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Manager  Joined: 09 Jun 2009
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a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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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

Originally posted by papillon86 on 08 Nov 2009, 13:53.
Last edited by Bunuel on 02 Jul 2013, 14:59, edited 1 time in total.
Veritas Prep GMAT Instructor V
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Posts: 9706
Location: Pune, India
Re: a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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4
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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Math Expert V
Joined: 02 Sep 2009
Posts: 58453

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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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Intern  Joined: 04 Jul 2014
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Re: a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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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.
Intern  Joined: 13 Mar 2011
Posts: 21
Re: a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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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

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 ) ?
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Please, press the +1 KUDOS button , if you find this post helpful Math Expert V
Joined: 02 Sep 2009
Posts: 58453
Re: a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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

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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

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Joined: 16 Oct 2010
Posts: 9706
Location: Pune, India
Re: a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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1
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

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Intern  Joined: 13 Mar 2011
Posts: 21
Re: a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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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 Non-Human User Joined: 09 Sep 2013
Posts: 13261
Re: a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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_________________ Re: a, b, c and d are four positive real numbers such that abcd=   [#permalink] 08 May 2019, 06:05
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