ildude02 wrote:Ian, as always a great explanation. I am wondering if there is any rule when to consider + integers, -ve intergers and maybe, fractions as well when picking numbers to solve arithmetic DS questions involving comparisions. Also, are there any pointers or some kind of rules you would suggest for certain kind of comparisions, so we don't need to pick numbers to solve them.
for eg;
y-1 is always < y; fo all values of y
x^2 < x whne x is between 0 and 1,
similarly, x+1 > x; for all values of x
also, if x >y then 1/x < 1/y , provided that x and y are of the same sign for all values of x and y.
I'm wondering if there are more of these hard core rules that can save us time when we come across such comparisions. I would appreciate your response.
What numbers we need to pick, if we do pick numbers, really depends on the question being asked. In some questions, we might know that x is not negative- perhaps it measures a distance, or the number of oranges in a box. But, if x is allowed to be negative,
always consider that possibility- otherwise you'll get into a lot of trouble on the GMAT!
We do often see statements that give inequalities involving powers of x- an inequality like "x^3 > x". If you are going to test numbers here, then you should always consider four different possible values for x:
x > 1
0 < x < 1
-1 < x < 0
x < -1
So you could try x = 2, x = 1/2, x = -1/2 and x = -2, for example. You'll see, with the example I've given, that if x^3 > x is true, then x might be larger than 1, but it might also be true that x is between -1 and 0. As a useful exercise, you might try looking at the inequality "x^3 > 1/x" in the same way- what does this tell you about x?
That type of statement is common enough that it's worthwhile learning how to decode it. There are, however, many other kinds of statements we can see on the GMAT, so it's impossible to give simple universal rules for what values need to be checked. In problems involving ratios, I'll normally consider extreme values -- what if x is very large? What if x is very small? In number theory questions (involving divisors or evens/odds), the guidelines would be completely different, and I'm not sure what to suggest there since I'd never choose numbers for a question involving evens and odds, or divisibility.
I'm sure you were hoping for a shorter answer, but it's not possible to give one, I'm afraid. I would strongly suggest, as you prepare, keeping notes of those DS questions you answer incorrectly because you didn't consider enough possible values for x. You may notice a pattern- perhaps you aren't considering negative values. for example- and you may then be able to learn from that.
