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Math Expert V
Joined: 02 Sep 2009
Posts: 59624
Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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6
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43 00:00

Difficulty:   55% (hard)

Question Stats: 63% (01:48) correct 37% (01:47) wrong based on 1756 sessions

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Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

(1) y > 3x
(2) y > x > 3

Diagnostic Test
Question: 32
Page: 25
Difficulty: 650

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Math Expert V
Joined: 02 Sep 2009
Posts: 59624
Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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9
7
SOLUTION

Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

Given integers are: {3, 4, 5, 6, x, y}

(1) y > 3x. If $$x=1$$ and $$y=4$$ then the range=6-1=5<9 but if $$x=100$$ then the range>9. Not sufficient.

(2) y > x > 3. If $$x=4$$ and $$y=5$$ then the range=6-3=3<9 but if $$x=100$$ then the range>9. Not sufficient.

(1)+(2) From $$x > 3$$ we have that the least value of $$x$$ is 4, and from $$y > 3x=12$$ we have that the least value of $$y$$ is 13, hence the least value of the range is 13-3=10>9. Sufficient.

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##### General Discussion
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GMAT 1: 640 Q50 V26 GMAT 2: 660 Q50 V28 GMAT 3: 730 Q50 V38 Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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2
1
Hi,

Range = Largest value - smallest value.

6, 3, y, 4, 5, and x, where x & y are integers

Using (1),
y>3x, if x=1, then y = 4, 5,..100....
in each case range can be 5, 6,....So, range is greater than 5. Insufficient.

Using (2),
y>x>3.
Minimum value of x = 4,
y=5,6,7... We can't say whether range is greater than 9.

Combining both statements;
$$x_{min} = 4$$
& since, y > 3x, $$y_{min}=13,$$
thus, 3, 4, 4, 5, 6, & 13 has range (13-3)=10, which is greater than 9
and on increasing x, range will also increase.

Regards,
Math Expert V
Joined: 02 Sep 2009
Posts: 59624
Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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1
SOLUTION

Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

Given integers are: {3, 4, 5, 6, x, y}

(1) y > 3x. If $$x=1$$ and $$y=4$$ then the range=6-1=5<9 but if $$x=100$$ then the range>9. Not sufficient.

(2) y > x > 3. If $$x=4$$ and $$y=5$$ then the range=6-3=3<9 but if $$x=100$$ then the range>9. Not sufficient.

(1)+(2) From $$x > 3$$ we have that the least value of $$x$$ is 4, and from $$y > 3x=12$$ we have that the least value of $$y$$ is 13, hence the least value of the range is 13-3=10>9. Sufficient.

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Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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1
Here is a visual that should help.
Attachments Screen Shot 2016-03-27 at 7.23.57 PM.png [ 93.38 KiB | Viewed 16083 times ]

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Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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Bunuel wrote:
Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

(1) y > 3x
(2) y > x > 3

Diagnostic Test
Question: 32
Page: 25
Difficulty: 650

pretty easy for a 700 level question...

1. x can be 1, and y can be 4 - so the answer is NO
x can be 4, and y can be 13 - so the answer is YES.
1 alone is insufficient. A and D are out.

2. y>x>3.
x can be 4, y =5 - answer is no
x can be 4, y can be 20 - answer is YES.
2 alone is not sufficient. B is out.

1+2.
x>3
y>3x
minimum value for x is 4.
minimum value for y is 13
yes, the range is greater than 9.

sufficient.
Target Test Prep Representative G
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Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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Bunuel wrote:
Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

(1) y > 3x
(2) y > x > 3

Statement One Alone:

y > 3x

Statement one is not sufficient to answer the question. For example, if x = 0, y could be 1 and the range is 6 - 0 = 6, which is less than 9. However, if x = 10, y could be 31 and the range is 31 - 3 = 28, which is greater than 9.

