Doing the Math on the Ever-Slowing Drive

Oct 23, 2006

RAY: You're driving your car on the highway at 75 mph, and you notice a sign that says you are 75 miles from your destination. So if you continue driving at that speed, you'd be there in an hour.

But, you're not going to do that, because then it wouldn't be a puzzler. When you have driven one mile and you are now 74 miles from your destination, you drop your speed down to 74 mph.

So, you drive that first mile at 75 mph; when you are 74 miles from your destination, you drop your speed down to 74 mph; and then 73 mph, 72 mph… and so on. Until, finally, you get down to 1 mile from your destination and you're driving at one mile per hour.

And the question is, if you do this, how long is it going to take you to travel the entire 75 miles?

I made this puzzler multiple choice, with four choices:

1st choice: approximately 2 hours.
2nd choice: approximately 5 hours.
3rd choice: approximately 8 hours
4th choice: approximately forever.
RAY: What you wind up with is a series of numbers. If you start counting from your destination of one hour, plus a half an hour, plus a third of an hour, plus a fourth, plus a fifth, plus a sixth, and when you're driving at 60 miles an hour, that mile takes you one sixtieth of an hour obviously which is a minute.

I got the answer by adding up one plus a half, plus a third, and so on. I converted them all to decimals.

There's no discrete answer, the best you can get is an approximation. Here's how you get it: you have 75 terms. You're going to look up the natural log of 75, which happens to be 4.317, and you're going to add it to something called Euler's constant.

Euler was a Swiss mathematician who lived in the 1700s, and he came up with a bunch of interesting stuff, not the least of which is this. His constant is something like 0.57712. If you add the log of 75 and .577 you come up with 4.89.

TOM: So choice two, approximately five hours is the correct answer.

RAY: Yes. I should add that I didn't want to identify the person that sent the puzzler in, but his answer was bo-oh-oh-gus, and I didn't use it, but I believe that this answer is a lot better.

Who's our winner?

TOM: Our winner this week is Christian Schmidt from Fort Collins, Colorado, and for having his answer selected at random from among all the correct answers that we got, Christian gets a 26-dollar gift certificate to the Shameless Commerce Division at, with which he can get our collection of favorite stories called, Once Upon a Car Fire.

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