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DS: q 99 from 198 700 Level GMAT Prep Question

by tallazndood » Wed Jan 27, 2010 11:47 pm
After going through 20 pages of search results, I figured it's time to post this question.

Question:
If x is a positive number less than 10, is z greater than average of x and 10?
1. z is closer to 10 than x on the number line.
2. z = 5x

OA: A

Comment: I disagree with the OA and below is my reasoning. Perhaps someone can shed some insights and be the judge.

Attempted Solution:
Question stem translation: 0 < x < 10, is z > (x + 10) / 2?
Condition 1:
Test case 1: z = 8.5, x = 8, (x + 10) / 2 = 9 > z
Test case 2: z = 9, x = 7, (x + 10) / 2 = 8.5 < z
Since test case 1 and 2 yield different results, Condition 1 is INSUFFICIENT.
Condition 2:
Test case 1: z = 5, x = 1, (x + 10) / 2 = 5.5 > z
Test case 2: z = 9, x = 2, (x + 10) / 2 = 6 < z
Since test case 1 and 2 yield different results, Condition 2 is INSUFFICIENT.
Condition 1 & 2 together:
Since z is closer to 10 than x is in both test cases (required by Condition 1) for Condition 2, hence, Condition 1 & 2 together are INSUFFICIENT.
Answer: E
Last edited by tallazndood on Thu Jan 28, 2010 3:00 am, edited 1 time in total.
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by ajith » Thu Jan 28, 2010 12:11 am
tallazndood wrote:After going through 20 pages of search results, I figured it's time to post this question.

Question:
If x is a positive number less than 10, is z greater than average of x and 10?
1. z is closer to 10 than x on the number line.
2. z = 5x

OA: A

The average of x and 10 is = x+10/2 = x/2+5

1. Implies a) z >x (otherwise it cannot be closer to 10)

Now there are two cases z>10 and z<10

If z>10 it is greater than the average of 10 and x anyway

So we concentrate on z<10 and z>x

according to 1. 10-z <z-x

adding z+x to both sides

10+x<2z

5+ x/2 <z

Hence sufficient

2. z=5x

say x= 0.2
z = 1

in this case z< avg of x and 10

say x =2
and z =10

in this case z> avg of x and 10

hence not sufficient
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by tanviet » Thu Jan 28, 2010 2:45 am
draw the number line, you solve it quickly

by the way, where to get 198 700 level question, pls tell me
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by tallazndood » Thu Jan 28, 2010 3:28 am
ajith wrote:according to 1. 10-z <z-x
Ajith,

Thank you for the explanation. I think our solutions are basically the same except for this quoted part in your Condition 1. How do you know that 10 - z < z - x? The question only indicates that z is closer to 10 then x. But x can be anywhere from 0.000001 to 9.9999999 as long as it's less than z. So using 2 test cases:

Test case 1: x = 1 & z = 9, 10 - z = 1 & z - x = 8, hence 10 - z < z - x.
Test case 2: x = 8.5 & z = 9, 10 - z = 1 & z - x = .5, hence 10 - z > z - x.

Both test cases conform to the two required conditions found in the Question and Condition 1: 1) 0 < x < 10; 2) |10 - z| < |10 - x|. But they yield two different results. Am I missing something?

duongthang,

Here's the link: https://www.beatthegmat.com/198-level-70 ... 43783.html
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by ajith » Thu Jan 28, 2010 4:32 am
tallazndood wrote:
ajith wrote:according to 1. 10-z <z-x
Ajith,

Thank you for the explanation. I think our solutions are basically the same except for this quoted part in your Condition 1. How do you know that 10 - z < z - x? The question only indicates that z is closer to 10 then x. But x can be anywhere from 0.000001 to 9.9999999 as long as it's less than z.
1. z is closer to 10 than x on the number line.

This is the original statement

this says

|z-10| > |z-x| (z is closer to 10 than it is to x and NOT z is closer to 10 than x is to 10 )

Now, we are concerned when z is in between x and 10 (if z is more than 10; it is more than the avg of 10 and x)

z-10 > z-x (both are positive if x<z<10)
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by dmitriyaleyev » Thu Jan 28, 2010 7:45 am
guys,
i think the only way to do it within 2 mins is to:
1. draw a line with x and 10.
2. understand that x doesnt have to be an integer
3. z doesnt have to be in between x and 10

1) z more than mean of x and 10. therefore Z must be closer to ten, wherever Z is. (simply think that 2z > x+10) - SUF
2) 5z could be 0.2*5 or could be 3*5. INSUF.
a it is!
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by Ian Stewart » Thu Jan 28, 2010 10:37 am
tallazndood wrote:
Question:
If x is a positive number less than 10, is z greater than average of x and 10?
1. z is closer to 10 than x on the number line.
2. z = 5x

Comment: I disagree with the OA and below is my reasoning. Perhaps someone can shed some insights and be the judge.

Attempted Solution:
Question stem translation: 0 < x < 10, is z > (x + 10) / 2?
Condition 1:
Test case 1: z = 8.5, x = 8, (x + 10) / 2 = 9 > z
Test case 2: z = 9, x = 7, (x + 10) / 2 = 8.5 < z
Since test case 1 and 2 yield different results, Condition 1 is INSUFFICIENT.
From Statement 1, you know that z is closer to 10 than to x. If you are going to come up with example numbers, you need to be sure to use this information. You might notice in your 'test case 1', z is not closer to 10 than to x, so you cannot use these numbers.

