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# a, b, c and d are four positive real numbers such that abcd=

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a, b, c and d are four positive real numbers such that abcd= [#permalink]

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08 Nov 2009, 12:53
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a, b, c and d are four positive real numbers such that abcd=1, what is the minimum value of (1+a)(1+b)(1+c)(1+d)?

A. 4
B. 1
C. 16
D. 18
[Reveal] Spoiler: OA

Last edited by Bunuel on 02 Jul 2013, 13:59, edited 1 time in total.
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08 Nov 2009, 14:28
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papillon86 wrote:
a,b,c and d are four +ve real nubers such that abcd=1, what is the minimum value of
(1+a)(1+b)(1+c)(1+d)?

a) 4
b) 1
c) 16
d) 18

what is the best way to solve such question where in we need to calculate the min or max values?

Think there is no catch in this question. As the numbers are positive and their product is 1: either 2,3, or all 4 numbers are reciprocals and rest is 1 OR all numbers are equal to 1.

Minimum value will be when a=b=c=d=1, hence (1+a)(1+b)(1+c)(1+d)=16. (You can try reciprocals to see that the product will be greater)

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Re: a, b, c and d are four positive real numbers such that abcd= [#permalink]

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14 Jul 2014, 13:06
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Re: a, b, c and d are four positive real numbers such that abcd= [#permalink]

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14 Jul 2014, 21:40
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papillon86 wrote:
a, b, c and d are four positive real numbers such that abcd=1, what is the minimum value of (1+a)(1+b)(1+c)(1+d)?

A. 4
B. 1
C. 16
D. 18

As Bunuel said, abcd = 1 implies that either the numbers are equal to 1 or there are pairs of reciprocals e.g. (1, 1, 1, 1) or (1, 1, 2, 1/2) or (3, 1/3, 4, 1/4) etc.

If a and b are 1 and 1,
(1+a)(1+b) = (1+1)(1+1) = 4

If a and b are 2 and 1/2,
(1+a)(1+b) = (1+2)(1+1/2) = 9/2 = 4.5

If a and b are 3 and 1/3,
(1+a)(1+b) = (1+3)(1+1/3) = 16/3 = 5.3

As you keep taking higher reciprocals, the value of (1+a)(1+b) keeps increasing.

So taking reciprocals is a bad idea and all numbers must be 1 giving us the minimum value of 16.

