If x y^2 z^4, which of the following statements could be

Question

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

I.  x>y>z

II.  z>y>x

III.  x>z>y

A. I only
B. I and II only
C. 1 and III only
D. II and III only
E. I, II and III

Bunuel · Accepted Answer

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

I. x>y>z
II. z>y>x
III. x>z>y

A. I only
B. I and II only
C. I and III only
D. II and III only
E. I, II and II

As this is a COULD be true question then even one set of numbers proving that statement holds true is enough to say that this statement should be part of correct answer choice.

Given:  x > y^2 > z^4 .

1.  x>y>z  --> the easiest one: if  x=100 ,  y=2  and  z=1  --> this set satisfies  x > y^2 > z^4  as well as given statement  x>y>z . So 1 COULD be true.

2.  z>y>x  --> we have reverse order than in stem ( x > y^2 > z^4 ), so let's try fractions: if  x=\frac{1}{5} ,  y=\frac{1}{4}  and  z=\frac{1}{3}  then again the stem and this statement hold true. So 2 also COULD be true.

3.  x>z>y  --> let's make  x  some big number, let's say 1,000. Next, let's try the fractions for  z  and  y  for the same reason as above (reverse order of  y  and  z ):  y=\frac{1}{3}  and  z=\frac{1}{2} . The stem and this statement hold true for this set of numbers. So 3 also COULD be true.

Answer: E.

KarishmaB · Answer

Here is a video solution to this problem: https://youtu.be/Ouz8Z-MhZkU I think the actual question is: Which of the following could be true? Plugging in numbers work best for such questions. The only thing to keep in mind is that you should plug in the right numbers. How do you know the right numbers? When I see x > y^2 > z^4 , I think that y^2 and z^4 are non negative. Since y^2 > z^4 , y^2 cannot be 0. Only z can be 0. x has to be positive. Also, I have to take into account two ranges: 0 to 1 and 1 to infinity. The powers behave differently in these two ranges. I will consider negative numbers only if I have to since with powers, they get confusing to deal with. The question says: "Which of the following could be true?" We have to find examples where each relation holds. I. x > y > z This is the most intuitive of course. z = 0, y = 1 and x = 2 2 > 1^2 > 0^4 II. z > y > x Let me consider the 0 to 1 range here. Say z = 1/2, y = 1/3 and x = 1/4 1/4 > 1/9 > 1/16 III. x > z > y Let's stick to 0 to 1 range. z > y as in case II above but x has to be greater than both of them. Say z = 1/2, y = 1/3 and x = 1 1>1/9 > 1/16 So all three statements could be true.

dabral · Answer

This is a really tough problem. Here is my video explanation: https://gmatquantum.com/gmatprep-algebr ... tatements/ Dabral

fluke · Answer

We just need to remember that 1. the number decreases in value with increment in the power of the number if 0< number< 1; if x=0.1; x>x^2>x^3>x^(100) because x is between 0 and 1. 2. the number increases in value with increment in the power of the number if number>1 if x=2; x

imhimanshu · Answer

Hi Bunuel/Karishma,
Thanks for the earlier response.. I think, I am very weak in Inequalities..
Could you please post how to go about this question in algebraic way.. ...also if you could let me know how do you make sure about the "Range of the values", that will also work.. 
Thanks
H

Bunuel · Answer

Plug-in method is really the best way to handle such kind of questions. No need to look for some kind of textbook or algebraic ways. Notice that there are are certain GMAT questions which are pretty much only solvable with plug-in or trial and error methods (well at leas in 2-3 minutes). Many difficult inequality problems will often require some sort of plug-ins, as part of your technique or else you'll spend too much time solving them with algebra. Which means that you MUST make plug-in methods part of your arsenal if you want to get a decent score. Inequality questions to practice. DS: search.php?search_id=tag&tag_id=184 PS: search.php?search_id=tag&tag_id=189 Hope it helps. P.S. I'm not sure understood the following part of your post: "how do you make sure about the "Range of the values", that will also work.. "

imhimanshu · Answer

Thanks Bunuel for your response..
This is what I mean when I said range - Red Part in Karishma's response" "

Bunuel · Answer

First notice that since x>z^4 (x is greater than some nonnegative value) then x>0. Now, as Karishma correctly noted, numbers in powers behave differently in the range {0. 1} and {1. +infinity}. For example: If 01 then a, a^2 and a^4 will be ordered as follows: 1--(a)--(a^2)--(a^4)-- So, we should take the above difference in ordering into account when picking numbers for x, y, and z, since we need to find the values which satisfy 3 different statements. Hope it's clear.

suhail91 · Answer

Algebraic solution: In the question we are given: x>y2>z4, hence from concepts of inequalities we break it into 2 parts: x>y2 and y2>z4. 1. x>y2 means -x(1/2)z4 means -y1, we also take such cases for all three of them. 1. 01 2. 01 3. 01 Hence looking at the combinations we find we have 8 possibilities (2*2*2). taking the 2 general ones: 1. x>1 y>1 z>1. In the general formula we simply put x,y,z and get x>y>z. (Would have figured initially). 2. 0y>x. 3. x>1 01/y>1/z. since only 1/y>1/z are in reciprocal hence z>y by inequalities. thus x>z>y. Therefore we can get 8 possibilities and the fact is all of them are correct.

dabral · Answer

Hi vinnisatija,

Yes, if you see a "could be" question, then just one example that satisfies the given condition is sufficient. In case of "must be" questions, the required condition must hold true under all circumstances along with whatever additional constraint is given in the problem.

