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

Question

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

(1) The positive difference between z and x is greater than the positive difference between z and 10.
(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: \(0<x<10\). Question: is z greater than the average of x and 10? Or: is \(z>\frac{10+x}{2}\)? --> \(2z>10+x\)?

(1) The positive difference between z and x is greater than the positive difference between z and 10 --> \(|z-x|>|z-10|\), which means that the distance between z and x is greater than the distance between z and 10:

x-----average-----10-----
(average of x and 10 halfway between x and 10).

Now, as the distance between z and x is greater than the distance between z and 10, 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 questions to practice:
https://gmatclub.com/forum/if-x-is-a-po ... 67734.html (OG)
https://gmatclub.com/forum/if-y-is-a-ne ... 95138.html
https://gmatclub.com/forum/if-x-is-a-po ... 31322.html
https://gmatclub.com/forum/if-a-is-a-po ... 05650.html
https://gmatclub.com/forum/if-x-and-z-a ... 55651.html

Hope it helps.

enigma123 · Answer

Thanks Bunuel - I am struggling to understand this:

Now, as the distance between z and x is greater than the distance between z and 10, then z is either in the blue area, so more than average OR in the green area, so also more than average.

Bunuel · Answer

x ----- average ----- 10 -----

Just try to place z in the green or blue area and you'll see that the distance between z and x will be  greater  than the distance between z and 10;

Now, place z in the red area (or to the left of x) and you'll see that the distance between z and x will be  less  than the distance between z and 10.

So we have that in order the distance between z and x to be greater than the distance between z and 10 it must be either in blue or red areas so in any case z is more than the average.

Hope it's clear.

devinawilliam83 · Answer

Hi, while solving statement A, I took the following critical points to solve the eqn |z-x|>|z-10| z10: solving this I got x<10. which is information given in the question stem. I dint know how this can answer the ques. hence, considered A as insufficient .. Pl help

Bunuel · Answer

You don't need to work with absolute values here. Again, in order  the distance between z and x to be greater than the distance between z and 10  (in order first statement tot hold true) z MUST be to the right of the average, so more than it.

Gmatdecoder · Answer

If the question said Z is less than the average of X & 10, would be get a yes answer only from the second statement as X<10/9 which would be 1 and avg would be 5.5. Z would be 5 and less than average?

kajaldaryani46 · Answer

——
Thanks for linking the similar Questions. I was wondering if there’s a specific name for this Question type?

Posted from my mobile device

daniformic · Answer

Hi  @Bunuel ,
could i resolve in this way?

z-x>z-10
at the same time:
z-x>0
z-10>0
Sum of them:
2z-x-10>0
z> \frac{x+10}{2} ?

Thank you in advance

bumpbot · Answer

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

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

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