Anyway, in any minimum-maximum question, it is a good idea to check on equality. Often, the point of equality is a transition point.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 04 Jul 2014 Posts: 1 Followers: 0 Kudos [?]: 0 [0], given: 64 Re: a, b, c and d are four positive real numbers such that abcd= [#permalink] ### Show Tags 22 Oct 2014, 00:20 Hello All, My first reply on this site. Another approach can be - For constant sum, product is minimum when terms are equal. 1+1/a = 1+1/b implies a=b=c=d. gives a hint that all can be 1. GMAT Club Legend Joined: 09 Sep 2013 Posts: 12834 Followers: 559 Kudos [?]: 157 [0], given: 0 Re: a, b, c and d are four positive real numbers such that abcd= [#permalink] ### Show Tags 28 Dec 2015, 09:14 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ Intern Joined: 13 Mar 2011 Posts: 21 Followers: 0 Kudos [?]: 2 [0], given: 106 Re: a, b, c and d are four positive real numbers such that abcd= [#permalink] ### Show Tags 24 Mar 2016, 03:18 papillon86 wrote: a, b, c and d are four positive real numbers such that abcd=1, what is the minimum value of (1+a)(1+b)(1+c)(1+d)? A. 4 B. 1 C. 16 D. 18 In terms of quality of the problem, isn't different names for the variables implies ( implicitly mean ) that the variables is different ? What is the probability that such problem can actually appears in the actual test ? In other words, does creators of the GMAT exam when naming of the different variables with different names assumes ( by default ) that the variables can be equals to each other on special circumstances ( like the problem above ) ? _________________ I’m not afraid of the man who knows 10,000 kicks and has practiced them once. I am afraid of the man who knows one kick & has practiced it 10,000 times! - Bruce Lee Please, press the +1 KUDOS button , if you find this post helpful Math Expert Joined: 02 Sep 2009 Posts: 35837 Followers: 6827 Kudos [?]: 89636 [1] , given: 10380 Re: a, b, c and d are four positive real numbers such that abcd= [#permalink] ### Show Tags 24 Mar 2016, 03:21 1 This post received KUDOS Expert's post leeto wrote: papillon86 wrote: a, b, c and d are four positive real numbers such that abcd=1, what is the minimum value of (1+a)(1+b)(1+c)(1+d)? A. 4 B. 1 C. 16 D. 18 In terms of quality of the problem, isn't different names for the variables implies ( implicitly mean ) that the variables is different ? What is the probability that such problem can actually appears in the actual test ? In other words, does creators of the GMAT exam when naming of the different variables with different names assumes ( by default ) that the variables can be equals to each other on special circumstances ( like the problem above ) ? Unless it is explicitly stated otherwise, different variables CAN represent the same number. _________________ Intern Joined: 13 Mar 2011 Posts: 21 Followers: 0 Kudos [?]: 2 [0], given: 106 Re: a, b, c and d are four positive real numbers such that abcd= [#permalink] ### Show Tags 24 Mar 2016, 03:25 Bunuel wrote: leeto wrote: papillon86 wrote: a, b, c and d are four positive real numbers such that abcd=1, what is the minimum value of (1+a)(1+b)(1+c)(1+d)? A. 4 B. 1 C. 16 D. 18 In terms of quality of the problem, isn't different names for the variables implies ( implicitly mean ) that the variables is different ? What is the probability that such problem can actually appears in the actual test ? In other words, does creators of the GMAT exam when naming of the different variables with different names assumes ( by default ) that the variables can be equals to each other on special circumstances ( like the problem above ) ? Unless it is explicitly stated otherwise, different variables CAN represent the same number. Many thanks, your answer get rid of a lot of doubts. _________________ I’m not afraid of the man who knows 10,000 kicks and has practiced them once. I am afraid of the man who knows one kick & has practiced it 10,000 times! - Bruce Lee Please, press the +1 KUDOS button , if you find this post helpful Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7059 Location: Pune, India Followers: 2081 Kudos [?]: 13234 [1] , given: 221 Re: a, b, c and d are four positive real numbers such that abcd= [#permalink] ### Show Tags 28 Mar 2016, 21:21 1 This post received KUDOS Expert's post leeto wrote: papillon86 wrote: a, b, c and d are four positive real numbers such that abcd=1, what is the minimum value of (1+a)(1+b)(1+c)(1+d)? A. 4 B. 1 C. 16 D. 18 In terms of quality of the problem, isn't different names for the variables implies ( implicitly mean ) that the variables is different ? What is the probability that such problem can actually appears in the actual test ? In other words, does creators of the GMAT exam when naming of the different variables with different names assumes ( by default ) that the variables can be equals to each other on special circumstances ( like the problem above ) ? Think from a conceptual point of view: A variable is not a stand in for a single value. We put in a variable when the actual value is not known. It is possible that two variables end up having the same value. Often, a variable could take multiple values (e.g. in a quadratic). It is possible that one of its values matches one of the values that another variable can take. Hence, there is no such restriction that two variables cannot take the same value. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: a, b, c and d are four positive real numbers such that abcd= [#permalink]

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28 Mar 2016, 23:13
VeritasPrepKarishma wrote:
leeto wrote:
papillon86 wrote:
a, b, c and d are four positive real numbers such that abcd=1, what is the minimum value of (1+a)(1+b)(1+c)(1+d)?

A. 4
B. 1
C. 16
D. 18

In terms of quality of the problem, isn't different names for the variables implies ( implicitly mean ) that the variables is different ? What is the probability that such problem can actually appears in the actual test ? In other words, does creators of the GMAT exam when naming of the different variables with different names assumes ( by default ) that the variables can be equals to each other on special circumstances ( like the problem above ) ?

Think from a conceptual point of view:
A variable is not a stand in for a single value. We put in a variable when the actual value is not known. It is possible that two variables end up having the same value. Often, a variable could take multiple values (e.g. in a quadratic). It is possible that one of its values matches one of the values that another variable can take. Hence, there is no such restriction that two variables cannot take the same value.

I like you analogy with quadratic equation, thank you for this idea.
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I’m not afraid of the man who knows 10,000 kicks and has practiced them once. I am afraid of the man who knows one kick & has practiced it 10,000 times! - Bruce Lee

Please, press the +1 KUDOS button , if you find this post helpful

Re: a, b, c and d are four positive real numbers such that abcd=   [#permalink] 28 Mar 2016, 23:13
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