Problem 104 - Elementary Algebra

Question

A veteran drives three times a month to a VA Hospital that is 125 miles away from his home. Find the amount that the veteran will be reimbursed for his travel. The current allowable reimbursement amount is \(28.5\) cents per mile with a $$\$ 7.77$$ deductible for each one-way visit, or $$\$ 15.54$$ for each round- trip visit. If your out-ofpocket costs exceed $$\$ 46.62$$ in a given month, you can be reimbursed for the entire expense for any other authorized travel for that month. (Source: www.benefits.gov, Jan. 2012)

Step-by-Step Solution

Answer

The veteran will be reimbursed 213.75 dollars.

Step 1: Calculate Total Monthly Mileage

Determine the total distance the veteran drives in one month: \(3 \text{ trips/month} \times 2 \text{ ways/trip} \times 125 \text{ miles/way} = 750 \text{ miles/month}\).

Step 2: Calculate Monthly Reimbursement Before Deductions

Find the reimbursement amount before any deductions: \(750 \text{ miles} \times 0.285 \text{ dollars/mile} = 213.75 \text{ dollars/month}\).

Step 3: Calculate Deductibles for Round-Trips

Calculate the total deductibles for the three round trips: \(3 \text{ trips/month} \times 15.54 \text{ dollars/trip} = 46.62 \text{ dollars/month}\).

Step 4: Calculate Out-of-Pocket Costs

Subtract the total deductible from the monthly reimbursement: \(213.75 \text{ dollars} - 46.62 \text{ dollars} = 167.13 \text{ dollars}\).

Step 5: Determine Excess Out-of-Pocket Costs

Check if the out-of-pocket costs exceed 46.62 dollars. Since they do: \(167.13 > 46.62\) dollars, the entire travel cost will be covered.

Key Concepts

Mileage Calculation

Mileage calculation is straightforward but very important. To understand the concept, think about each trip as a single unit.
For each trip, you need to consider the total distance traveled. In this example, the veteran drives 125 miles to the VA Hospital and 125 miles back home, making each trip 250 miles in total.
Since the veteran goes three times a month, you simply multiply the round-trip distance by the number of trips. So, you get:

• 3 trips/month
• 250 miles/trip

Multiplying these together, you get the total monthly mileage:\[ 3 \times 250 = 750 \text{ miles/month} \]Understanding the total distance isn't just a number; it’s the start for calculating reimbursement and other costs.

Deductibles

Deductibles represent the amount you need to cover before getting any reimbursement. Think of it as your share of the expense.
For each round-trip visit, there's a fixed deductible of \(15.54. This means for each trip, you don't get reimbursed for the first \)15.54 you spend.
In our example, there are three trips per month, so you need to calculate the total deductibles for these three trips. It's simple multiplication:

• 3 trips/month
• 15.54 dollars/trip deductible

Then the monthly deductible cost is:\[ 3 \times 15.54 = 46.62 \text{ dollars/month} \]Deductibles reduce the amount that you will first be reimbursed, but they are essential to know for accurate calculations.

Reimbursement Policy

The reimbursement policy specifies how much you get paid back. For this exercise, the veteran's travel reimbursement rate is 28.5 cents per mile. This isn't cash given for the miles driven but compensation for the expense.
After calculating the total miles driven in a month, you can find the reimbursement amount before deductibles. In our case, we calculated 750 miles. Multiply this by the reimbursement rate:

• 28.5 cents or 0.285 dollars per mile

Then, calculate the monthly reimbursement before deductible:\[ 750 \times 0.285 = 213.75 \text{ dollars/month} \]This amount is what you could be reimbursed if there were no deductibles. After deductibles, you determine the final reimbursement.

Monthly Travel Expenses

Monthly travel expenses include not just the actual traveling cost but also your deductible and any unclaimed expenses. First, determine the out-of-pocket costs excluding deductible reimbursement.
In our example, the reimbursable amount after deductibles is:

• Reimbursement before deductibles: \(213.75
• Total deductibles: \)46.62

Subtract the deductible from the pre-deductible reimbursement:\[ 213.75 - 46.62 = 167.13 \text{ dollars} \]If the out-of-pocket costs exceed \(46.62, the reimbursement covers the full travel cost. Here,\[ 167.13 > 46.62 \]Therefore, the entire travel cost of \)213.75 is reimbursed. This is crucial for budgeting and planning your monthly expenses.