To solve the problem, we need to determine how many points Ellsworth needs to improve his credit score from the current score of 498 in order to afford a \[tex]$150,000 mortgage with a maximum annual mortgage payment of \$[/tex]14,000.
1. Convert the maximum annual mortgage payment to a monthly mortgage payment:
[tex]\[
\text{Monthly mortgage payment} = \frac{\$14,000}{12} = \$1166.67
\][/tex]
2. Identify the monthly mortgage payments for different FICO score brackets from the given table:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{FICO score} & \text{Interest rate} & \text{Monthly payment} \\
\hline
720-850 & 5.59\% & \$860 \\
700-719 & 5.71\% & \$872 \\
675-699 & 6.25\% & \$924 \\
620-674 & 7.40\% & \$1039 \\
560-619 & 8.53\% & \$1157 \\
500-559 & 9.29\% & \$1238 \\
\hline
\end{array}
\][/tex]
3. Determine the FICO score bracket that Ellsworth must reach to afford a monthly payment within \[tex]$1166.67:
Checking each bracket in descending order:
- The \$[/tex]860 monthly payment for the 720-850 bracket is affordable.
- The \[tex]$872 monthly payment for the 700-719 bracket is affordable.
- The \$[/tex]924 monthly payment for the 675-699 bracket is affordable.
- The \[tex]$1039 monthly payment for the 620-674 bracket is affordable.
- The \$[/tex]1157 monthly payment for the 560-619 bracket is affordable.
- The \[tex]$1238 monthly payment for the 500-559 bracket exceeds \$[/tex]1166.67, so it's not affordable.
Thus, Ellsworth needs to fall within the 560-619 FICO score bracket.
4. Calculate the minimum number of points Ellsworth needs to improve:
To qualify for the 560-619 bracket, Ellsworth's current score of 498 needs to be improved to at least 560.
[tex]\[
\text{Points to improve} = 560 - 498 = 62
\][/tex]
Therefore, Ellsworth needs to improve his credit score by 62 points.
[tex]\[
\boxed{62}
\][/tex]
