jplats wrote:I came across this question on a GMATPrep practice exam. Does anyone know of a shortcut beyond actually multiplying it out?

(1001^2 - 999^2)/(101^2 - 99^2) = ?

[^2 = squared and the answer is 10]

whoa whoa already people.

this is an absolutely classic, canonical presentation of the difference of squares pattern. it's not obscured, hidden, or even rearranged in any way; both the top and the bottom of the fraction are plain differences of squares, in all their glory.

so, ALL of you should be able to come up with the following solution. if you can't, then review the difference of squares formula, x^2 - y^2 = (x+y)(x-y), until you can.

no excuses.

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top of fraction = (1001 - 999)(1001 + 999) = (2)(2000)
bottom of fraction = (101 - 99)(101 + 99) = (2)(200)

fraction = 2000/200
= 10

xilef wrote:didn't I use the same method?

no, you used the expansion of a perfect square binomial. while that method is certainly viable, it has the following disadvantages vis-a-vis the solution i posted:
* it's not obvious
- meaning that you have to come up with the idea to express 99 as (100-1) and so on.
* it requires a great deal more work
- instead of factoring two differences of squares, both of which are already in canonical form, you have to discover, and then factor, four perfect square trinomials.

with that said, though, two comments:

(1) if you have a method that's going to work, don't hesitate; use it.
- this is super important. xilef's method is not the shortest method available - but it definitely solves the problem, and does so in under two minutes. therefore, if you come up with a method that WILL work, then just use it. NEVER just stare at a problem.
(2) drill the common factoring forms into your head until you can recognize them IMMEDIATELY.
- as stated above