one thing i feel compelled to add:
there is really no such thing as "cancel".
in other words, there are exactly 6 operations of which you should be aware on the gmat: add, subtract, multiply, divide, raise to a power, take a root.
that is all.
if you have any other, more exotic operations, such as "cancel" or "cross-multiply", YOU NEED TO UNDERSTAND WHICH OPERATION YOU'RE REALLY DOING when you do those.
examples:
if you "cancel" the b in abc = b^3, to give ac = b^2, then you're actually dividing both sides by b.
if you "cancel" the x's in x + y = x - y to give y = -y, then you're actually subtracting x from both sides.
if you "cross multiply" a/b = c/d to give ad = bc, then you're actually multiplying both sides by (bd).
you MUST understand what operation you're really doing in every case; otherwise, you won't understand the restrictions on those operations.
for instance, if you think of the first operation as "cancelling", then you won't understand why it's not ok. on the other hand, if you understand the fact that you're dividing by b on both sides, it should be abundantly clear why you need to make sure that b is non-zero before performing this operation.
similarly, if you're given a/b > c/d, you may want to "cross multiply" to give ad > bc. however, if you understand that "cross multiplication" is really multiplication by the product of both denominators (bd), then you'll realize that you can't do this, either, unless you know whether bd is positive or negative.
basically, what it boils down to is "always know what you're doing when you're doing what you're doing".
Ron has been teaching various standardized tests for 20 years.
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