There are a few ways to approach this question. You could draw a number line; the average "A" of x and 10 is exactly halfway between x and 10:

------x------A-------10-----------

If z is closer to 10 than to x, then it must be greater than A, so Statement 1 is sufficient.

For an algebraic solution, you could consider cases; this lets us avoid awkward expressions with absolute values in them. If z > 10, then the answer to the question is clearly 'yes'. What if z < 10? Using Statement 1, we know that the distance from z to 10, which is 10-z since z < 10, is less than the distance from x to z, which is z - x since x < z. So

10 - z < z - x
10 + x < 2z
(10 + x)/2 < z

which says exactly that z is greater than the average of 10 and x, which is what we were hoping to prove.

And of course you could try picking numbers, hoping not to miss some exception. For any numbers you try that agree with the information in Statement 1, you should find that the answer to the question is yes, which, while not a mathematical proof, is persuasive evidence that the Statement is sufficient.
For online GMAT math tutoring, or to buy my higher-level Quant books and problem sets, contact me at ianstewartgmat at gmail.com

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by tallazndood » Thu Jan 28, 2010 10:58 am
ajith wrote:(z is closer to 10 than it is to x and NOT z is closer to 10 than x is to 10 )
That's the hammer I needed to smack my head with. Question resolved. Thanks for being patient with me, Ajith!
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by Sm1520 » Thu Sep 30, 2010 12:20 am
hey..

sorry..need a bit of a clarification on this:

Is the statement 1 : "z is closer to 10 than x on the number line." the same as "z is closer to 10 than it is to x on the number line. " ??
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by Ian Stewart » Thu Sep 30, 2010 11:02 am
Sm1520 wrote:hey..

sorry..need a bit of a clarification on this:

Is the statement 1 : "z is closer to 10 than x on the number line." the same as "z is closer to 10 than it is to x on the number line. " ??
The original question was badly copied, leaving the meaning of Statement 1 open to interpretation (and turning it into a decent Sentence Correction question :) ). Statement 1 in the original question reads like this:

1. On the number line z is closer to 10 than to x

With that wording, Statement 1 is no longer ambiguous.
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by GMATGuruNY » Thu Sep 30, 2010 2:30 pm
ajith wrote:
tallazndood wrote:After going through 20 pages of search results, I figured it's time to post this question.

Question:
If x is a positive number less than 10, is z greater than the average of x and 10?
1. z is closer to 10 than to x on the number line.
2. z = 5x

OA: A
The average of any 2 numbers is the value right in the middle (equidistant from each number):

0 and 10: Average = 5, which is 5 away from 0 and 5 away from 10.
1/8 and 5/8: Average = 3/8, which is 1/4 from 1/8 and 1/4 from 5/8.

Statement 1:
if Z is closer to 10 than to X, then Z is greater than the value right in the middle. Thus, Z is greater than the average. Sufficient.

Statement 2:
If X=2, then Z=10, which is closer to 10 than to 2. Thus, Z is greater than the average of 2 and 10.
If X=1/2, then Z=5/2, which is closer to 1/2 than to 10. Thus, Z is less than the average of 1/2 and 10.
Z can be both greater than and less than the average. Insufficient.

The correct answer is A.
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by Sm1520 » Fri Oct 01, 2010 3:07 am
Ian Stewart wrote:
The original question was badly copied, leaving the meaning of Statement 1 open to interpretation (and turning it into a decent Sentence Correction question :) ). Statement 1 in the original question reads like this:

1. On the number line z is closer to 10 than to x

With that wording, Statement 1 is no longer ambiguous.
haha!
Grt!! thnx Ian! :)
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by sumit.sinha » Fri Oct 01, 2010 4:47 am
tallazndood wrote:After going through 20 pages of search results, I figured it's time to post this question.

Question:
If x is a positive number less than 10, is z greater than average of x and 10?
1. z is closer to 10 than x on the number line.
2. z = 5x

OA: A

Comment: I disagree with the OA and below is my reasoning. Perhaps someone can shed some insights and be the judge.

Attempted Solution:
Question stem translation: 0 < x < 10, is z > (x + 10) / 2?
Condition 1:
Test case 1: z = 8.5, x = 8, (x + 10) / 2 = 9 > z
Test case 2: z = 9, x = 7, (x + 10) / 2 = 8.5 < z
Since test case 1 and 2 yield different results, Condition 1 is INSUFFICIENT.
Condition 2:
Test case 1: z = 5, x = 1, (x + 10) / 2 = 5.5 > z
Test case 2: z = 9, x = 2, (x + 10) / 2 = 6 < z
Since test case 1 and 2 yield different results, Condition 2 is INSUFFICIENT.
Condition 1 & 2 together:
Since z is closer to 10 than x is in both test cases (required by Condition 1) for Condition 2, hence, Condition 1 & 2 together are INSUFFICIENT.
Answer: E

The statement "z is closer to 10 than x on the number line." says:
Z is closer to 10 than it is to x.
and NOT
z is more closer to 10 between x and z.
Cheers,
Sumit
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