On every GRE Math section, the test makers try to come up with a few extremely difficult problems that will leave even the cleverest students scratching their heads. The really evil part, though, is that even these problems can be solved in under a minute without a calculator – if you know what to do. This means that once you “figure out the trick,” these difficult problems become easy. So, while those test makers are busy cackling with sadistic glee, let’s see if we can’t beat them at their own game.

Consider the following problem:

Out of 50 space cadets, 30 speak English, 20 speak Klingon, and 5 speak neither English nor Klingon. What is the ratio of those who speak both English and Klingon to those who speak neither?

A) 1:1

B) 1:2

C) 2:3

D) 1:3

E) 5:1

We know that the total number of space cadets is 50, and that the number of cadets who only speak English plus the number of cadets who only speak Klingon plus the number of cadets who speak both plus the number of cadets who speak neither should equal 50. The trouble is in figuring out how many cadets speak both so that we don’t count them twice. If we just add the 30 who speak English to the 20 who speak Klingon, we will have counted the ones who speak both twice. So, how can we prevent this?

By subtracting them. If we add the cadets who speak English to those who speak Klingon and those who speak neither but subtract the number who speak both, then we will have made up for double counting and the result should equal the 50 total cadets. If we let *b* be the number of cadets who speak both, then:

50 = 30 + 20 – *b *+ 5

50 = 55 – *b*

*b* = 5

Since the problem asked for the ratio of those who speak both English and Klingon to those who speak neither, the ratio will be 5 to 5, or simplified, 1:1, which is choice A. If you know what to do, it takes only about 30 seconds to solve this problem. So you see, with practice, even the hardest problems on the GRE become easy. Check back here each week for more extra hard problems and the tricks you need to solve them! Also, remember that you can find out all the tricks from experts like me with a Test Masters course or private tutoring. Until then, keep up the good work and happy studying!