### GRE Math Riddle: Doubling the Average Rate

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Here’s a GRE math riddle:

This isn’t exactly a GRE question, but solving it tells you something absolutely crucial about “average rate” questions—which definitely are on the GRE. So take a minute and think about it before you read any further.

(I trust you’re doing some good thinking on your own first. It wouldn’t really be a GRE math riddle if I just gave you the answer. 😊)

….

(Still thinking? Cool. Keep scrolling when you’re ready to discuss.)

Alright, now let’s talk. Like I said, you’d never get something like this on the GRE. But it’s so worth thinking about, I say we play around with it for the rest of this blog entry. First, let’s discuss an obvious answer. An obvious answer that also happens to be a wrong answer: 150 mph.

If you took a direct average of 150 and 50, you’d get 100.

In the good old-fashioned average formula, that looks like:

But, like I said, 150 mph is incorrect for our question. Average speed is not a direct average of two numbers. And it actually follows a slightly more nuanced formula:

Let’s keep playing around with 150 mph to see why it fails. If we plug our given information into the rate formula, we’ll be able to crank out everything else we need.

For the first lap:

Dividing both sides of the equation by 50, you can solve for T. The time it takes for the first lap is 2 hours.

And for the 2nd lap:

We divide both sides by 100 mph and solve for T. The time it takes for the 2nd lap is 100/150… or about .66 hours.

Since each lap was 100 miles, the journey totals 200 miles.

Since the first lap took 2 hours and the second lap took .66 hours, the total time is 2.66 hours.

Not only does 150 mph fail to double our average speed for the whole trip, it actually falls quite a bit short of the overall average of 100 mph we were shooting for. Just for reference, here’s the riddle again:

Doubling the car’s speed for the overall trip requires it to go quite a bit faster on the 2nd lap than it went on the first. Let’s try plugging in a pretty outlandish guess to see what happens. Say the car burns some serious rubber and comes zipping through the 2nd lap at a speed of 400 miles per hour.

The car would make quick work of 100 miles, finishing in just ¼ of an hour.

Plug that in with the time and distance of the first lap, just like we did before:

That’s definitely faster, but it’s still not even close to the 100 miles per hour we were shooting for.

Now let’s cut to the chase. What’s the answer to the GRE math riddle???

The car would need to go infinitely fast for the 2nd lap in order to obtain double its average speed. Even if it was travelling at 5,000 miles per hour, the car wouldn’t quite be able to double its average speed. That’s because it would have to be done with the 2nd lap at the exact moment the lap started. To get an average speed of 100 miles per hour, we’d need this:

Pretty crazy, right? Think about that the next time you’re driving to work.

Actually, don’t. Focus on driving safely. Driving and mathematics are probably a somewhat dangerous pair to combine.

If you’ve got the kind of nerdy friends who are open to it, try asking them this same math riddle. I’m admittedly pretty nerdy, and I tend to hang with a somewhat nerdy crowd, but I’ve found that this question gets some pretty reliable mental fireworks going.

Enjoy! And maybe now you’ve got a good way to truly remember that average rate formula (which definitely is tested on the GRE):

Happy studying! 📝

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Tom Anderson is a Manhattan Prep instructor based in New York, NY. He has a B.A. in English and a master’s degree in education. Tom has long possessed an understanding of the power of standardized tests in propelling one’s education and career, and he hopes he can help his students see through the intimidating veneer of the GRE. Check out Tom’s upcoming GRE courses here.

1. knappi March 5, 2018 at 3:42 pm

Hi Po! Here’s Tom’s response. -Kim, MPrep

“Po,
I like the way that you’re thinking about this problem. And you’re right – the difference between the average speed and 100 mph diminishes bit by bit, the faster you go, until you’re within tenths of a mile per hour. Your overall strategy – plugging in an easy test case and then plugging in a very extreme case – is not just clever, but it’s good GRE thinking too.

In case you’re curious, the problem I wrote about works no matter what actual numbers we pick. Consider this variation:

” A runner runs around a track very slowly. How much faster would he have to run the 2nd lap so that his average speed for the two laps is twice as his speed for the first lap.”

To get a speed of exactly double, he’d have to go infinitely fast. Like you pointed out, though, once the 2nd lap is about 100 times as fast as the first lap, you’re within a small rounding margin of double.

Great comment!

-Tom”

2. Po February 19, 2018 at 10:42 pm

Although what you say is true, in real life we use approximations and rounding all the time. Even your calculator is only as accurate as it has to be. Consider going 10,000 miles an hour, not realistic today but entirely possible someday, somehow. 200 miles/2.01 hours = 99.5 mph, or 100 mph to the nearest mph, which is how we usually speak (rarely would anyone be more precise). The second lap took 0.6 minutes, or 36 seconds.
BTW, my initial guess was not 150, but 200 mph. I figured, if it did half a lap the first time, it would have to be double a lap the second time.