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# If x is a positive number less than 10, is z greater than th

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If x is a positive number less than 10, is z greater than th  [#permalink]

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20 Feb 2014, 00:15
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Question Stats:

59% (02:10) correct 41% (02:37) wrong based on 845 sessions

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x.
(2) z = 5x

Data Sufficiency
Question: 103
Category: Arithmetic; Algebra Statistics; Inequalities
Page: 160
Difficulty: 650

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If x is a positive number less than 10, is z greater than th  [#permalink]

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20 Feb 2014, 00:16
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If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

Given: $$0<x<10$$.
Question: is z greater than the average of x and 10? Or: is $$z>\frac{10+x}{2}$$? --> $$2z>10+x$$?

(1) On the number line, z is closer to 10 than it is to x --> $$|10-z|<|z-x|$$ --> as z is closer to 10 than it is to x, then z>x, so $$|z-x|=z-x$$ --> two cases for 10-z:

A. $$z\leq{10}$$ --> $$|10-z|=10-z$$ --> $$|10-z|<|z-x|$$ becomes: $$10-z<z-x$$ --> $$2z>10+x$$. Answer to the question YES.

B. $$z>{10}$$ --> in this case $$2z>20$$ and as $$x<10$$, then $$x+10<20$$, hence $$2z>10+x$$. Answer to the question YES.

OR another approach:
Given:
x-----average-----10----- (average of x and 10 halfway between x and 10).

Now, as z is closer to 10 than it (z) is to x, then z is either in the blue area, so more than average OR in the green area, so also more than average. Answer to the question YES.

Sufficient.

(2) z = 5x --> is $$2z>10+x$$? --> is $$10x>10+x$$? --> is $$x>\frac{10}{9}$$. We don't now that. Not sufficient. (we've gotten that if $$x>\frac{10}{9}$$, then the answer to the question is YES, but if $$0<x\leq{\frac{10}{9}}$$, then the answer to the question is NO.)

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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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20 Feb 2014, 02:18
3
If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x.
(2) z = 5x

On st1 i'll draw a number line. 0----------------------x-----------------z---10
We can see that z will always be greater than the average. Sufficient

St2 say that z=5x. we'll choose 2 cases:
Case 1: x=2, so z=10. avarge is 6 and z is greater than the avarage.
Case 2: x=1, so z=5 avarge is 5.5 and z is smaller than the avarage. Insufficient

So ans is A.
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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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20 Feb 2014, 23:06
Option A.
From S1:Is z>(x+10)/2 (1)
(x+10)2 is basically the middle value between x and 10.
So,if z is closer to 10 than x it would imply z>the mid value of x and 10 which is the mean.
Sufficient.

From S2:If z=5x
Replacing in (1) Is 5x>x+10/2
Or Is 9x>10
If x=1 ans is true
If x=2.
Ans is false.Not suff.
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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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21 Feb 2014, 02:59
2
Q Is z>(x+10)/2 ; provided that 0<x<10
(1) On the number line, z is closer to 10 than it is to x.

this means distance between z and x is greater than distance between z and 10 on the number line. hence we have
z-x>10-z
z>(x+10)/2
in the above case z is less than 10 and lies between x and 10. If z is greater 10, then it will definitely be greater than average of x+10. hence z is greater than average of x+10. therefore sufficient.

(2) z = 5x

z=5x assume x=1 average of 10 and 1 = 5.5 and z=5(1)
when x=3, average of 10 and 3= 6.5 and z=15

since two different answers are possible hence 2 alone is not sufficient hence correct answer is A
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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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23 Aug 2015, 01:37
Statement 1: 0---X----Z--10 The average is the middle point between X (which is 0<x<10) and 10, and we are told that Z is closer to 10 that means Z>Average

Statement 2: 10x>x+10 (plugin 5x=z here z>(x+10)/2 --> 9x>10 ? If x=1 9<10 No, and if x=2 18>10 Yes NOT Sufficient
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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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12 Dec 2015, 23:13
Hi,

i request your help to find out weather statement A is sufficient, when x=8 and Z =9 . Here z is closer to 10 than x, but the average of 10 and X will be equal to z.

