mortgage balance after each of the first three monthly payments.

Mortgage Details:
- Initial mortgage amount: \[tex]$100,000 - Annual Percentage Rate (APR): 4.25% - Monthly payment amount: \$[/tex]491.94

Payment Details:
- Payment Number 1:
- Interest Payment: \[tex]$354.17 - Principal Payment: \$[/tex]137.77

- Payment Number 2:
- Interest Payment: \[tex]$353.68 - Principal Payment: \$[/tex]138.26

- Payment Number 3:
- Interest Payment: \[tex]$353.19 - Principal Payment: \$[/tex]138.75

### Step-by-Step Calculation:

1. Initial Mortgage Balance:
[tex]\[ \text{Initial Balance} = \$100,000 \][/tex]

2. After First Payment:
- Principal Payment for first month: \[tex]$137.77 \[ \text{Balance after first payment} = \$[/tex]100,000 - \[tex]$137.77 = \$[/tex]99,862.23
\]

3. After Second Payment:
- Principal Payment for second month: \[tex]$138.26 \[ \text{Balance after second payment} = \$[/tex]99,862.23 - \[tex]$138.26 = \$[/tex]99,723.97
\]

4. After Third Payment:
- Principal Payment for third month: \[tex]$138.75 \[ \text{Balance after third payment} = \$[/tex]99,723.97 - \[tex]$138.75 = \$[/tex]99,585.22
\]

### Summarized Results:

- Balance after the 1st payment: \[tex]$99,862.23 - Balance after the 2nd payment: \$[/tex]99,723.97
- Balance after the 3rd payment: \[tex]$99,585.22 Thus, the balances after the first three payments are as follows: \[ (\text{Balance 1}, \text{Balance 2}, \text{Balance 3}) = (\$[/tex]99,862.23, \[tex]$99,723.97, \$[/tex]99,585.22)
\]

These are the mortgage balances after the first three monthly payments.
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