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If x, y, and z are positive numbers such that 3x < 2y < 4z , which of

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If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   20 May 2019, 04:56
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FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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


M37-15

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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   19 Nov 2021, 04:04
Expert Reply
Official Solution:


If \(x\), \(y\), and \(z\) are positive numbers such that \(3x < 2y < 4z\), which of the following statements could be true?

I. \(y = z\)

II. \(y > z\)

III. \(x > z\)


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


Notice that the question asks: "which of the following statements COULD be true?" NOT "which of the following statements MUST be true?"

I. \(y = z\). From the stem we know that \(2y < 4z\). So, \(y < 2z\). It's certainly possible that \(y = z\), for example, \(y = z=1\).

II. \(y > z\) From the stem we know that \(2y < 4z\). So, \(y < 2z\). It's certainly possible that \(y > z\), for example, \(y = 3\) and \(y=2\).

III. \(x > z\). From the stem we know that \(3x < 4z\). And again, it's possible that \(x > z\), for example, \(x=10\) and \(y=9\).


Answer: E
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   20 May 2019, 05:35
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Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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



Let us take each option..

I. y=z....
Let us compare y and z in the inequality 3x<2y<4z, so 2y<4z or y<2z..
So yes...let y=z=2 and x=1...3*1<2*2<4*2...3<4<8...

II. y>z....
Let us compare y and z in the inequality 3x<2y<4z, so 2y<4z or y<2z..
So yes, y can be between z and 2z..let y=3, z=2 and x=1...3*1<2*3<4*2...3<6<8...

III. x>z....
Let us compare x and z in the inequality 3x<2y<4z, so 3x<4z or x<4z/3..
So yes, x can be between z and 4z/3...let z=30 and x=31, and y=...3*31<2*50<4*30...91<100<120.

All three possible

E
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   20 May 2019, 09:24
chetan2u
thanks for the solution , I had a doubt that since the question has given that x,y,z are +ve numbers so shouldnt we check the given relations with a fraction value as well?


chetan2u wrote:
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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



Let us take each option..

I. y=z....
Let us compare y and z in the inequality 3x<2y<4z, so 2y<4z or y<2z..
So yes...let y=z=2 and x=1...3*1<2*2<4*2...3<4<8...

II. y>z....
Let us compare y and z in the inequality 3x<2y<4z, so 2y<4z or y<2z..
So yes, y can be between z and 2z..let y=3, z=2 and x=1...3*1<2*3<4*2...3<6<8...

III. x>z....
Let us compare x and z in the inequality 3x<2y<4z, so 3x<4z or x<4z/3..
So yes, x can be between z and 4z/3...let z=30 and x=31, and y=...3*31<2*50<4*30...91<100<120.

All three possible

E
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   08 Oct 2019, 03:18
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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


Asked: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

I. y = z
2y<4z
If y=z
2y<4y; Take y=1; 2<4
COULD BE TRUE

II. y > z
2y<4z
Take y = 1.5; z =1
2*1.5=3<4*1=4
COULD BE TRUE

III. x > z
3x<4z
Take x = 1.1
z = 1
3.3 < 4
COULD BE TRUE

IMO E
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   14 Oct 2019, 04:39
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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


Hello Bunuel,

For II condition y>z
if I take Y=3, Z=2 and X=1
Then,
3*1 < 2*3 < 4*2
3 < 6 < 8 ..........Correct
But if I choose values as below: -
X=1 Y=10 & Z=3
Then,
3*1 < 2*10 < 4*3
3 < 20 < 12 .......... Its wrong 20 > 12. So II. Y>Z is may be or may not be condition.

Similar for III. X>Z. It is also may be or may not be condition.

So don't you think that only I Y=Z i.e. Option A should be the answer.

Regards
Chetak
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   14 Oct 2019, 04:57
Expert Reply
chetaksatav wrote:
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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


Hello Bunuel,

For II condition y>z
if I take Y=3, Z=2 and X=1
Then,
3*1 < 2*3 < 4*2
3 < 6 < 8 ..........Correct
But if I choose values as below: -
X=1 Y=10 & Z=3
Then,
3*1 < 2*10 < 4*3
3 < 20 < 12 .......... Its wrong 20 > 12. So II. Y>Z is may be or may not be condition.

Similar for III. X>Z. It is also may be or may not be condition.

So don't you think that only I Y=Z i.e. Option A should be the answer.

Regards
Chetak


Notice that the question asks: "which of the following statements COULD be true?" NOT "which of the following statements MUST be true?"
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   14 Oct 2019, 05:14
Bunuel wrote:
chetaksatav wrote:
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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


Hello Bunuel,

For II condition y>z
if I take Y=3, Z=2 and X=1
Then,
3*1 < 2*3 < 4*2
3 < 6 < 8 ..........Correct
But if I choose values as below: -
X=1 Y=10 & Z=3
Then,
3*1 < 2*10 < 4*3
3 < 20 < 12 .......... Its wrong 20 > 12. So II. Y>Z is may be or may not be condition.

Similar for III. X>Z. It is also may be or may not be condition.

So don't you think that only I Y=Z i.e. Option A should be the answer.

Regards
Chetak


Notice that the question asks: "which of the following statements COULD be true?" NOT "which of the following statements MUST be true?"


Thanks Bunuel for clarifying.
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   14 Oct 2019, 23:39
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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

3x<2y<4z
3/2x<y<2z=1.5x<y<2z
Since all are positive numbers
x=1/2, y =1, z=1 then 0.75<1<2
x=0.61, y=0.99, z=0.6 then 0.91<0.99<1.2
E:)
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   01 Jul 2020, 09:28
chetan2u wrote:
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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



Let us take each option..

