If x > y2 > z4

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If x > y2 > z4

by adwaitkasbekar » Thu Dec 23, 2010 6:33 am
If x > y^2 > z^4, which of the following could be true?
x > y > z
z > y > x
x > z > y
a. I
b. I and II
c. I and III
d. II and III
e. I, II, and III

This problem states "could be scenario". Kindly anyone suggest how to tackle such problem.

OA is [spoiler]E
[/spoiler]

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by Anurag@Gurome » Thu Dec 23, 2010 6:52 am
adwaitkasbekar wrote:If x > y^2 > z^4, which of the following could be true?
x > y > z
z > y > x
x > z > y
(I) x > y^2
  • Certainly for integer values of x and y, x is always greater than y. But what about fractional values> f y is a fraction then y^2 is < y. Now always there is a fraction x, such that y > x > y^2. Thus for fractional values x may be less than y. Therefore regarding the relation between x and y all of the options could be true.
(II) x > z^4
  • Same logic as above.
(II) y^2 > z^4
  • This directly implies y > z^2. Now again apply same logic as above.
The correct answer is E.
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by adwaitkasbekar » Thu Dec 23, 2010 7:02 am
Thanks Anurag.

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by captcha » Thu Dec 23, 2010 10:26 am
If x > y^2 > z^4, which of the following could be true?
x > y > z

x=100; y=5; z=1;


z > y > x

x=.01 y = .02 z= .03


x > z > y

x=100; y=.2 z =.3

All 3 possible . Hence E

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by paml » Sun Apr 19, 2015 7:11 pm
Can anyone recommend ways to get familiar with what happens to inequalities and exponents? And can anyone recommend similar problems in the official guide to this one that we can price with?

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by Brent@GMATPrepNow » Sun Apr 19, 2015 8:08 pm
paml wrote:....can anyone recommend similar problems in the official guide to this one that we can price with?
I think this one is an official question: https://qa.www.beatthegmat.com/0-888-sq- ... 74282.html

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by Brent@GMATPrepNow » Sun Apr 19, 2015 8:11 pm
Here's my solution to the original question:
If x > y² > z�, 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 III
If we CAN find a set of values that satisfies a statement AND yields values such that x > y² > z�, then we'll keep that statement.

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

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

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

Answer: E

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by Matt@VeritasPrep » Sun Apr 19, 2015 10:54 pm
paml wrote:Can anyone recommend ways to get familiar with what happens to inequalities and exponents? And can anyone recommend similar problems in the official guide to this one that we can price with?
I've noticed this question a couple times today (both presumably from you), and wanted to advise you not to become too dependent on the OG. Most of the questions in those books, while helpful, are (i) very old and (ii) explained in great depth everywhere on the internet. That combination makes them very unlikely to to be asked again in anything that close to their original form. While you will see many similar ideas in modern questions, there are better sources to prep for the exam in 2015, I think.

As far as official sources go, the GMAT's Question Pack 1 and Exam Pack 1 are MUCH more reflective of the modern test than the OGs or the two free mba.com exams, in my opinion.

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by paml » Mon Apr 20, 2015 3:45 pm
Thanks Brent & Matt!

The reason I've been asking for similar OG questions was that my GMAT instructor had instructed that I look for questions in the OG that mirrored (as closely as possible) questions that I got wrong on my practice exams, for pattern recognition purposes. Thus, I've been looking up those questions that I got wrong and have been posting to ask for questions that mirror them. If you guys have any suggestions on how to get better with pattern recognition, let me know. I will check out GMATs question pack/exam pack -- are these the ones that come with the free GMATprep software, or are the ones you referred to the ones that you pay for? Thanks!

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by Scott@TargetTestPrep » Wed May 13, 2015 9:41 am
adwaitkasbekar wrote:If x > y^2 > z^4, which of the following could be true?
x > y > z
z > y > x
x > z > y
a. I
b. I and II
c. I and III
d. II and III
e. I, II, and III

This problem states "could be scenario". Kindly anyone suggest how to tackle such problem.

