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The maximum amount Bill Jones can claim for car expenses related to his medical mileage is $630. This figure is determined by applying the IRS standard mileage rate for medical travel, which is specifically set for the purpose of deducting expenses incurred while driving to and from medical appointments or related healthcare services.

To arrive at the correct amount, it's essential to calculate the total number of miles driven for medical purposes during the given tax period and then multiply that number by the applicable mileage rate established by the IRS for that tax year. The IRS updates the mileage rates annually, and for the context of this question, $630 is reflective of a scenario where the calculated expenses fall within the accepted parameters under IRS regulations.

This numeric outcome helps individuals like Bill recuperate some costs associated with necessary healthcare, thus providing financial relief. Understanding the rationale behind this deduction reinforces the importance of keeping accurate mileage records and staying informed about annual changes in tax regulations.