Dabral

JeffTargetTestPrep · Answer

We are given that x > y^2 > z^4 and need to determine which statements must be true. Let’s test each Roman Numeral.
 
I. x > y > z 
 
Notice that the order of arrangement of x, y, and z in the inequality x > y > z is the same as the order of arrangement of x, y^2, and z^4 in the inequality x > y^2 > z^4, so we want to test positive integers in this case.
 
x = 10
 
y = 3
 
z = 1
 
Notice that 10 > 3 > 1 for x > y > z AND 10 > 9 > 1 for x > y^2 > z^4.
 
We see that I could be true.
 
II. z > y > x 
 
Notice that the order of arrangement of x, y, and z in the inequality z > y > x differs from the order of arrangement of x, y^2, and z^4 in the inequality x > y^2 > z^4, so we want to test positive proper fractions in this case. This is because we need to decrease the value of y and z to make them work within the given inequality.
 
x = 1/5
 
y = 1/3
 
z = 1/2
 
Notice that 1/2 > 1/3 > 1/5 for z > y > x AND 1/5 > 1/9 > 1/16 for x > y^2 > z^4.
 
We see that II could be true.
 
III. x > z > y

Notice that the order of arrangement of y and z in the inequality x > z > y differs from the order of arrangement of y^2 and z^4 in the inequality x > y^2 > z^4, so we once again want to test positive proper fractions. This is because we need to decrease the value of z to make it work within the given inequality (that is, we want to swap the order of z4 and y2 even if z > y).
 
x = 1/2
 
y = 1/4
 
z = 1/3
 
Notice that 1/2 > 1/3 > 1/4 for x > z > y AND 1/2 > 1/16 > 1/81 for x > y^2 > z^4.
 
We see that III could be true.
 
Answer: E

BrentGMATPrepNow · Answer

If we CAN find a set of values that satisfies a statement AND yields values such that x > y² > z⁴, then we'll keep that statement.

Statement I. x > y > z  
If x = 2, y = 1, and z = 0, then x > y² > z⁴
KEEP statement I

Statement II. z > y > x  
If x = 1/4, y = 1/3, and z = 1/2, then x > y² > z⁴
KEEP statement II

Statement III. x > z > y  
If x = 2, y = -1, and z = 0, then x > y² > z⁴
KEEP statement III

Answer: E

Cheers,
Brent

nutella · Answer

VeritasKarishma 
 Bunuel 
Please help me understand where I am going wrong. 
If you choose x = 1/3 and y = 1/2 then y>x but y^​2<x
So how can we choose the options that include 1 - which is x>y>z. Because that might not be true. 
Thank you in advance!!

KarishmaB · Answer

Stmnt I will be true for some values of x, y and z. Note that only one of the three statements can be true for a given set of values of x, y and z. The three statements hold for different values of x, y and z as shown in my post above: https://gmatclub.com/forum/if-x-y-2-z-4 ... l#p1032823 Basically, the question should be: Which of the following "could be" true for some values of x, y and z? I. x > y > z would be true when z = 0, y = 1 and x = 2

CrackverbalGMAT · Answer

Its given  x > y^2 > z^4 
Since its a "Could be true " question , we assume values for x,y and z and try to prove that each statement could be true or not.

1. x > y > z

Assume x = 1, y = 1/2 , z = 1/4

Then,  1 > (1/2)^2 > (1/4)^4

Also 1 > 1/2 > 1/4

Hence x > y > z could be true

2. z > y > x
 Assume x= 1/8, y= 1/4 ,z = 1/3

Then  1/8 > (1/4)^2 > (1/3)^4   >>>  x > y^2 > z^4

Also 1/3 > 1/4> 1/8 
Hence z > y > x could be true

3. x > z >y

Assume x= 1, y= -1/2 , z = 1/2

Then  x > y^2 > z^4 
Also x > z > y
This statement also could be true.

