### GRE Math for People Who Hate Math: Average Speed

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On the GRE, you will never, ever, ever, ever have to average two speeds together. If a GRE Quant problem gives you two speeds (say, 40 mph and 60 mph), and you average them (ending up with 50 mph), you’ve just gotten that problem wrong.

Here’s an example:

Rajesh drove the 240 miles from Springfield to Greenville at a rate of 40 mph. Then, he returned along the same route at a rate of 60 mph. What was his average speed for the entire trip?

(A) 42 mph
(B) 48 mph
(C) 50 mph
(D) 54 mph
(E) 56 mph

If you average the two speeds, you’d get (40 mph + 60 mph)/2 = 50 mph. That answer choice is only there to trap you!

Here’s what you should do instead. Every time you see the term ‘average speed’ (or anything similar) in a problem, write the following on your scratch paper right away:

avg speed = total dist / total time

This formula works every single time. Break it down into pieces to keep yourself organized:

1. Calculate the total distance traveled. Rajesh traveled 240 miles in one direction, then turned around and traveled 240 miles in the other direction. His total distance was 240 + 240 = 480 miles.

2. Calculate the total time. For this one, you’ll need a famous formula: distance = rate * time. You’ll also need to calculate the time for each half of the trip separately. For the first half, the distance is 240 miles and the rate is 40 mph.

240 miles = 40 mph * time

time = 240 miles / 40 mph = 6 hours

For the second half of the trip, the distance is 240 miles again, and the rate is 60 mph.

240 miles = 60 mph * time

time = 240 miles / 60 mph = 4 hours

The total time for the entire trip is 10 hours.

Once you have those pieces, you’re ready to find the actual answer. Plug them into the formula:

avg speed = total dist / total time

avg speed = 480 miles / 10 hours = 48 mph

The correct answer is 48 mph.

Let’s dig deeper. Does it make sense that the answer is 48 mph? One useful skill for average speed problems is sanity checking your answers. Do this by comparing your answer to the other numbers in the problem. Here, 48 is between 40 and 60. That makes sense. It’s slightly below the midpoint of those two numbers: is that where it should be?

Well, Rajesh went more slowly during the first half of the trip, so he spent a longer time traveling at a lower speed. The longer you travel at a speed, the closer your average will be to that speed. If you walk across America, then hop on a plane and fly back the other way, your average speed for the whole trip will be much closer to your walking speed than to your flying speed. It makes sense that Rajesh’s average was closer to his slower speed of 40 mph. By the way, you could’ve used that to make a fast guess on that problem!

Here’s something else we can learn from that problem. What if we changed the distance that Rajesh traveled?

Rajesh drove the 300 miles from Springfield to Greenville at a rate of 40 mph. Then, he returned along the same route at a rate of 60 mph. What was his average speed for the entire trip?

(F) 42 mph
(G) 48 mph
(H) 50 mph
(I) 54 mph
(J) 56 mph

Does his average speed change? Try working it out, or at least take a guess, before you keep reading.

His total distance was 600 miles. The time for the first part of the trip was 7.5 hours, and the time for the second part was 5 hours. Plug (600 miles)/(7.5+5 hours) into your calculator to find that his average speed was 48 mph, yet again.

That would actually happen no matter what the distance was. The answer always comes out to 48 mph. Interestingly, the GRE knows this, so sometimes they won’t even tell you the distance. That’s okay, since the answer is always the same. But how do you calculate the answer? By choosing your own numbers!

Here’s the rule: if the GRE doesn’t tell you the distance, you’re allowed to choose it yourself.

Try it out on this more complicated problem:

Adrian drove from Springfield to Greenville at a rate of 100 kilometers per hour. Then, he halved his speed to 50 kilometers per hour, and drove from Greenville to Wilmington. If the distance between Greenville and Wilmington is twice the distance from Springfield to Greenville, what was Adrian’s average speed for the entire trip?

Got your answer? Here’s how I approached it. We don’t know the distances, so we can choose whatever we’re comfortable with, as long as it doesn’t contradict what the problem says. We’ll have to divide the distances by 50 or 100, so let’s choose values that are multiples of those numbers. Let’s say that Springfield and Greenville are 100 kilometers apart, and Greenville and Wilmington are 200 kilometers apart.

In that case, the total distance would be 300 kilometers. It takes Adrian 1 hour to travel the first 100 kilometers, and 4 hours to travel the remaining 200. His total travel time is 5 hours. Use the formula:

avg speed = 300 km / 5 hours = 60 km/hr

If you got the same answer, you should feel confident about GRE average speed problems! All you need is these three tools:

• The average speed formula
• Sanity checking