I. y=z....
Let us compare y and z in the inequality 3x<2y<4z, so 2y<4z or y<2z..
So yes...let y=z=2 and x=1...3*1<2*2<4*2...3<4<8...

II. y>z....
Let us compare y and z in the inequality 3x<2y<4z, so 2y<4z or y<2z..
So yes, y can be between z and 2z..let y=3, z=2 and x=1...3*1<2*3<4*2...3<6<8...

III. x>z....
Let us compare x and z in the inequality 3x<2y<4z, so 3x<4z or x<4z/3..
So yes, x can be between z and 4z/3...let z=30 and x=31, and y=...3*31<2*50<4*30...91<100<120.

All three possible

E


VeritasKarishma

In this ques, since we have to just tell whether a situation involving 2 variables is possible, do we also need to consider what value the 3rd variable can take? (Just want to confirm if doing so would add something)

Like for statement 3, I just considered x<4z/3. So take z=3/4=.75 ---> 1>x>.75 take x=.76 (Now do we need to see for what value of y the inequation 3x < 2y < 4z is true?)


Thanks in advance!
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   05 Jul 2020, 03:48
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GDT wrote:
chetan2u wrote:
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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



Let us take each option..

I. y=z....
Let us compare y and z in the inequality 3x<2y<4z, so 2y<4z or y<2z..
So yes...let y=z=2 and x=1...3*1<2*2<4*2...3<4<8...

II. y>z....
Let us compare y and z in the inequality 3x<2y<4z, so 2y<4z or y<2z..
So yes, y can be between z and 2z..let y=3, z=2 and x=1...3*1<2*3<4*2...3<6<8...

III. x>z....
Let us compare x and z in the inequality 3x<2y<4z, so 3x<4z or x<4z/3..
So yes, x can be between z and 4z/3...let z=30 and x=31, and y=...3*31<2*50<4*30...91<100<120.

All three possible

E


VeritasKarishma

In this ques, since we have to just tell whether a situation involving 2 variables is possible, do we also need to consider what value the 3rd variable can take? (Just want to confirm if doing so would add something)

Like for statement 3, I just considered x<4z/3. So take z=3/4=.75 ---> 1>x>.75 take x=.76 (Now do we need to see for what value of y the inequation 3x < 2y < 4z is true?)


Thanks in advance!


No, you do not need to worry about the third variable here.
Note that in stmnts I and II, I can just say x = 1/1000 and focus my energies on y and z.
Again, in statement III, for all values such that 3x < 4z, there will certainly be some values in between these two so it will work for some values of y for sure.
Note that we might want to be more careful if we were given that x, y and z are positive integers.
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   12 Nov 2020, 08:39
Expert Reply
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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

Solution:

If we let y = z = 2, then 2y = 4 and 4z = 8; so 2y < 4z is satisfied. This shows that y = z is possible. Notice that y = z = 1 is actually not a valid example since in that case, no positive integer value of x satisfies 3x < 2y.

If we let y = 3 and z = 2, then 2y = 6 and 4z = 8; so 2y < 4z is satisfied. This shows that y > z is possible.

Finally, if we let x = 10 and z = 9, then 3x = 30 and 4z = 36. Choosing y = 16, we see that 3x < 2y < 4z is satisfied. We see that x > z is also possible.

Answer: E
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   12 Nov 2020, 08:48
ScottTargetTestPrep wrote:
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

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

Solution:

If we let y = z = 2, then 2y = 4 and 4z = 8; so 2y < 4z is satisfied. This shows that y = z is possible. Notice that y = z = 1 is actually not a valid example since in that case, no positive integer value of x satisfies 3x < 2y.

If we let y = 3 and z = 2, then 2y = 6 and 4z = 8; so 2y < 4z is satisfied. This shows that y > z is possible.

Finally, if we let x = 10 and z = 9, then 3x = 30 and 4z = 36. Choosing y = 16, we see that 3x < 2y < 4z is satisfied. We see that x > z is also possible.

Answer: E


Hi ScottTargetTestPrep

You mentioned that "Notice that y = z = 1 is actually not a valid example since in that case, no positive integer value of x satisfies 3x < 2y."

But the question states that x,y and z are positive numbers (not integers) so we very well could have x = 0.5, y = z = 1. In that case
3x = 1.5
2y = 2
4z = 4

So 3x < 2y

Please could you correct me if I'm wrong


Thanks!

Posted from my mobile device
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   31 Aug 2021, 10:03
Bunuel wrote:

FRESH GMAT CLUB TESTS QUESTION



If x, y, and z are positive numbers such that 3x < 2y < 4z , which of the following statements could be true?

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

Substituting diferent values should do the trick

I. y = z
when y=z=1 the eqn is satisfied

II. y > z
y=1/2 and z=1/3 eqn is satisfied

III. x > z
x=1/9 and z=1/10 theeqn is satisfied

Therefore IMO E
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink] New post   31 Aug 2021, 23:52
Expert Reply
Hey guys,

The key to this problem is to notice the wording

It says that they are numbers and not specifically integers as is often the case on most questions

Keeping this in mind:

I. can be true because y = z = 1 yields 2 < 4, which is true

II. can be true because y = 1.1 and z = 1 yields 2.2 < 4, which is true

III. can be true because x = 1.1 and z = 1 yields 3.3 < 4, which is true

The answer is E
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Re: If x, y, and z are positive numbers such that 3x < 2y < 4z , which of [#permalink]
31 Aug 2021, 23:52
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