At the end of every SAT Math section, the test makers try to come up with an extremely difficult problem 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:
If x and y are integers and y < 20, for exactly how many ordered pairs (x, y) will x^2 = y?
This one actually doesn’t seem so bad, does it?
1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25 > 20
So we’ve got (1,1), (2,4), (3,9), and (4,16). Answer choice A, right? Not so fast! You forgot that the square of a negative number is also positive, so for ever y, there are two x values: one positive and one negative. So really our list should look like this:
(1,1) and (-1,1)
(2,4) and (-2,4)
(3,9) and (-3,9)
(4,16) and (-4,16)
So the answer is D, right? Wrong again! There’s one last square you forgot:
0^2 = 0
Thus, there are in fact 9 pairs: the eight already mentioned, plus (0,0). Thus, the correct answer is actually choice E.
Was there actually anything hard about this question? Not really. However, if you were going fast and running out of time, you might have easily made one of the careless errors above. Note that 4 and 8 are traps set for students who see this problem, think it’s easy, and then blow through it too fast without thinking carefully (if you forgot the negatives but remembered 0, there’s also choice B, 5). If you get toward the end of a math section and see a problem that looks really easy, be careful – there’s probably more to it than meets the eye. Sometimes it’s just as bad to spend too little time on a problem as it is to spend too much, so make sure you don’t go too fast through any “easy” problems at the end of a math section.
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!