OA is [spoiler]E
[/spoiler]
Solution:

The first thing to notice in this problem is that we are being asked which of the following could be true. This means that we can create different scenarios for the values of x, y, and z, to determine whether each Roman numeral could be true. Another thing to consider is that any time we are presented with a problem with inequalities and exponents, there is a high likelihood that we are being tested on the way in which different types of numbers react to being raised to positive integer exponents. For example, when a number greater than 1 is raised to an integer exponent greater than 1, the resulting value increases (e.g., 32 > 3). When a number between 0 and 1 (i.e., a positive proper fraction) is raised to an integer exponent greater than 1, the resulting value decreases (e.g., (1/3)^2 < 1/3). Let's keep that in mind as we test convenient numbers for each Roman numeral. Remember, too, that whatever convenient numbers we choose to use must fulfill the statement: x > y² > z�.

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 AND 10 > 9 > 1

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 it work within the given inequality.

x = 1/5

y = 1/3

z = ½

Notice that ½ > 1/3 > 1/5 AND 1/5 > 1/9 > 1/16.

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 in this case. 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 z^4 and y^2 even if z > y).

x = 1/2

y = 1/4

z = 1/3

Notice that ½ > 1/3 > 1/4 AND ½ > 1/16 > 1/81.

We see that III could be true.

The answer is E

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by gmat_for_life » Tue Apr 12, 2016 8:46 am
Hello Experts,

Is it really possible to solve this question within 2 minutes? How do we find out the exact numbers which need to be plugged into the answer choices?

Regards,
Amit

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by DavidG@VeritasPrep » Tue Apr 12, 2016 9:30 am
gmat_for_life wrote:Hello Experts,

Is it really possible to solve this question within 2 minutes? How do we find out the exact numbers which need to be plugged into the answer choices?

Regards,
Amit
Of course it's possible to do this in 2 minutes! If it weren't, it wouldn't be on the test. Typically, when you see questions that involve inequalities and exponent, values between 0 and 1 will factor into your thinking. Otherwise, approach this strategically.

The first statement is easy to prove.

I) x = 10, y = 1 and z = 0.

Now notice that z > y in both II and III. If z = .8 and y = .79, clearly (.79)^2 > (.8)^4. So we can use those numbers for both statements. So now all we have to do is pick x's to satisfy the conditions. Not too hard.

II) z = .8, y = .79, x = .78
III) x = 100, z = .8, y = .79

They all work.
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by [email protected] » Tue Apr 12, 2016 9:45 am
Hi Amit,

From the way you've written your post, it sounds like you don't TEST VALUES very often. That Tactic is remarkably useful in the Quant section of the GMAT (and can sometimes be used in IR and rare CR prompts), so learning it and honing those skills would likely pay off huge on Test Day. The Roman Numerals are also designed to help you narrow down your thinking (you're not TESTing "random" VALUES, you're TESTing ones that 'fit' the given restrictions). The 'math work' involved in most Quant questions typically takes less than a minute to complete - the rest of the time is spent reading the prompt, taking notes, deciding how to approach the question, etc. If you're spending lots of time on individual questions, then it's likely that 'your way' of approaching the prompt is the 'long way.'

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by Matt@VeritasPrep » Tue Apr 12, 2016 6:24 pm
gmat_for_life wrote:Hello Experts,

Is it really possible to solve this question within 2 minutes? How do we find out the exact numbers which need to be plugged into the answer choices?

Regards,
Amit
For sure, though maybe not on the first try!

When testing numbers, there are a few ranges to consider:

between 0 and 1
between 0 and -1
greater than 1
less than -1
even
odd
if primes are mentioned, 2
if nonnegative is mentioned, 0 *for sure*

After you've seen these numbers come up in other, similar situations, you'll feel more comfortable plugging in the appropriate values; but at first it's a real headache.

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by gmat_for_life » Fri Apr 15, 2016 8:42 am
Hello Rich and Matt,

I do face issues while trying to plug in the numbers because I go for random plugging of numbers. Could you please let me know if the list of number sets mentioned in the post below can be applied to all types of inequality questions?

Regards,
Amit