GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 27 Jun 2019, 01:43

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

 
Manager
Manager
avatar
Joined: 25 Jul 2010
Posts: 102
If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 05 Sep 2010, 13:17
26
167
00:00
A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

51% (01:56) correct 49% (02:01) wrong based on 3184 sessions

HideShow timer Statistics


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 II
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 05 Sep 2010, 14:06
56
62
Orange08 wrote:
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.
_________________
Most Helpful Community Reply
Retired Moderator
avatar
Joined: 20 Dec 2010
Posts: 1743
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 28 Jun 2011, 13:17
6
Baten80 wrote:
If x > y^2 >z^4, which of the following statements could be true?

I. x > y > z

x=10000
y=10; y^2=100
z=1; z^4=1
x>y^2>z^4

II. z > y > x
z=0.5; z^4=0.0625
y=0.4; y^2=0.16
x=0.3
x>y^2>z^4

III. x > z > y
x=0.5
z=0.2; z^4=0.0016
y=0.1; y^2=0.01
x>y^2>z^4

a. I only
b. I and II only
c. I and III only
d. II and III only
e. I, II and III


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<x^2<x^(100) because x is more than 1.


Ans: "E"
_________________
General Discussion
Director
Director
User avatar
B
Affiliations: GMATQuantum
Joined: 19 Apr 2009
Posts: 609
If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post Updated on: 17 Sep 2018, 00:51
6
2
This is a really tough problem. Here is my video explanation:
https://gmatquantum.com/gmatprep-algebr ... tatements/

Dabral

Originally posted by dabral on 28 Jun 2011, 12:19.
Last edited by dabral on 17 Sep 2018, 00:51, edited 1 time in total.
Veritas Prep GMAT Instructor
User avatar
D
Joined: 16 Oct 2010
Posts: 9369
Location: Pune, India
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 23 Jan 2012, 01:52
21
17
arps wrote:
1) x > y2 > z4

which of the following is 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 III


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.
_________________
Karishma
Veritas Prep GMAT Instructor

Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >
Senior Manager
Senior Manager
avatar
Joined: 07 Sep 2010
Posts: 254
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 28 Feb 2012, 02:31
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
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 28 Feb 2012, 02:46
5
1
imhimanshu wrote:
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


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.. "
_________________
Senior Manager
Senior Manager
avatar
Joined: 07 Sep 2010
Posts: 254
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 28 Feb 2012, 04:01
1
Thanks Bunuel for your response..
This is what I mean when I said range - Red Part in Karishma's response"
VeritasPrepKarishma wrote:
arps wrote:
1) x > y2 > z4

which of the following is 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 III


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.
"
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 28 Feb 2012, 05:59
imhimanshu wrote:
Thanks Bunuel for your response..
This is what I mean when I said range - Red Part in Karishma's response"
VeritasPrepKarishma wrote:


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 0<a<1 then a, a^2 and a^4 will be ordered as follows: 0--(a^4)--(a^2)--(a)--1

If a>1 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.
_________________
Manager
Manager
avatar
Joined: 17 Oct 2012
Posts: 61
Location: India
Concentration: Strategy, Finance
WE: Information Technology (Computer Software)
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 08 Sep 2013, 07:15
Restrictions are not provided on the variables so I planned to check different values I and used x=y=z=1/2.
As if I take 1/2 for each variables, its given condition would be satisfied and it will become 1/2>1/4>1/8.

So according to me none of the conditions are satisfied.
Am I doing anything wrong here?
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 08 Sep 2013, 07:40
chetan86 wrote:
Restrictions are not provided on the variables so I planned to check different values I and used x=y=z=1/2.
As if I take 1/2 for each variables, its given condition would be satisfied and it will become 1/2>1/4>1/8.

So according to me none of the conditions are satisfied.
Am I doing anything wrong here?


Notice that the question asks "which of the following statements could be true" NOT "which of the following statements must be true"
_________________
Intern
Intern
avatar
Joined: 19 Oct 2013
Posts: 1
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 07 Nov 2013, 02:10
Bunuel wrote:
Orange08 wrote:
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.




Isn't it stated in the exam that assume all numbers are integers? We can't try fractions unless they say they are not integers.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 07 Nov 2013, 02:19
1
SaramiR wrote:
Bunuel wrote:
Orange08 wrote:
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.




Isn't it stated in the exam that assume all numbers are integers? We can't try fractions unless they say they are not integers.


No that's not true at all. All numbers on the test represent real numbers: Integers, Fractions and Irrational Numbers. You cannot assume a variable is integer if you are not explicitly told so.
_________________
Intern
Intern
avatar
Joined: 31 Jul 2015
Posts: 3
GMAT 1: 720 Q50 V37
Reviews Badge
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 07 Aug 2015, 15:45
1
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)<y<x(1/2)
2. y2>z4 means -y<z2<y, but a square cannot be negative so 0<z2<y, this implies -y(1/2)<z<y(1/2).

