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# If w > x > y > z > 0, is z < 4?

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If w > x > y > z > 0, is z < 4? [#permalink]

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20 Mar 2011, 13:14
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If w > x > y > z > 0, is z < 4?

(1) $$\frac{1}{w} +\frac{1}{x} +\frac{1}{y} +\frac{1}{z} = 1$$.

(2) $$\frac{1}{w} > \frac{1}{4}$$.

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-w-x-y-z-0 ... 29991.html
[Reveal] Spoiler: OA

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Re: If w > x > y > z > 0, is z < 4? [#permalink]

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20 Mar 2011, 13:25
(1) We could transform the initial condition:
0<1/W<1/X<1/Y<1/Z
If Z>4, then 1/Z<1/4 and three other number 1/W, 1/Y, 1/X are also less than 1/4, so their sum could not be equal to 1. So (1) is sufficient to say that Z<4.

(2) alone does not say us anything about Z
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Re: If w > x > y > z > 0, is z < 4? [#permalink]

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20 Mar 2011, 17:12
Given 0 < Z < y<X <w

1. sufficient
if z >4 => 1/z <0.25 , then the given sum will never be 1. Hence this is enough to say z cannot be greater than 4.

2. sufficient

w<4 =>z<w<4 => z<4, enough to answer the question.

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Re: If w > x > y > z > 0, is z < 4? [#permalink]

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20 Mar 2011, 18:05
@Spidy001, I've a query here :

if z >4 => 1/z <0.25 , then the given sum will never be 1. Hence this is enough to say z cannot be greater than 4.

How is the above true ? If z > 4 => z < 0.25 then z can have max value(s) as 0.24, 0.249 etc or lesser values as 0.05 etc. and the values of y, x and w can be more than this and hence the total may be 1, right ?

(2) is clearly sufficient as we have > W, and w > z so clearly z < 4.

Not sure if I'm doing anything wrong in the analysis of (1), someone please enlighten me.

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Re: If w > x > y > z > 0, is z < 4? [#permalink]

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20 Mar 2011, 18:23
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No.

As W>X>Y>Z

1/W < 1/X < 1/Y < 1/Z

Also as 1/Z < 1/4 ,

=> 1/W < 1/X < 1/Y < 1/Z < 1/4

so with this condition their sum will never be equal to 1.

Hope its clear now.

subhashghosh wrote:
@Spidy001, I've a query here :

if z >4 => 1/z <0.25 , then the given sum will never be 1. Hence this is enough to say z cannot be greater than 4.

How is the above true ? If z > 4 => z < 0.25 then z can have max value(s) as 0.24, 0.249 etc or lesser values as 0.05 etc. and the values of y, x and w can be more than this and hence the total may be 1, right ?

(2) is clearly sufficient as we have > W, and w > z so clearly z < 4.

Not sure if I'm doing anything wrong in the analysis of (1), someone please enlighten me.

Regards,
Subhash

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Re: If w > x > y > z > 0, is z < 4? [#permalink]

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20 Mar 2011, 19:10
OA is D.

for 1) as Spidy said, if z >= 4 then, 1/z =< 0.25 and since w,x, and y are greater than z, 1/w, 1/x, 1/y and 1/z will all be less than 0.25 and their sum will never amount to 1. 4*(# less than .25)<1
Thus z has to be less than 4.

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Re: If w > x > y > z > 0, is z < 4? [#permalink]

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20 Mar 2011, 21:21
Yes, I had mistaken this bit :

As W>X>Y>Z

1/W < 1/X < 1/Y < 1/Z

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Re: If w > x > y > z > 0, is z < 4? [#permalink]

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20 Mar 2011, 21:58
if W>X>Y>Z>0, for positive integers 1/W<1/X<1/Y<1/Z

1. 1/W + 1/X + 1/Y + 1/Z = 1
To check whether Z > 4 or not, lets put Z = 5
this means 1/W + 1/X + 1/Y = 1-1/5= 4/5 or 0.8. average value of each term would be 0.266. But 1/z = 0.2. We know from question stem that 1/Z > 1/X , 1/Y, 1/Z. Thus we know z is not greater than 4. Hence 1 is sufficient to answer this question

2. 1/W >1/4 this implies that 1/Z >1/4. or Z<4. This also correctly answers the question.

Since both statements are sufficient by themselves, D

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Re: If w > x > y > z > 0, is z < 4? [#permalink]

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12 Dec 2017, 21:42
Given W>X>Y>Z>0

Stat#1:
1/W +1/X +1/Y +1/Z = 1 and W>X>Y>Z>0 can only possible if w,x,y,z are fractions so suff.

Stat#2:
1/W > 1/4 => w>4 { as w>0 } so suff.

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Re: If w > x > y > z > 0, is z < 4? [#permalink]

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12 Dec 2017, 21:56
If w > x > y > z > 0, is z < 4?

We can rewrite given inequality as: $$\frac{1}{z}>\frac{1}{y}>\frac{1}{x}>\frac{1}{w}$$ and the question becomes whether $$\frac{1}{z}>\frac{1}{4}$$?

(1) $$\frac{1}{w} +\frac{1}{x} +\frac{1}{y} +\frac{1}{z} = 1$$. Now, if $$\frac{1}{z}\leq{\frac{1}{4}}$$ then all other three fractions will also be less than $$\frac{1}{4}$$ and their sum (the sum of four numbers each of which is less than 1/4) can not add up to 1, hence $$\frac{1}{z}>\frac{1}{4}$$. Sufficient.

(2) $$\frac{1}{w} > \frac{1}{4}$$. Since $$\frac{1}{z}>\frac{1}{w}$$ then $$\frac{1}{z}$$ is also more than $$\frac{1}{4}$$. Sufficient.

OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/if-w-x-y-z-0 ... 29991.html
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Re: If w > x > y > z > 0, is z < 4?   [#permalink] 12 Dec 2017, 21:56
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