## Tuesday, June 12, 2012

### How much does it cost to drive to work?

A question few of us think about it on my mind a lot more recently -- especially as the summer approaches. I often will do school work during the summer -- which in most cases I can do from anywhere with a laptop and internet access. Some of that is planning, reorganizing, and some of that is helping build the schedule for next years classes. As nice as it is to be home during the summer, I often work more efficiently from school, simply because I am focused and don't have to worry about distractions. But what does it cost to drive in?

Budgeting has become a monthly chore for our household and one thing we have tried to do is plan our driving to and from Grand Rapids -- about a 30 mile hike one way. This post describes how we make these calculations.

A few assumptions have to be made, or we'll kill each other squabbling over the details. The assumptions I proposed were:

• Distance to work: 30 miles one way or 60 miles round trip
• Miles Per Gallon: 25 in my Ford Taurus
• Price of Gas: $4. A little high, but I'd rather over budget than under budget, and I also not considering things like oil changes, wiper fluid refills, etc. This sort of calculation is easily performed using my favorite mathematical tool -- Unit conversion by multiplying by unit fractions. The process is to start with some given quantity and convert it to a desired unit in one chain of fractions. Every time you convert from one unit to another you must either multiply or divide, and in this technique the units are your guide for which operation to perform. When you write each fraction down, you must be sure to write it in a way that cancels out the last unit you had. Specifically, if the units you currently have are in miles on top, the next fraction I should write will have to have miles in the bottom, because a unit in the denominator cancels out a unit in the numerator. So, lets get started. I'll begin by considering how much it costs to make one round trip, and so I'll write: The next factor I need to include is the distance of one trip. This can be a fraction that I can include in one of two different ways: The first option represents needing to divide by 60, and the second option represents multiplication. Which operation do I perform? I need to write the second operation, because the roundtrip that is in the numerator will cancel with the round trip in the denominator. Continuing, I need to write the next conversion factor, 25 miles per gallon in such a way that miles is in the denominator to cancel out miles, so it looks like: Finally, including the price of gas so that gallons cancels out suggests writing: $\frac{1 roundtrip}{ } \frac{60 miles}{roundtrip} \frac {gallon}{25 miles} \frac {\ 4}{gallon}$ Ultimately, any numbers that are on top need to be multiplied, and any on the bottom need to be divided, so this chain of fractions suggests the cost of a round trip is 60 * 4 / 25 =$9.60.

Now rather than guessing at how much money we're going to spend on gas, we can plan on which days we'll be going into Grand Rapids, and calculate how much we'll have to spend. For the month of June, we anticipated 12 trips into Grand Rapids - four mandatory trips for school, 2 for doctors appointments, 4 for softball, and 2 days of open houses. This alone will cost $115 (12*9.60). Sparing you the remaining details regarding shorter trips around town and to the lakehouse, we ended up assuming we would spend$200 on gas.

A few remaining related calculations, just for fun:
• How much does it cost per mile:     $\frac {1 gallon}{25 miles}\frac{\ 4}{gallon} = \0.16$

• How much would it cost to drive to the moon? $\frac {\238,000 miles}{ }\frac{gallon}{25 miles}\frac{\ 4}{gallon} = \ 38,000$

1. So you'd need to make $10 working in GR to make the trip break even. That's why I didn't like summers working at FCS where I'd be scheduled for 2-3 hours and make$20. Half of my paycheck would be going toward gas!