Now we plot all these points on number line with the intersection of there ranges. But before that we need to understand that we will only be taking x,y,z as positive since if we take y as negative for example(easiest one) the value z(2) becomes negative, whereas a square can never be negative.

hence we plot all of them on the positive x-axis. From above 1 & 2 point we get a general range as such 0<z<y(1/2)<y<x(1/2)<x.
Now, we need to see that we haven't in reality considered various values of x,y,z but have come up with a general idea of how they look on the number line.
Now we define the ranges, since we know about a^x graph varies for values 0<a<1 and a>1, we also take such cases for all three of them.
1. 0<x<1 and x>1
2. 0<y<1 and y>1
3. 0<z<1 and z>1

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. 0<x<1, 0<y<1 and 0<z<1. In this possibility put x as 1/x, y as 1/y and z as 1/z in general formula we get z>y>x.
3. x>1 0<y<1 and 0<z<1. In this put y as 1/y and z as 1/z. Keep x as x in general formula, we see x>1/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.
Director
Director
User avatar
B
Affiliations: GMATQuantum
Joined: 19 Apr 2009
Posts: 609
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 22 Aug 2015, 23:59
1
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
Target Test Prep Representative
User avatar
G
Status: Head GMAT Instructor
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2823
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 10 Mar 2017, 10:41
1
Quote:
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


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
_________________

Jeffrey Miller

Head of GMAT Instruction

Jeff@TargetTestPrep.com
TTP - Target Test Prep Logo
122 Reviews

5-star rated online GMAT quant
self study course

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager
Manager
User avatar
G
Status: Not Applying
Joined: 27 Apr 2009
Posts: 53
Location: India
Schools: HBS '14 (A)
GMAT 1: 730 Q51 V36
CAT Tests
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 11 May 2017, 05:35
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

Solution:
To answer questions like this, use ZONEF.
Z = zero
O = one
N = negative integers
E = Extreme Integers (Read as Positive Integers > 1)
F = Fractions (think both positive and negative)

For x > y^2 > z^4, the following numbers work: x = 10, y = 3, z = 1. So , I is true.
Eliminate (D).

The easiest way to prove II true is to multiply the above numbers by -1, but as you are dealing with even exponents, negative numbers will not work.
That leaves us with F(Fractions).

Let z = 0.9, y = 0.7 and x = 0.1
These numbers keep both x > y^2 > z^4 and z > y > x true.
Eliminate (A) and (C).

Now, proving the third is easy. Just take x = any big positive number, say 10
Now, you have x = 10, y = 0.7 and z = 0.9
These values keep both x > y^2 > z^4 and x > z > y true.

So, all three are true.
The answer is (E).
_________________
http://www.wizius.in
Better Prep. Better Scores. Better Schools

Guaranteed Admission to Top-50 MBA Programs
You either get-in or get your money-back.
Retired Moderator
avatar
P
Joined: 17 Jun 2016
Posts: 503
Location: India
GMAT 1: 720 Q49 V39
GMAT 2: 710 Q50 V37
GPA: 3.65
WE: Engineering (Energy and Utilities)
GMAT ToolKit User Reviews Badge
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 25 Jun 2017, 12:13
Orange08 wrote:
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 II


Question is about "COULD BE" true ..so we just need one set of numbers which ensures that the above conditions are true..

Take x = 100, y = 5 and z = 2
So statement 1 and the statement in the question stem are true..

Take x = 0.4, y = 0.5 and z = 0.6
So statement 2 and the statement in the question stem are true

Take x = 100, y = 0.5 and z = 0.6
So statement 3 and the statement in the question stem are true

So option E
_________________
CEO
CEO
User avatar
V
Joined: 12 Sep 2015
Posts: 3787
Location: Canada
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 17 Jan 2018, 16:37
Top Contributor
1
Orange08 wrote:
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 II


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
_________________
Test confidently with gmatprepnow.com
Image
Intern
Intern
avatar
B
Joined: 16 Apr 2014
Posts: 16
Re: If x > y^2 > z^4, which of the following statements could be  [#permalink]

Show Tags

New post 16 Apr 2018, 20:49
Dear Moderator,

Pl. clear how is it possible to pick different numbers for all three stem. just because it says " COULD BE"?
I am considering one value for x, one for y and one for z and try to find out answer for all three.

Thanks!
GMAT Club Bot
Re: If x > y^2 > z^4, which of the following statements could be   [#permalink] 16 Apr 2018, 20:49

Go to page    1   2    Next  [ 21 posts ] 

Display posts from previous: Sort by

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

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne