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

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If x is a positive number less than 10, is z greater than [#permalink] New post 08 Nov 2009, 20:32
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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?

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

(2) z = 5x
[Reveal] Spoiler: OA

Last edited by Bunuel on 23 Mar 2012, 00:57, edited 1 time in total.
Edited the question and added the OA
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Re: x is a positive number less than 10 [#permalink] New post 08 Nov 2009, 20:37
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amitgovin wrote:
if x is a positive number less than 10 is z greater than the average of x and 10?

1) on the number line, z is closer to 10 than it is to x

2) z=5x

*********************************
please explain. thanks.


IMO A

1) The distance between Z and 10 is lesser than distance between Z and X. So Z is greater than the center of the line segment X to 10. That means Z is greater than the average of X and 10. Sufficient.
2) Z = 5X
If X=1, then Z = 5. Avg of X and 10 = 5.5. So Z<Avg.
If X=2, then Z = 10. Avg of X and 10 = 6. So Z>Avg.
Statement is Insufficient.
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Re: x is a positive number less than 10 [#permalink] New post 09 Nov 2009, 07:01
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Yup A.

stmt1:
|10-z| < |z-x| , here x and z are +ve ( as Z is closer to 10 than to x, and x is +ve )
10-z < z-x
=> z > Av of 10 and x.

stmt2:
x=1, z=5, 5 < 5.5
x=2, z=10, 10 > 6
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Re: DS Problem [#permalink] New post 21 Oct 2010, 15:14
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Hi,

Can anyone please suggest me the solution. I arrived at B, but unfortunately it was wrong. Thanks.


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

Given: \(0<x<10\).
Q: 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.)

Answer: A.

Similar question: 600-level-question-95138.html?hilit=number%20line%20closer
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Re: DS Problem [#permalink] New post 19 Apr 2011, 13:07
hello Bunuel,

how did u get Z>10 ? above in red. from your inequality i got 10>x

thx
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Re: x is a positive number less than 10 [#permalink] New post 19 Apr 2011, 15:29
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If x is a positive number less than 10 is z greater than the average of x and 10?

1) on the number line, z is closer to 10 than it is to x

2) z=5x

Drawing a number line can help you see the answer quickly.

Attachment:
Ques2.jpg
Ques2.jpg [ 4.71 KiB | Viewed 11994 times ]

So x is somewhere between 0 and 10 and average of x and 10 is between x and 10.
z can be in any of the 5 regions as shown.

1) on the number line, z is closer to 10 than it is to x

In which regions is z closer to 10 than to x? Only the two rightmost regions. There, z is greater than the average of x and 10. Sufficient.

2) z = 5x
Put x = 1
z = 5 but average of 1 and 10 is 5.5 so z is less than the average. Actually, this is where I stop and mark (A) as the answer without checking to see if there is a value of x for which z is greater than the average. Think, why?
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Re: x is a positive number less than 10 [#permalink] New post 19 Apr 2011, 20:50
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@Karishma
We stop in (2) because we've found something contrary to the answer in (1), while in earlier choice we had a definite answer. So the answer can't be B,C,D oe E.

The goal in DS is to get No as answer first for the choices, or to find the inconsistencies.

(1)
---------------0--------x-----Av--------10----z------------

---------------0--------x-----Av------z--10----------------

From the diagram above, it's evident that the answer is Yes

(2) is insufficient

if x is very close to 0, then z may not be closer to 10

e.g, x = 0.1, then z = 0.5

if x = 2, then z = 10 (greater than average)

Answer - A
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Re: x is a positive number less than 10 [#permalink] New post 20 Apr 2011, 03:28
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subhashghosh wrote:
@Karishma
We stop in (2) because we've found something contrary to the answer in (1), while in earlier choice we had a definite answer. So the answer can't be B,C,D oe E.

The goal in DS is to get No as answer first for the choices, or to find the inconsistencies.

(1)
---------------0--------x-----Av--------10----z------------

---------------0--------x-----Av------z--10----------------

From the diagram above, it's evident that the answer is Yes

(2) is insufficient

if x is very close to 0, then z may not be closer to 10

e.g, x = 0.1, then z = 0.5

if x = 2, then z = 10 (greater than average)

Answer - A


Yes, it is not possible to get inconsistent answers from the two statements i.e. we cannot have 'Yes. z is greater. Sufficient Alone.' from statement 1 and 'No. z is smaller. Sufficient Alone.' from statement 2.
From statement 2, we will either get 'Yes. z is greater. Sufficient Alone.' or 'z can be smaller or greater so not sufficient alone'

From statement 1, we got that z is greater than the average.
From statement 2, we tried a value and we got that z is less than the average. Then there must be a value for which z is greater than the average too in statement 2. We don't even need to try it. Definitely statement 2 alone is not sufficient and answer is (A).
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If x is a positive number less than 10, is z greater than th [#permalink] New post 24 Apr 2012, 08:18
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.

Please explain your answer choice.
Thanks
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Re: If x is a positive number less than 10, is z greater than th [#permalink] New post 24 Apr 2012, 09:28
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gayathriz 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.

