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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?

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.)

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

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25 Feb 2012, 02:07

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. _________________

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.

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

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27 Feb 2012, 21:12

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.)

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.)

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.
_________________

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

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21 Jun 2015, 00:49

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.)

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?

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

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17 Dec 2017, 22:47

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