Problem Solving

Hi,

how do I pick numbers in such sums & what all do I need to test here...
+ve , -ve , fractions / decimals.....square roots ?

Is there an easier method or a formulae to deal with such sums ?
[spoiler]OA = option E[/spoiler]

Hi,
x>y^2>z^4

The rule here is:

x>y^2 then
x>y if x>1
or x<y if x<y<1

the same for y^2 and z^4

If y^2>z^4 then
y^2>z^2 when y^2>1
or y^2<z^2 when y^2<z^2<1 ( this is a little messy, but it's the same rule as above when x>y^2)

The inequal sign can be change freely, it depends on the value of x,y,z then all I,II,III are possile

I'll plug in some example:

I. x = 100000, y = 10, z=3
x>y^2>z^4 and x>y>z

II. x = 0.3; y =0.4; z= 0.5
z>y>x and x>y^2>z^4

III. x = 10; y = 0.4 ; z = 0.5
x>z>y and x>y^2>z^4

Moreover, only x needs to be positive. y and z can be either negative or positive, so they can switch their orders freely.

Hey Skywalker,

Good question - both your question regarding this problem and the problem itself. The strategy I'd recommend on any problems that seem to require you to plug in numbers is this:

Know your goal first.

On many Data Sufficiency questions, for example, your goal is to get two different answers (e.g. YES and NO) to prove that the statement is insufficient. Here, your goal is to determine "could this be true?" - you want to find a situation in which each option is satisfied.

The nice thing about having a goal is that you can then determine strategically how to accomplish it. The types of numbers that are often useful in these cases include those that you suggested: negatives, fractions, 0, 1, something approaching infinity, etc. Most importantly, though, you should think about the situation and ask how you'll accomplish that goal so that you can efficiently pick the kind of number likely to do it for you.

Here, since our goal is to try to make each statement true, let's attack them one at a time:

1) x > y > z

If we know that x > y^2 > z^4, our strategy should be to make x as big as possible so that there's no way that we need to worry about it...it will just stay huge and out of our way, larger than the rest. If x = 1,000,000 for example, it's huge and we don't need to worry about it - it's much bigger than any of the single-digit numbers we'll try for y and z.

Now, to keep z small without much trouble, we can also fix that one at 1...z^4 will then be 1 and z will still be 1, so that gives us the entire range of 1 to a million for y^2 and y. If we call y, say, 2, then y^2 fits in that range and so does y, and 1 should be pretty easy to prove.


2) z > y > x

Here, the challenge is that we know that z^4 is the smallest of these numbers, but now we need z to become the biggest. We also know that x is fixed so we won't change that one at all, so z needs to be bigger than z^4. Well, what kind of number becomes smaller when you take it to an exponent? That's your strategy...the only number that will work is one that when you take it to a larger power, it becomes smaller. That should tip you off...you need a fraction between 0 and 1.

Because we also need the same performance out of y (y^2 is less than x but y is greater than x), we'll need a fraction there, too. If we fix x at something easy to visualize like 1/2, then we need to find numbers that will go from >1/2 to < 1/2 when we square them or take them to the power of 4:

3/4 > 2/3 > 1/2 works

and if we take z^4 we get a really small fraction (81/256) that's obviously less than 1/2, so we can definitely see that becoming the smallest.

Taking 2/3 and squaring it we'll get 4/9, which we know is very close to but still not quite 1/2, and the inequality is satisfied, so statement 2 works, too.


3) x > z > y

Now our goal is a little different - the first time we just needed to keep the same order and the second time we had to completely reverse it. Now we need x to stay the biggest, but we need to flip-flop y and z. Our goal, then, is to keep x big (which we know how to do...make it 1,000,000), but pick numbers for y and z for which:

y^2 > z^4

BUT

z > y

What kind of numbers will work? Well, when using even exponents it should be pretty natural to use negative numbers, because taking a negative number to an even power makes it positive, so it's easy to make that transposition of z and y. If we fix z and z^4 at 1 again, our real goal is to get y to move...when squared it's bigger than 1 and when not it's less than. What kind of number works? A negative, so let's call y = -2. y is then the smallest number (1,000,000 > 1 > -2) but y^2 is greater than 1 (it's 4), so we also have 1,000,000 > 4 > 1.



Hopefully that lengthy reply is more helpful than distracting. My goal was just to walk through the whole thought process - the GMAT is testing whether you can think strategically to focus on a goal and set up the steps to get there. That checklist of negative/positive/fraction/zero is a great one to give you the tools to think strategically, but overall the focus should be on trying to accomplish whatever your goal is.

Thanks Brian!