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Intern  Joined: 15 Dec 2015
Posts: 34
For any numbers W and Z, W•Z=Wˆ3 * Z(8-Wˆ2)  [#permalink]

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4 00:00

Difficulty:   95% (hard)

Question Stats: 38% (02:18) correct 62% (02:12) wrong based on 85 sessions

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For any numbers W and Z, W∙Z=Wˆ3 * Z(8-Wˆ2) . If both Z and W∙Z are positive numbers, which of the following could be a value of W ?

(A) 9

(B) 3

(C) 0

(D) -2

(E) -9

Originally posted by sara86 on 24 Dec 2015, 16:51.
Last edited by sara86 on 24 Dec 2015, 21:09, edited 3 times in total.
Math Expert V
Joined: 02 Aug 2009
Posts: 7987
Re: For any numbers W and Z, W•Z=Wˆ3 * Z(8-Wˆ2)  [#permalink]

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sara86 wrote:
For any numbers W and Z, $$W•Z=Wˆ3 Z(8-Wˆ2)$$ . If both Z and W•Z are positive numbers, which of the following could be a value of W ?

(A) 9

(B) 3

(C) 0

(D) -2

(E) -9

Hi,
it is given that both Z and W•Z are positive numbers and $$W•Z=Wˆ3 Z(8-Wˆ2)$$..
this means Wˆ3 Z(8-Wˆ2) must be positive..
we know Z is positive, so Wˆ3 *(8-Wˆ2) must be positive...
in other words, Wˆ3 and (8-Wˆ2) must be of the same sign..
lets see the two cases ..

1) Both positive..
W^3 is positive, so 8-W^2>0..
or W^2<8....or W < 3 as 3^2=9...
eliminate 9 and 3 that is A and B...

2) both negative..
W^3 is negative, so 8-W^2<0..
or W^2>8....or W <- 2 or W>2..
W>2 can be discarded as here W is taken as negative
D can be eliminated since it is -2,
E is the answer..

Note :- C which is 0 can be eliminated initially itself as W.Z will become 0 in that case.
and after initial steps We can substitute the values and find the answer, especially so, since it is asking " which of the following could be a value of W ?".
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Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
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Re: For any numbers W and Z, W•Z=Wˆ3 * Z(8-Wˆ2)  [#permalink]

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chetan2u wrote:
sara86 wrote:
For any numbers W and Z, $$W•Z=Wˆ3 Z(8-Wˆ2)$$ . If both Z and W•Z are positive numbers, which of the following could be a value of W ?

(A) 9

(B) 3

(C) 0

(D) -2

(E) -9

Hi,
it is given that both Z and W•Z are positive numbers and $$W•Z=Wˆ3 Z(8-Wˆ2)$$..
this means Wˆ3 Z(8-Wˆ2) must be positive..
we know Z is positive, so Wˆ3 *(8-Wˆ2) must be positive...
in other words, Wˆ3 and (8-Wˆ2) must be of the same sign..
lets see the two cases ..

1) Both positive..
W^3 is positive, so 8-W^2>0..
or W^2<8....or W < 3 as 3^2=9...
eliminate 9 and 3 that is A and B...

2) both negative..
W^3 is negative, so 8-W^2<0..
or W^2>8....or W <- 2 or W>2..
W>2 can be discarded as here W is taken as negative
D can be eliminated since it is -2,
E is the answer..

Note :- C which is 0 can be eliminated initially itself as W.Z will become 0 in that case.
and after initial steps We can substitute the values and find the answer, especially so, since it is asking " which of the following could be a value of W ?".

IMO, W can not be <0 in this question as it is given that both z and w*z are >0 ---> w>0. The question needs to be reformatted as it is mentioning W and w and the RHS is not clearly written.
Math Expert V
Joined: 02 Aug 2009
Posts: 7987
Re: For any numbers W and Z, W•Z=Wˆ3 * Z(8-Wˆ2)  [#permalink]

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1
Engr2012 wrote:
chetan2u wrote:
sara86 wrote:
For any numbers W and Z, $$W•Z=Wˆ3 Z(8-Wˆ2)$$ . If both Z and W•Z are positive numbers, which of the following could be a value of W ?

(A) 9

(B) 3

(C) 0

(D) -2

(E) -9

Hi,
it is given that both Z and W•Z are positive numbers and $$W•Z=Wˆ3 Z(8-Wˆ2)$$..
this means Wˆ3 Z(8-Wˆ2) must be positive..
we know Z is positive, so Wˆ3 *(8-Wˆ2) must be positive...
in other words, Wˆ3 and (8-Wˆ2) must be of the same sign..
lets see the two cases ..

1) Both positive..
W^3 is positive, so 8-W^2>0..
or W^2<8....or W < 3 as 3^2=9...
eliminate 9 and 3 that is A and B...

2) both negative..
W^3 is negative, so 8-W^2<0..
or W^2>8....or W <- 2 or W>2..
W>2 can be discarded as here W is taken as negative
D can be eliminated since it is -2,
E is the answer..

Note :- C which is 0 can be eliminated initially itself as W.Z will become 0 in that case.
and after initial steps We can substitute the values and find the answer, especially so, since it is asking " which of the following could be a value of W ?".

IMO, W can not be <0 in this question as it is given that both z and w*z are >0 ---> w>0. The question needs to be reformatted as it is mentioning W and w and the RHS is not clearly written.

Hi Engr2012,
The Question is correct as it is and yes it is possible to read the Q the way you are reading it as I too did think the same way for a minute..
W•Z does not mean W*Z but the "•" is basically a symbol telling us a relation.. it could have been #, @ etc to avoid confusion..
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Intern  Joined: 15 Dec 2015
Posts: 34
Re: For any numbers W and Z, W•Z=Wˆ3 * Z(8-Wˆ2)  [#permalink]

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Hi,

Yes, 'W' and 'w' are the same variable. Both should be in capital letters. It's just a typo. My bad.

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Intern  Joined: 15 Dec 2015
Posts: 34
Re: For any numbers W and Z, W•Z=Wˆ3 * Z(8-Wˆ2)  [#permalink]

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Thanks a lot Chetan2u ! You answered it in a very clever way!

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Joined: 14 Jul 2014
Posts: 158
Location: United States
Schools: Duke '20 (D)
GMAT 1: 600 Q48 V27 GMAT 2: 720 Q50 V37 GPA: 3.2
Re: For any numbers W and Z, W•Z=Wˆ3 * Z(8-Wˆ2)  [#permalink]

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W∙Z=Wˆ3 * Z(8-Wˆ2)

If both Z and WZ are positive, then w^3(8-w^2) must be positive. Only E will satisfy this condition. (-9)^3 is negative and (8-81) is negative so this will result in a positive product. D does not satisfy this condition.
Intern  Joined: 23 Sep 2018
Posts: 1
Re: For any numbers W and Z, W•Z=Wˆ3 * Z(8-Wˆ2)  [#permalink]

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sara86 wrote:
For any numbers W and Z, W∙Z=Wˆ3 * Z(8-Wˆ2) . If both Z and W∙Z are positive numbers, which of the following could be a value of W ?

(A) 9

(B) 3

(C) 0

(D) -2

(E) -9

Looks like you have a mistake.
It is said, "both Z and W∙Z are positive numbers".
If Z is positive, then W*Z can be positive only when W also positive.
-9 * [positive] != positive. Re: For any numbers W and Z, W•Z=Wˆ3 * Z(8-Wˆ2)   [#permalink] 30 Sep 2018, 12:10
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