Please explain your answer choice.
Thanks


Merging similar topics. Please ask if anything remains unclear.
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Re: If x is a positive number less than 10, is z greater than [#permalink] New post 25 Apr 2012, 16:22
just wondering about statement 1, am i misunderstanding the question?

"on the number line, z is closer to 10 than it is to x"

z = 11
x = 8
so z is closer to 10 than x is

is 11 > (8+10)/2
11 > (18/2) becomes 11 > 9 so the answer is YES

z = 9
x = 8
so z is closer to 10 than x is

is 9 > (8+10)/2
9 > (18/2) becomes 9 > 9 so the answer is NO

or is the question stating z is closer to 10 than z is to x? i interpreted it as z is closer to 10 than 10 is to x.
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Re: If x is a positive number less than 10, is z greater than [#permalink] New post 25 Apr 2012, 20:18
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chamisool wrote:
just wondering about statement 1, am i misunderstanding the question?

"on the number line, z is closer to 10 than it is to x"

z = 11
x = 8
so z is closer to 10 than x is

is 11 > (8+10)/2
11 > (18/2) becomes 11 > 9 so the answer is YES

z = 9
x = 8
so z is closer to 10 than x is

is 9 > (8+10)/2
9 > (18/2) becomes 9 > 9 so the answer is NO

or is the question stating z is closer to 10 than z is to x? i interpreted it as z is closer to 10 than 10 is to x.


(1) says: "On the number line, z is closer to 10 than it is to x". Here "it" refers to z.

So, it's: "On the number line, z is closer to 10 than z is to x".

Similar question to practice: if-x-is-a-positive-number-less-than-10-is-z-less-than-the-131322.html

Hope it helps.
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Re: If x is a positive number less than 10, is z greater than [#permalink] New post 13 Oct 2013, 06:57
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Re: If x is a positive number less than 10, is z greater than [#permalink] New post 17 Dec 2014, 15:50
Hmm...I don't understand why the answer is A.

for Statement A it states that "On the number line, z is closer to 10 than it is to x". Let's say Z is 9 and X is 8...then the average of X and 10 is 9. This equals Z... OR, let's say Z is 2 and X is 1...then Z < avg (x & 10). How can statement A be sufficient?
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Re: If x is a positive number less than 10, is z greater than [#permalink] New post 17 Dec 2014, 20:01
GMATAnnihilator wrote:
Hmm...I don't understand why the answer is A.

for Statement A it states that "On the number line, z is closer to 10 than it is to x". Let's say Z is 9 and X is 8...then the average of X and 10 is 9. This equals Z... OR, let's say Z is 2 and X is 1...then Z < avg (x & 10). How can statement A be sufficient?



The case you are considering with z=9,x=8 then z is equidistant from x and 10 both....So this is not valid.

Case 2 where z=2 ie. average of x+10 ----> (x+10)/2=2 or x+10=4 or x =-6...But we are told x is positive number less than 10...so x=-6 is not possible...

Hope it is clear
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Re: If x is a positive number less than 10, is z greater than [#permalink] New post 18 Dec 2014, 06:59
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GMATAnnihilator wrote:
Hmm...I don't understand why the answer is A.

for Statement A it states that "On the number line, z is closer to 10 than it is to x". Let's say Z is 9 and X is 8...then the average of X and 10 is 9. This equals Z... OR, let's say Z is 2 and X is 1...then Z < avg (x & 10). How can statement A be sufficient?



The case you are considering with z=9,x=8 then z is equidistant from x and 10 both....So this is not valid.

Case 2 where z=2 ie. average of x+10 ----> (x+10)/2=2 or x+10=4 or x =-6...But we are told x is positive number less than 10...so x=-6 is not possible...

Hope it is clear


Thanks Wounded Tiger...now where do I find that facepalm emoticon. I clearly misread the question... one of more deficiencies with timed exams... This will have to do: http://vignette3.wikia.nocookie.net/car ... 1217081420
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Re: If x is a positive number less than 10, is z greater than [#permalink] New post 04 Jan 2015, 09:26
Quote:
Hmm...I don't understand why the answer is A.

for Statement A it states that "On the number line, z is closer to 10 than it is to x". Let's say Z is 9 and X is 8...then the average of X and 10 is 9. This equals Z... OR, let's say Z is 2 and X is 1...then Z < avg (x & 10). How can statement A be sufficient?


For semplicity sake: rephrase the question into is 2z > 10 + x?

1) We know a few interesting things about x: it has to be smaller than 10; it can be a rational number; it is positive.
According to this statement --> if z=9 then x<8, say x=7,9 so that if 7,9 works any smaller value of x will work too. 2z=18 > 10+x. Thus z is always going to be greater than the average of 10 and x.

hope it helps :P
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Re: If x is a positive number less than 10, is z greater than [#permalink] New post 18 May 2015, 05:36
Hi Karishma,

This is regarding the number line approach that you have explained.Please share some problems where i could apply this technique.

Regards,

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Re: If x is a positive number less than 10, is z greater than [#permalink] New post 18 May 2015, 05:49
Hi Karishma,

How can we be sure that the average is not more than z.The pictorial representation suits your explanation but somewhere i am not able to convince myself with it.

Would be glad if we could discuss

Regards
Re: If x is a positive number less than 10, is z greater than   [#permalink] 18 May 2015, 05:49

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