Statement Two Alone:

y > x > 3

Statement two is not sufficient to answer the question. For example, if x = 4, y could be 5 and the range is 6 - 0 = 6, which is less than 9. However, if x = 14, y could be 15 and the range is 15 - 3 = 12, which is greater than 9.

Statements One and Two Together:

From the two statements, we know that x > 3 and y > 3x. So the smallest integer x can be is 4 and the smallest integer y can be is 3(4) + 1 = 13. Thus the smallest range of the integers is 13 - 3 = 10, which is greater 9. Since 10 is the smallest range, any other range of the integers will be greater than 10 and hence greater than 9.

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Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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In the solution of these questions it is not asked to take minimum value and x and y can take any numbers so why we have solved this by taking minimum values
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GMAT 1: 740 Q50 V40 GMAT 2: 770 Q51 V42 Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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2
arpitalewe wrote:
In the solution of these questions it is not asked to take minimum value and x and y can take any numbers so why we have solved this by taking minimum values

Hi arpitalewe,

Welcome to the GMAT Club!

So the logic behind trying to take the minimum permissible values for x and y is: we want to see if the minimum permissible value of y is such that range of the given set is greater than 9. If for minimum value this condition holds then for all other values it will hold as well.

Let me know if you need further clarification or if the above solutions make sense with the above logic.

Regards,
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Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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1
Statement 1)

y > 3x

if x = 1 then y must be at least 4 but we cannot tell any further. 6-1 = 5 > 9 no.

If x = 5 y must be at least 16. 16 - 3 = 13 > 9 yes

Insufficient.

Statement 2)

y > x > 3 we can try the x = 4 and y = 5 we get an answer 6-3 = 3 > 9 no.

Try x = 4 y = 16, 16-3 = 13 > 9 yes.

Insufficient.

Now combine (1 and 2)

If x = 4 then y at least 13 (given that we took the minimum possible value for x)

13 - 3 = 10 > 9

Try x = 10 then y at least 31

31 - 3 = 28 > 9

arpitalewe check the above solution it might help.

Also as Gladiator59 explained that the logic is to test if the minimum value satisfies the condition (range > 9) then the maximum will satisfy it too since y > 3x

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Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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Hi everyone,

This thread is very helpful and a good example of something that is currently holding me back a bit. I understand the process to get to the answer, I can get to the answer by myself but in around 3.15 minutes and not under 2 minutes. There are many problems like this one where I get to the solution but it takes me a lot of time to gather my thoughts. Any thoughts on how I can improve my thinking process?

Kind regards,

Majinn
Manager  S
Joined: 15 Dec 2016
Posts: 118
Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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Bunuel wrote:
SOLUTION

Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

Given integers are: {3, 4, 5, 6, x, y}

(1) y > 3x. If $$x=1$$ and $$y=4$$ then the range=6-1=5<9 but if $$x=100$$ then the range>9. Not sufficient.

(2) y > x > 3. If $$x=4$$ and $$y=5$$ then the range=6-3=3<9 but if $$x=100$$ then the range>9. Not sufficient.

(1)+(2) From $$x > 3$$ we have that the least value of $$x$$ is 4, and from $$y > 3x=12$$ we have that the least value of $$y$$ is 13, hence the least value of the range is 13-3=10>9. Sufficient.

Hi Bunuel -- Between C and E , I chose E unfortunately

Just wondering why does this not work ?

S1 : y > 3x
S2 : y > x > 3.

2y > 4x > 3

dividing by 2

y > 2x > (3/2)

Hence i said per above ....okay x = 2 and y = 6 satisfies above inequality ..

if x = 2 and y can be =6 - the range is below 9

if x = 2 and y can be = 600 -- the range is above 9

Hence i chose E
Math Expert V
Joined: 02 Sep 2009
Posts: 59624
Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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jabhatta@umail.iu.edu wrote:
Bunuel wrote:
SOLUTION

Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

Given integers are: {3, 4, 5, 6, x, y}

(1) y > 3x. If $$x=1$$ and $$y=4$$ then the range=6-1=5<9 but if $$x=100$$ then the range>9. Not sufficient.

