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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, 13:53
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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
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Last edited by Bunuel on 02 Jul 2013, 14:59, edited 1 time in total.
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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 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 minimummaximum question, it is a good idea to check on equality. Often, the point of equality is a transition point.
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Re: a, b, c and d are four positive real numbers such that abcd= [#permalink]
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22 Oct 2014, 01:20
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.



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Re: a, b, c and d are four positive real numbers such that abcd= [#permalink]
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24 Mar 2016, 04:18
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 ) ?
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Re: a, b, c and d are four positive real numbers such that abcd= [#permalink]
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24 Mar 2016, 04:21
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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24 Mar 2016, 04:25
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.
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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, 22:21
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.
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Re: a, b, c and d are four positive real numbers such that abcd= [#permalink]
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29 Mar 2016, 00:13
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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