Is this correct?

Thanks
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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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13 Sep 2016, 01:50
syamukrishnan wrote:
Hi,

i request your help to find out weather statement A is sufficient, when x=8 and Z =9 . Here z is closer to 10 than x, but the average of 10 and X will be equal to z.

Is this correct?

Thanks

No, the Statement 1 says that - Z is closer to 10 than it is to x and not that z is closer to 10 than x is. Your's is the later case.
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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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05 Oct 2016, 10:25
2
Top Contributor
Bunuel wrote:
If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x.
(2) z = 5x

Here's my solution:

Target question: Is z greater than the mean of x and 10?

Statement 1: On the number line, z is closer to 10 than it is to x.
IMPORTANT: On the number line, the mean of two numbers will lie at the midpoint between those two numbers.
So, the mean of x and 10 will lie halfway between x and 10.
So, if z is closer to 10 than it is to x, then z must lie to the right of the midpoint between x and 10.
This means that z must be greater than the mean of x and 10
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: z = 5x
There are several pairs of numbers that meet this condition. Here are two:
Case a: x=1, z=5, in which case z is less than the mean of x and 10 (mean = 5.5)
Case b: x=4, z=20, in which case z is greater than the mean of x and 10 (mean = 7)
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Cheers,
Brent

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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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05 Oct 2016, 12:23
If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x.
(2) z = 5x

I first rephrased the question to
z>(x+10)/2
2z>x+10
2z-10>x?
(This didn't help me w/ the question)

(1) On the number line, z is closer to 10 than it is to x.

I just used guess and check for this one.
x=8 then (10+8)/2=9 if z is closer to 10 than to 8 it has to be more than 9 so yes
x=1 then (10+1)=11/2=5.5 5.5 is the midpoint between 1 and 10 (i.e. 5.5 is the average). In order for z to be closer to 10 it need to be more than 5.5. yes
suff

(2) z = 5x
again I plugged in numbers
x=1, z=5 then no z is not greater than the average
x=2, z=10 z is greater than the average

insuff
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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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31 Jul 2017, 17:23
Given:
x is a +ve integer and x< 10.

Is z > (x+10)/2 ?

Using the values of x we can find the range of the inequality.

case 1: when x = 1 (minimum value of x), z > 11/2 => z > 5.5

case 2: when x = 9 (maximum value of x ), z >19/2 => z > 9.5

From case 1 and case 2, we can rephrase the question as Is z > 5.5 ?

Taking Statement 2 first: -----------> NOT Sufficient
For x = 1, z = 5 is z >5.5 ---> No
For x = 2, z = 10 is z > 5.5. --> yes

Statement 1: It say's z is closer to 10 i.e. z> 5.5, therefore sufficient. Answer is A
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Re: If x is a positive number less than 10, is z greater than th  [#permalink]

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27 Aug 2018, 07:10
Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x.
(2) z = 5x

0 < x < 10
Question : is z greater than the average (arithmetic mean) of x and 10?

Question : is z > (x+10)/2?
OR
Question : is z > Midpoint of x and 10 ?

Statement 1: On the number line, z is closer to 10 than it is to x.

i.e. z is to the right of "midpoint of 10 and x"

i.e. z is GREATER than x

SUFFICIENT

Statement 2: z = 5x

@x = 1, Midpoint = (1+10)/2 = 5.5 and z = 5 i.e. z < AVerage

@x = 2, Midpoint = (2+10)/2 = 6 and z = 10 i.e. z > AVerage

NOT SUFFICIENT

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Re: If x is a positive number less than 10, is z greater than th &nbs [#permalink] 27 Aug 2018, 07:10
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