# Math puzzle: Miles per gallon

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I filled up my Prius's gas tank tonight; a while later, on the way home from the East Bay, I noticed that the MPG indicator (showing miles per gallon since the most recent gas-tank fill-up) said I was getting 50 miles per gallon, and the odometer (which appears right next to the MPG indicator) said I had gone 51 miles.

Neat, I thought to myself. The MPG indicator has nearly caught up to the odometer. I wonder what gas mileage I would need to get in the next few minutes for the MPG indicator to pass the odometer—that is, for the display to show a number on the MPG indicator that's greater than the number showing at the same time on the odometer. (Not just greater than 51; greater than whatever the odometer will say at that future time.)

It didn't take long to figure out the answer, but I think it's kind of a neat puzzle, and thought some of you might enjoy it.

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Since there've been no responses to this, I'll just post the answer:

It's a trick question.

Having gone 51 miles at 50 MPG means that I had already burned a little more than 1 gallon of gas.

To go n miles at >n MPG requires burning less than 1 gallon of gas. For example, to go 53 miles at 54 MPG, you would burn 0.98 gallons of gas.

Since I had already burned more than 1 gallon, it wasn't possible to reach a future state of having burned less than 1 gallon.

(It would be possible to maintain a tie (like 50 miles at 50 MPG) by continuing forward without burning any gas, as one could do in a conventional car by coasting with the engine off, or in a hybrid like the Prius by switching to the electrical motor. But once you've burned more than 1 gallon, you can't go back to having burned less.)

Another part of the trick here is the framing. I thought of it as the MPG indicator having almost caught up to the odometer, because I had been carefully managing my driving to increase MPG. But in fact, it would be entirely possible to be showing 40 MPG at, say, 20 miles. In other words, the fact that the two numbers happened to be similar when I looked at them was pretty much a coincidence; it wasn't the case that MPG started out much lower and was gradually getting closer to caught up.

This puzzle reminds me of the old one about speed:

You're going on a 60-mile trip. So far, you've gone 30 miles and it's taken you an hour, so you're averaging 30 MPH. How fast would you have to drive on the second half of the trip to average 60 MPH for the whole trip?

The answer, of course, is that you can't; you'd have to teleport instantaneously. Averaging 60 MPH for a 60-mile trip means you'd have to complete the whole trip in an hour, and you've already spent that hour on the first half.

But I think I may like this MPG puzzle better; it feels a little more elegant in some ways, and a little trickier in others.