{"id":11286,"date":"2018-01-31T17:39:43","date_gmt":"2018-01-31T17:39:43","guid":{"rendered":"https:\/\/www.manhattanprep.com\/gre\/?p=11286"},"modified":"2019-08-30T16:40:21","modified_gmt":"2019-08-30T16:40:21","slug":"gre-math-riddle-doubling-average-rate","status":"publish","type":"post","link":"https:\/\/www.manhattanprep.com\/gre\/blog\/gre-math-riddle-doubling-average-rate\/","title":{"rendered":"GRE Math Riddle: Doubling the Average Rate"},"content":{"rendered":"

\"Manhattan<\/p>\n

You can attend the first session of any of our online or in-person GRE courses absolutely free. Crazy, right? <\/i><\/b>Check out our upcoming courses here<\/i><\/b><\/a>.<\/i><\/b><\/p>\n

\n

<\/i><\/b>Here\u2019s a GRE math riddle:<\/span><\/p>\n

\"Manhattan<\/p>\n

This isn\u2019t exactly a GRE question, but solving it tells you something absolutely crucial about \u201caverage rate\u201d questions\u2014which definitely are on the GRE. So take a minute and think about it before you read any further.<\/span><\/p>\n

\u2026<\/span><\/p>\n

(I trust you\u2019re doing some good thinking on your own first. It wouldn\u2019t really be a GRE math riddle if I just gave you the answer. <\/span>?<\/span>)<\/span><\/p>\n

\u2026.<\/span><\/p>\n

(Still thinking? Cool. Keep scrolling when you\u2019re ready to discuss.)<\/span><\/p>\n

\u2026<\/span><\/p>\n

Alright, now let\u2019s talk. Like I said, you\u2019d never get something like this on the GRE. But it\u2019s so worth thinking about, I say we play around with it for the rest of this blog entry. First, let\u2019s discuss an obvious answer. An obvious answer that also happens to be a wrong answer: 150 mph.<\/span><\/p>\n

If you took a direct average of 150 and 50, you\u2019d get 100.<\/span><\/p>\n

In the good old-fashioned average formula, that looks like:<\/span><\/p>\n

\"Manhattan<\/p>\n

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:<\/span><\/p>\n

\"Manhattan<\/p>\n

Let\u2019s keep playing around with 150 mph to see why it fails. If we plug our given information into the rate formula, we\u2019ll be able to crank out everything else we need.<\/span><\/p>\n

For the first lap:<\/span><\/p>\n

\"Manhattan<\/p>\n

Dividing both sides of the equation by 50, you can solve for T. The time it takes for the first lap is 2 hours. \u00a0<\/span>
\n<\/span>
\n<\/span>And for the 2<\/span>nd<\/span> lap:<\/span><\/p>\n

\"Manhattan<\/p>\n

We divide both sides by 100 mph and solve for T. The time it takes for the 2<\/span>nd<\/span> lap is 100\/150\u2026 or about .66 hours. \u00a0<\/span><\/p>\n

Since each lap was 100 miles, the journey totals 200 miles.<\/span><\/p>\n

Since the first lap took 2 hours and the second lap took .66 hours, the total time is 2.66 hours.<\/span><\/p>\n

\"Manhattan<\/p>\n

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\u2019s the riddle again:<\/span><\/p>\n

\"Manhattan<\/p>\n

Doubling the car\u2019s speed for the overall trip requires it to go quite a bit faster on the 2<\/span>nd<\/span> lap than it went on the first. Let\u2019s try plugging in a pretty outlandish guess to see what happens. Say the car burns some serious rubber and comes zipping through the 2<\/span>nd<\/span> lap at a speed of 400 miles per hour.<\/span><\/p>\n

\"Manhattan<\/p>\n

The car would make quick work of 100 miles, finishing in just \u00bc of an hour.<\/span><\/p>\n

Plug that in with the time and distance of the first lap, just like we did before:<\/span><\/p>\n

\"Manhattan<\/p>\n

That\u2019s definitely faster, but it\u2019s still not even close to the 100 miles per hour we were shooting for. \u00a0<\/span>
\n<\/span>
\n<\/span>Now let\u2019s cut to the chase. What\u2019s the answer to the GRE math riddle???<\/span>
\n<\/span>
\n<\/span>The car would need to go <\/span>infinitely fast<\/i><\/b> for the 2<\/span>nd<\/span> lap in order to obtain double its average speed. Even if it was travelling at 5,000 miles per hour, the car wouldn\u2019t quite be able to double its average speed. That\u2019s because it would have to be done with the 2<\/span>nd<\/span> lap at the exact moment the lap started. To get an average speed of 100 miles per hour, we\u2019d need this:<\/span><\/p>\n

\"Manhattan<\/p>\n

Pretty crazy, right? Think about that the next time you\u2019re driving to work. \u00a0<\/span><\/p>\n

Actually, don\u2019t. Focus on driving safely. Driving and mathematics are probably a somewhat dangerous pair to combine.<\/span><\/p>\n

If you\u2019ve got the kind of nerdy friends who are open to it, try asking them this same math riddle. I\u2019m admittedly pretty nerdy, and I tend to hang with a somewhat nerdy crowd, but I\u2019ve found that this question gets some pretty reliable mental fireworks going.<\/span><\/p>\n