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# For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po

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Math Expert
Joined: 02 Sep 2009
Posts: 46191
For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po [#permalink]

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25 Aug 2016, 01:32
00:00

Difficulty:

55% (hard)

Question Stats:

59% (01:39) correct 41% (01:23) wrong based on 191 sessions

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For any numbers w and z, $$w#z = w^3z(8 - w^2)$$. 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

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For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po [#permalink]

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25 Aug 2016, 10:06
1
Bunuel wrote:
For any numbers w and z, $$w#z = w^3z(8 - w^2)$$. 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

Since w#Z and w are positive, we need to take the value of w such that w#z or w^3z(8 - w^2) is positive.
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Re: For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po [#permalink]

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25 Aug 2016, 21:50
1
Bunuel wrote:
For any numbers w and z, $$w#z = w^3z(8 - w^2)$$. 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

Given z is positive and w#z is positive

Now we need to find the value of w, let's assume z = 1.

From options.

A. 9 if we sub w as 9 in the equation we get w#z as -ve.
B. 3 if we sub w as 3 in the equation we get w#z as -ve
C. 0 Here we get result as 0, 0 is even but not positive. so w#z is not positive.
D. −2 if we sub w as -2 in the equation we get w#z as -ve
E. −9 if we sub w as 3 in the equation we get w#z only +ve value.

Option E is correct.
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Re: For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po [#permalink]

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26 Aug 2016, 01:11
Bunuel wrote:
For any numbers w and z, $$w#z = w^3z(8 - w^2)$$. 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

To satisfy the condition we have two scenarios:
1. w^3 & (8-w^2) both need to be +ve. So A & B are out

2. w^3 & (8-w^2) both need to be -ve. -2 makes (8-w^2) +ve, so D is out. -9 satisfies the requirements, E is the correct answer
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Re: For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po [#permalink]

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20 Feb 2017, 20:41
How did you guys figure out that in order for the answer to be correct the result must be positive? I kind of thought that but didn't understand how you could derive that from the question.
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Joined: 17 Apr 2016
Posts: 100
Re: For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po [#permalink]

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21 Feb 2017, 06:49
Nunuboy1994 wrote:
How did you guys figure out that in order for the answer to be correct the result must be positive? I kind of thought that but didn't understand how you could derive that from the question.

Hi Nunuboy1994,

It is mentioned in the question that w#z is positive. Also we are given the equation w#z = w^3z(8 - w^2) to figure out the value of w. Hence using we have to use the fact that w#z should be positive and only one option will satisfy this condition. Hence the option that satisfies the condition , is the correct answer.

Hope it is clear.

Thanks,
Manager
Joined: 17 Apr 2016
Posts: 100
Re: For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po [#permalink]

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21 Feb 2017, 06:55
Hi @Bunnuel,

I used the following approach.
Since w#z is positive,w#z = w^3z(8 - w^2),
8w^3z- w^5z is positive.

Solving we get the following inequality--> w^2<8. Only Option D fulfils this condition.

What am I missing? Kindly help.
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Re: For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po [#permalink]

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21 Feb 2017, 08:22
1
Hi @Bunnuel,

I used the following approach.
Since w#z is positive,w#z = w^3z(8 - w^2),
8w^3z- w^5z is positive.

Solving we get the following inequality--> w^2<8. Only Option D fulfils this condition.

What am I missing? Kindly help.

You have considered only one aspect i.e. (8 - w^2) > 0. However, (8 - w^2) can be negative too.
w^3z(8 - w^2) is positive.
Since we know that z is positive, the contributing expressions are w^3 and (8 - w^2). Either both can be negative or both can be positive.
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Re: For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po [#permalink]

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23 Feb 2017, 10:35
Bunuel wrote:
For any numbers w and z, $$w#z = w^3z(8 - w^2)$$. 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

We have to go through the answer choices. Remember, we want w#z to be positive.

A. 9

If w = 9, then 9#z = (9^3)z(8 - 9^2) is negative, since 9^3 and z are positive, but 8 - 9^2 is negative.

B. 3

If w = 3, then 3#z = (3^3)z(8 - 3^3) is negative, since 3^3 and z are positive, but 8 - 3^2 is negative.

C. 0

If w = 0, then 0#z = (0^3)z(8 - 0^2) is zero, since 0^3 = 0.

D. -2

If w = -2, then -2#z = ((-2)^3)z(8 - (-2)^2) is negative, since z and 8 - (-2)^2 are positive, but (-2)^3 is negative.

E. -9

If w = -9, then -9#z = ((-9)^3)z(8 - (-9)^2) is positive, since z is positive, but (-9)^3 and 8 - (-9)^2 are negative. Notice that negative x positive x negative is positive.

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Re: For any numbers w and z, w#z = w^3z(8 - w^2). If both z and w#z are po   [#permalink] 23 Feb 2017, 10:35
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