Problem Solving

Q) If x > y^2 > z^4, which of the following could be true?

1) x>y>z
2) z>y>x
3) x>z>y

A) 1 only
B) 1 and 2 only
C) 1 and 3 only
D) 2 and 3 only
E) 1,2, and 3

OA:E

x>y^2>z^4
1)x>y>z => true when x=10, y=2, z=-.5
2) z>y>x true when z=0.8,y=.7,x=.5
3)x>z>y true when x=10,z=-.5,y=-1
IMO E

Love the question - thanks for sharing!

Any PS problem that uses the phrase "could be true" or "must be true" should trigger that same "they're testing logic" notion in your mind that you have for Data Sufficiency questions. "Could be true" is asking "can you find at least one set of numbers for which this works?". Inversely, "must be true" is asking "can you find at least one set of numbers for which this WON'T work?" as you go through process of elimination. And we know that Data Sufficiency is asking "will you always get this same answer" or "is this answer a 'must be true'?". So the same logic holds.

When dealing with logic in math this way, you should really be pushing yourself to find "unique" or "special case" numbers that you can use as your one example of when a unique situation will occur. Negative numbers, nonintegers (especially those 0 < x < 1) and zero tend to be your greatest tools in this regard.

Hopefully the first case here (x > y > z) comes fairly naturally to you. Positive, well-spread-out integers should solve this nicely (10, 2, and 1, for example).

For the second, we reverse the order. How could z be the greatest value and, when taken to the fourth power, end up then being the least? Well, fractions between 0 and 1 get much, much smaller each time you take them to another exponent. So if we started with x = 1/2, y = 2/3 and z = 3/4, then going x^1, y^2 and z^4 would reverse the order: (2/3)^2 becomes 4/9, or slightly less than 1/2. And (3/4)^4 becomes really small at 81/256. So the order does reverse.

But you probably don't even need to plug in numbers there if you get the idea that fractions will become much, much smaller when taken to larger exponents. That's the real concept they're testing there - can you think of a number type that will behave that way?

For the third one, we need to keep x the largest and then flip-flop z and y. We can accomplish this by making y negative; y^2 would become positive and take its rightful place in the middle, but y on its own would be lessened because it's negative and that provides us the flip-flop. So let's make x huge to keep it out of the way (x = 10,000). And if y is -10 and z is 1, then x > z > y, but y^2 would become positive and leapfrog z to get its spot in the middle.

Here, again, you may not even need to test numbers - the key is "are you considering how a negative number will react?".

So, in summary, look at these "must be true" / "could be true" / Data Sufficiency problems in terms of "types of numbers" or "number properties. Often that's just enough to solve without having to work that hard. They're just testing whether you'll consider all the possibilities, so keep in mind that different types of numbers behave quite differently.

Thanks Brian !!
Nice post.