One of my favorite books / movies is "The Princess Bride" - it's just such a classic! I know most of the great lines and one of them is appropriate for today's blog: it occurs when Inigo Montoya says to Vizzini about a word he keeps using, "I don't think it means what you think it means." Here's why. Most of us think we know what stuff means; for example, if I tell you my truck gets about 18 miles to the gallon (18 mpg), you know what that means. You understand that it's going to cost me more to drive my truck than it would if I had a small car that got, say, 30 miles to the gallon. You also understand that, for a four-wheel-drive truck, 18 mpg is not as bad as it could be, since many trucks get way less than that. But all it takes is a simple problem to illustrate that it doesn't mean what you think it means.
Here's your problem:
Consider two car owners who want to reduce their fuel costs: Adam switches from a gas-guzzler that gets 12 mpg to a slightly-less voracious guzzler that gets 14 mpg. The environmentally-virtuous Beth switches from a 30 mpg to one that gets 40 mpg. Suppose they both drive equal distances per year. Who will save more gas by switching?
Now, I ask you, which one did you pick?
I can almost guarantee that you chose Beth. I mean, it's obvious, right? Switching from 30 to 40 mpg is a larger increase than the switch from 12 to 14 mpg and larger increase must equal larger savings, right? WRONG! Don't believe me? Ok, let's do the math.
Suppose Adam and Beth both drive 10,000 miles in a year.
WIth Adam's old vehicle, he needed about 833 gallons of fuel (rounded to the nearest gallon). Calculation is distance / MPG or 10,000 miles divided by 12 mpg for the crazy-expensive 833 gallons of gas he'd need for the year. For his new vehicle, he'd need about 714 gallons, still a crazy-expensive amount, but it's a significant savings of 119 gallons. Calculations are 10,000 / 14 for about 714 gallons (rounded to the nearest unit) and then 833-714 to calculate the difference (savings) of 119 gallons.
What about Beth? Well, now you can do the math with me. For her old car, she'd need about 333 gallons of fuel for the year and for the new one, she'd need 250 gallons. That's a difference of only 83 gallons!
In other words, Adam's change would yield the highest savings (of both fuel and money). How can this be? How can something so intuitively right be so wrong in reality? In a word, "framing". Framing is like the context within which you think about something; in this case, the "frame" revolves around the rate of measure: miles per gallon (MPG). Recall from algebra class that MPG is a rate which is a special kind of ratio involving incommensurable items (e.g. items with no common measure). Quick refresher: A ratio is a comparison of two things that have the same unit of measure (e.g. student to teacher ratio for a school, both items being measured in the unit of people) while a rate is a comparison of two things that do not have the same unit of measure (e.g. miles per gallon, where one item, miles, is measured in distance units, and the other item, gallon, is measured in volume units). Now the rate MPG, which is common here in the US, is very useful for calculating how far you can travel on a given amount of fuel; as in, if my car gets 20 mpg and I have 100 gallons, then I can travel 2000 miles. The MPG rate is not so helpful though in comparing differences between cars with various fuel economies (like the question about Adam and Beth).
Two questions might occur to you right now: Why isn't the MPG a good rate for these types of problems? and What IS a good rate to use in these types of problems?
I can answer both questions in this sentence: The MPG rate isn't good for these types of problems because we're not trying to compare miles per gallon, we're trying to compare gallons per mile. Think about it: to answer the Adam vs Beth problem, you needed to find out how many gallons of fuel each would need to go the 10,000 miles distance for the year. So the rate we really need is gallons per mile (GPM). In fact, since here in the US we tend to drive at least 10,000 miles per year, an even better unit of measure would be gallons per 10,000 miles (GP10KM?) ... hmm, that probably won't catch on. Ok, let's stick with GPM for the moment. Now, why is MPG so counter-intuitive for us in trying to answer the Adam-Beth type problems? Well, it's because of the nature of the relationship between the two variables.
Let's explore this a bit more. As we've already mentioned, miles per gallon is a rate. Fuel (e.g. gasoline) consumed is the inverse of that rate; in other words, as the MPG goes up, the fuel needed goes down (and as the MPG goes down, the fuel needed goes up). A rate and its inverse do not have a linear relationship; in other words, a one unit difference in MPG is NOT equivalent to a one unit difference in fuel needed -- the slope is not constant; instead, it varies.
You look confused. It's ok. Math sometimes sounds more complicated when you try to use words than if you just look at a picture. Here's a graph comparing miles per gallon with gallons per 10,000 miles:
Click on the graph to see a larger image.
The points in the graph above are 5 mpg units apart, but as you can clearly see, they do not correspond to the same vertical distance (on the gallons per 10,000 miles axis) since their graph is not a line.
Let's explore some specific points on the graph to "see" what's going on here. The first entry in the table is (10,1000) representing a vehicle that only gets 10 mpg and therefore needs 1,000 gallons of fuel to travel the 10,000 mile distance for the year. That vehicle will have the largest cost to run the given distance of all the various fuel economies shown here. Notice that a change of 5 units at the low end of the MPG rating (e.g. from 10 mpg to 15 mpg) yields a much larger difference in gallons needed than a similar 5-unit change at the high end of the MPG rating (e.g. from 50 mpg to 55 mpg). Specifically, a 5 mpg change at the low end (10 to 15 mpg) yields a savings of about 333 gallons (1000-667), whereas a 5 mpg change at the high end (50 to 55 mpg) only yields a savings of 18 gallons (200-182).
The "illusion" gets even more clear if we "zoom in" and focus on the part of the graph from 10 mpg to 20 mpg:
Click on the graph to see a larger image.
Now it is really clear that even a small change yields big savings if your current vehicle gets < 20 mpg. Here's a table to illustrate what I mean:
This problem is now one of my new favorites. It epitomizes what NCTM refers to as a "worthwhile mathematical task" and I think it can provide for many rich discussions. If you try it out with your children or students, please drop me a line and let me know how it goes. I'd love to hear about it!
References:
"Thinking, Fast and Slow" by Daniel Kahneman
"The MPG Illusion" by Richard Larrick & Jack Soll (from 2008 Science magazine)
The MPG Illusion website
"Nudge" by Cass Sunstein with Richard Thaler
US Office of Information and Regulatory Affairs, 2013 "fuel economy and environment" auto sticker
National Council of Teachers of Mathematics (NCTM) website.
The Solver Blog
Author: Dr. Diana S. Perdue