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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  [#permalink]

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New post 24 Feb 2012, 16:50
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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

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

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New post 24 Feb 2012, 22:15
4
3
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.

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Hope it helps.
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Director
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Re: If x is a positive integer less than 10, is z greater than  [#permalink]

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New post 25 Feb 2012, 03:07
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.
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Re: If x is a positive integer less than 10, is z greater than  [#permalink]

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New post 25 Feb 2012, 03:17
enigma123 wrote:
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.


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

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New post 27 Feb 2012, 22:12
Hi,
while solving statement A, I took the following critical points to solve the eqn |z-x|>|z-10|
z<x : discard
x<z<10 : True .. Also the answer
z>10: 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 wrote:
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 question: 600-level-question-95138.html

Hope it helps.
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Re: If x is a positive integer less than 10, is z greater than  [#permalink]

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New post 27 Feb 2012, 23:07
devinawilliam83 wrote:
Hi,
while solving statement A, I took the following critical points to solve the eqn |z-x|>|z-10|
z<x : discard
x<z<10 : True .. Also the answer
z>10: 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 wrote:
enigma123 wrote:
If x is a positive integer less than 10, is z less 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

Any idea guys how come the answer is A?


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 question: 600-level-question-95138.html

Hope it helps.


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.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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

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New post 21 Jun 2015, 01:49
Bunuel wrote:
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 question: 600-level-question-95138.html

Hope it helps.




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

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