(2) y > x > 3. If $$x=4$$ and $$y=5$$ then the range=6-3=3<9 but if $$x=100$$ then the range>9. Not sufficient.

(1)+(2) From $$x > 3$$ we have that the least value of $$x$$ is 4, and from $$y > 3x=12$$ we have that the least value of $$y$$ is 13, hence the least value of the range is 13-3=10>9. Sufficient.

Hi Bunuel -- Between C and E , I chose E unfortunately

Just wondering why does this not work ?

S1 : y > 3x
S2 : y > x > 3.

2y > 4x > 3

dividing by 2

y > 2x > (3/2)

Hence i said per above ....okay x = 2 and y = 6 satisfies above inequality ..

if x = 2 and y can be =6 - the range is below 9

if x = 2 and y can be = 600 -- the range is above 9

Hence i chose E

Adding inequalities in this case gives you broader ranges. How can x be 2, if we are given that x > 3?
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Manager  S
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Posts: 118
Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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Hi Bunuel

Just wondering where my understanding about the math fundamentals is wrong here ...I have seen answer solutions where inequalities that are facing the same way are added

Why doesnt it work in this case ?

So if x and y are both positive

equation 1) x > 3y
equation 2) x > y

1 + 2

2x > 4 y or x > 2y

x > 2y does not make sense when equation 1 is saying x > 3y

Why doesnt this tactic of adding equations in this case work out
Intern  B
Joined: 02 Jul 2019
Posts: 12
Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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Bunuel wrote:
SOLUTION

Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

Given integers are: {3, 4, 5, 6, x, y}

(1) y > 3x. If $$x=1$$ and $$y=4$$ then the range=6-1=5<9 but if $$x=100$$ then the range>9. Not sufficient.

(2) y > x > 3. If $$x=4$$ and $$y=5$$ then the range=6-3=3<9 but if $$x=100$$ then the range>9. Not sufficient.

(1)+(2) From $$x > 3$$ we have that the least value of $$x$$ is 4, and from $$y > 3x=12$$ we have that the least value of $$y$$ is 13, hence the least value of the range is 13-3=10>9. Sufficient.

Hello Bunuel,

In no explaination i see anyone considering x or y to be negative. In quesion it is mentioned they are integers, so will be right to consider negative values? Am i missing somethinghere?
Also i assumed all the integers would be distinct, but unless stated , we can consider that integers are repeating, something i missed.

Math Expert V
Joined: 02 Sep 2009
Posts: 59624
Re: Is the range of the integers 6, 3, y, 4, 5, and x greater  [#permalink]

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Rishbha wrote:
Bunuel wrote:
SOLUTION

Is the range of the integers 6, 3, y, 4, 5, and x greater than 9?

Given integers are: {3, 4, 5, 6, x, y}

(1) y > 3x. If $$x=1$$ and $$y=4$$ then the range=6-1=5<9 but if $$x=100$$ then the range>9. Not sufficient.

(2) y > x > 3. If $$x=4$$ and $$y=5$$ then the range=6-3=3<9 but if $$x=100$$ then the range>9. Not sufficient.

(1)+(2) From $$x > 3$$ we have that the least value of $$x$$ is 4, and from $$y > 3x=12$$ we have that the least value of $$y$$ is 13, hence the least value of the range is 13-3=10>9. Sufficient.

Hello Bunuel,

In no explaination i see anyone considering x or y to be negative. In quesion it is mentioned they are integers, so will be right to consider negative values? Am i missing somethinghere?
Also i assumed all the integers would be distinct, but unless stated , we can consider that integers are repeating, something i missed.

(2) says that y > x > 3, so neither of them can be negative.
_________________ Re: Is the range of the integers 6, 3, y, 4, 5, and x greater   [#permalink] 02 Dec 2019, 08:49
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