Since all the Statements could be true, Answer is OPTION E

Thanks,
Clifin J Francis,
GMAT SME

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=2BOz0WoEAZM

AADINAATH · Answer

Bunuel , I face issues in DS questions like these. How do you come up with cases?

egmat · Answer

Hello AADINAATH, This is not a DS question, but I understand what your doubt and concern is regarding. It gets difficult, especially under a time-controlled exam environment, to think of cases or examples which would satisfy or not satisfy different conditions. Through this post, I will help you identify certain ranges in which all numbers follow some fixed patterns. Knowing these ranges and the patterns within them will help you easily think of examples for every condition in such questions. Let’s get started! (1, ∞) This is the range of all numbers greater than 1. For example, 20 and 10 . PATTERN 1: Observe that 10 < 10 ^2 < 10 ^3 < 10 ^4 < …. and 20 < 20 ^2 < 20 ^3 < 204 < …. Using this observation, we can generalize that if x > 1, then x < x ^2 < x ^3 < x ^4 < …… That is, as power (n) increases, the value of x ^n also increases for all n > 0. PATTERN 2: Observe that 20 > 10; (20)2 > (10)2, (20)3 > (10)3, and so on. Using this observation, we can generalize that if two numbers x and y belong to the range (1, ∞), then if x > y, then x ^n > y ^n for all n > 0. (0, 1) This is the range of all positive numbers less than 1. For example, 0.4 and 0.5 . PATTERN 1: Observe that 0.4 > 0.4 ^2 > 0.4 ^3 > 0.4 ^4 > …. and 0.5 > 0.5 ^2 > 0.5 ^3 > 0.5 ^4 >…. Using this observation, we can generalize that if 0< x < 1, then x > x ^2 > x ^3 > x ^4 > …… That is, as power (n) increases, the value of x ^n decreases for all n > 0. PATTERN 2: Observe that 0.5 > 0.4; (0.5) ^2 > (0.4) ^2 , (0.5) ^3 > (0.4) ^3 , and so on. Using this observation, we can generalize that if two numbers x and y belong to the range (0, 1), then if x > y, then x ^n > y ^n for all n > 0. (-∞, -1) This is the range of all numbers less than -1. For example, -10 and -20. PATTERN 1: Observe that 0 > -10 > (-10) ^3 > (-10) ^5 > …. and 0 < (-10) ^2 < (-10) ^4 < (-10) ^6 <…. Using this observation, we can generalize that if x < -1, then 0 > x > x ^3 > x ^5 > …. and 0 < x ^2 < x ^4 < x ^6 < … That is, for odd powers n, as power increases, the value of x ^n decreases for all integer n > 0. And for even powers (m), as power increases, the value of x ^m also increases for all integer m > 0. PATTERN 2: Observe that -10 > -20; (-10) ^3 > (-20) ^3 and (-10) ^5 > (-20) ^5 and so on but (-10) ^2 < (-20) ^2 and (-10) ^4 < (-20) ^4 and so on. Using this observation, we can generalize that if two numbers x and y belong to the range (-∞, -1), then if x > y, then x ^n > y ^n for all odd integers n > 0 and x ^m < y ^m for all even integers m > 0. (-1, 0) This is the range of all negative numbers greater than -1. For example, -0.5 and -0.4 . PATTERN 1: Observe that -0.5 < (-0.5) ^3 < (-0.5) ^5 < …. and (-0.5) ^2 > (-0.5) ^4 > (-0.5) ^6 >…. Using this observation, we can generalize that if -1 < x < 0, then x < x ^3 < x ^5 < …. and x ^2 > x ^4 > x ^6 > … That is, for odd powers (n), as power increases, the value of x ^n also increases for all n > 0. And for even powers (m), as power increases, the value of x ^m decreases for all n > 0. PATTERN 2: Observe that -0.4 > -0.5; (-0.4) ^3 > (-0.5) ^3 and (-0.4) ^5 > (-0.5) ^5 and so on but (-0.4) ^2 < (0.5) ^2 and (0.4) ^4 < (0.5) ^4 and so on. Using this observation, we can generalize that if two numbers x and y belong to the range (-1, 0), then if x > y, then x ^n > y ^n for all odd integers n > 0 and x ^m < y ^m for all even integers m > 0. These are the ranges that you should consider while thinking of cases and examples. Having these ranges in mind and using a little bit of inference will make it easier for you to solve questions of such kind. TAKEAWAYS: c) For positive numbers, if x > y, all positive powers of x will be greater than the same power of y. So, if x > y, x ^n > y ^n where n > 0. d) For negative numbers, if x < y, x ^n < y ^n where n is odd but x ^m > y ^m where m is even. Hope this helps! Best Regards, Ashish Arora Quant Expert, e-GMAT

sarthak1701 · Answer

This is more of a semantic problem than a quant one lol. I took x =1/5, y = 1/3 and z = 1/2

These numbers satisfy the condition: x>y^2>z^4

But since the numbers do not satisfy the 1st condition, I rejected the 4 choices containing that condition and selected the one choice that does not contain the 1st condition but alas I was wrong.

KarishmaB  For such questions that say COULD BE true, we only need one set of numbers to satisfy a condition to accept it, and for the questions that say MUST BE true, we only need one set of numbers that DO NOT Satisfy the condition to reject it. Correct?

If x > y^2 > z^4, which of the following statements could be

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