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Intern  Joined: 06 Oct 2010
Posts: 23
For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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Difficulty:   15% (low)

Question Stats: 80% (01:47) correct 20% (02:12) wrong based on 104 sessions

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For all real numbers v, the operation is defined by the equation v* = v - v/3. If (v*)* = 8, then v=

(A) 15
(B) 18
(C) 21
(D) 24
(E) 27

My problem is that I don't know how to interpret the following symbols: (v*)*. Hopefully someone can help. Thanks.
Manager  Joined: 06 Jun 2009
Posts: 233
Location: USA
WE 1: Engineering
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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niheil wrote:
Please explain how to solve the following question. It was listed in one of the GMAT Paper Tests.

16. For all real numbers v, the operation is defined by the equation v* = v - v/3. If (v*)* = 8, then v=

(A) 15
(B) 18
(C) 21
(D) 24
(E) 27

My problem is that I don't know how to interpret the following symbols: (v*)*. Hopefully someone can help. Thanks.

v* = v - v/3

For sake of understanding, let v* = B

B = v - v/3

Therefore, (v*)* = B* = B-B/3

Or, (v*)* = [v- v/3] - [v- v/3]/3

8 = 2v/3 - 2v/9
v = 18
##### General Discussion
Math Expert V
Joined: 02 Sep 2009
Posts: 64951
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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niheil wrote:
Please explain how to solve the following question. It was listed in one of the GMAT Paper Tests.

16. For all real numbers v, the operation is defined by the equation v* = v - v/3. If (v*)* = 8, then v=

(A) 15
(B) 18
(C) 21
(D) 24
(E) 27

My problem is that I don't know how to interpret the following symbols: (v*)*. Hopefully someone can help. Thanks.

Given: $$v*=v-\frac{v}{3}=\frac{2}{3}v$$ and $$(v*)*=8$$.
Question: $$v=?$$

$$(v*)*=(\frac{2}{3}v)*=\frac{2}{3}*(\frac{2}{3}v)=8$$ --> $$\frac{4}{9}v=8$$ --> $$v=18$$.

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Intern  Joined: 06 Oct 2010
Posts: 23
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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Bunuel, how did you know that (2v/3)* = 2/3 x (2v/3)?
Math Expert V
Joined: 02 Sep 2009
Posts: 64951
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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niheil wrote:
Bunuel, how did you know that (2v/3)* = 2/3 x (2v/3)?

$$v*=v-\frac{v}{3}=\frac{2}{3}v$$ so $$v*$$ is $$\frac{2}{3}$$rd of $$v$$. So $$(\frac{2}{3}v)*$$ is $$\frac{2}{3}$$rd of $$\frac{2}{3}v$$.
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Intern  Joined: 06 Oct 2010
Posts: 23
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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Awesome! I understand. Thanks Bunuel! And thanks again adishail!
Intern  Joined: 31 Oct 2010
Posts: 28
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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ywilfred wrote:
For all real numbers v, the operation v* is defined by the equation v* = v - v/3 . If (v*)* = 8, what is the value of v.

Since v* = v - v/3,

(v*)* = (v - v/3) - (v - v/3)/3
= (3v-v)/3 - [(3v-v)/3]/3
= 2v/3 - (2v/3)/3
= 2v/3 - 2v/9
= (6v - 2v)/9
= 4v/9

We also know that (v*)* = 8, so (v*)* = 8 = 4v/9 and v = 18.

(v-v/3) did you just multiply by three to just make the v/3 go away? but then why does the first part get divided by three, its late here so the synapses arent firing at full speed scotty.

im losing the progression from (v-v/3) - (v-v/3)/3 --------> (3v-v)/3 - [(3v-v)/3]/3
Senior Manager  Status: Bring the Rain
Joined: 17 Aug 2010
Posts: 333
Location: United States (MD)
Concentration: Strategy, Marketing
Schools: Michigan (Ross) - Class of 2014
GMAT 1: 730 Q49 V39
GPA: 3.13
WE: Corporate Finance (Aerospace and Defense)
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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Thanks for the explanation.
Intern  B
Joined: 16 Oct 2017
Posts: 35
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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Is this also a correct way to solve?

Step 1:
V* = V/1 - V/3
= 3V/3 - V/3 = 2V/3 = 8
2V = 24
V = 12

Step 2: 2V/3 = 12
2V/2 = 36/2
2V = 36
V = 18
Manager  B
Joined: 16 Jul 2016
Posts: 52
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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$$v-\frac{v}{3}-([v-\frac{v}{3}]/3)$$
after multiplying everything by 3 we obtain $$3v-v-v+\frac{v}{3}=24$$

$$4v/3=24$$

$$v=18$$
Intern  B
Joined: 16 Jan 2017
Posts: 2
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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Bunuel wrote:
niheil wrote:
Bunuel, how did you know that (2v/3)* = 2/3 x (2v/3)?

$$v*=v-\frac{v}{3}=\frac{2}{3}v$$ so $$v*$$ is $$\frac{2}{3}$$rd of $$v$$. So $$(\frac{2}{3}v)*$$ is $$\frac{2}{3}$$rd of $$\frac{2}{3}v$$.

Thanks for the explanation. But why (2/3V)* doesn't have a (2/3)* x V* but simply only give 2/3 x V*?
Senior Manager  P
Joined: 10 Apr 2018
Posts: 267
Location: United States (NC)
Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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flowertown wrote:
Bunuel wrote:
niheil wrote:
Bunuel, how did you know that (2v/3)* = 2/3 x (2v/3)?

$$v*=v-\frac{v}{3}=\frac{2}{3}v$$ so $$v*$$ is $$\frac{2}{3}$$rd of $$v$$. So $$(\frac{2}{3}v)*$$ is $$\frac{2}{3}$$rd of $$\frac{2}{3}v$$.

Thanks for the explanation. But why (2/3V)* doesn't have a (2/3)* x V* but simply only give 2/3 x V*?

Hi

V*= V-$$\frac{V}{3}$$
(V*) = $$\frac{2V}{3}$$

$$(V*)^{*}$$= $$\frac{2V}{3}$$ - $$\frac{\frac{2V}{3} }{3}$$
$$(V*)^{*}$$=$$\frac{2V}{3}$$-$$\frac{2V}{9}$$

We are told that $$(V*)^{*}$$= 8
hence
we solve for V = 18

Hope this helps
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Re: For all real numbers v, the operation is defined by the equation v* =  [#permalink]

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_________________ Re: For all real numbers v, the operation is defined by the equation v* =   [#permalink] 16 May 2020, 13:10

# For all real numbers v, the operation is defined by the equation v* =   