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

Question

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

Bunuel · Accepted Answer

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

Given:

Question:

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

This means that \(|10-z|<|z-x|\). Since z is closer to 10 than it (z) is to x (and from the stem we know that x < 10), then \(z > x\), thus \(|z-x|=z-x\). We can have the following two cases for 10-z:

OR another approach:
Given:

x-----average-----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 \(2z>10+x\)?

(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.

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liranmaymoni · Answer

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.

manpreetsingh86 · Answer

Q Is z>(x+10)/2 ; provided that 010-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

KarishmaB · Answer

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 [ 3.17 KiB | Viewed 49210 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?

syamukrishnan · Answer

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

umg · Answer

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.

BrentGMATPrepNow · Answer

I received a PM about this question. 
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

Answer:  A

Cheers,
Brent

RELATED VIDEO
 https://www.youtube.com/watch?v=Y9f_0QJU1ug

GMATinsight · Answer

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 Answer: Option A

TheUltimateWinner · Answer

VeritasKarishma 
From your explanation:
Statement 1 says:  YES 
Statement 2 says:  NO  (before finding YES value)
We should not try for finding YES value in statement 2. Without finding YES value in statement 2, we can confidently say that the correct choice is A, because  both statements can't contradict each other! 
Is it the right way to choose A?
Thanks__

KarishmaB · Answer

Yes, absolutely! The two statements won't contradict each other so your answer will be (A). But if I am not running out of time, I will probably get the Yes answer too for stmnt 2, just for my satisfaction!!

JackieO · Answer

How can the answer be A? what if x = 1/2 and z = 5? in this case the average between x and 10 is 5.25 and z is 5, thus less than the average. Also, z is closer to 10 than it is to x. Please advice on this question.

Bunuel · Answer

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

If x = 1/2 and z = 5, then the distance between z and 10 is 5, while the distance between z and z is 4.5. So, those numbers violate the statement.

Teitsuya · Answer

Hi Bunuel

In (1) what if Z= 9 and X = 8, then both are equal as 9.

Bunuel · Answer

Those numbers do not comply with (1):

9 (z) is NOT closer to 10 than 9 (z) is to 8 (x). The distances are equal.

gmatpro15 · Answer

what if x = 8.99 and z = 10

statement 1 is not sufficient

Bunuel   KarishmaB

Bunuel · Answer

How?  Is z greater than the average (arithmetic mean) of x and 10?  z = 10 is obviously greater than the average of x = 8.99 and z = 10.

gmatpro15 · Answer

Sorry typo 
If z =9

And average will be ~9.499

Bunuel 
   Posted from my mobile device

Bunuel · Answer

z =9 and x = 8.99 do not satisfy (1):  On the number line, z is closer to 10 than it is to x .

z = 9 IS not closer to 10 than it (z = 10) is to x = 8.99.

You won't be able to come up with a counterexample, because (1) IS sufficient.

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

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

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