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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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New post Updated on: 02 Jul 2013, 14:59
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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.
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Re: a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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New post 14 Jul 2014, 22:40
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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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Re: Algebra....aproach?  [#permalink]

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New post 08 Nov 2009, 15:28
1
4
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)

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

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New post 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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New post 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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New post 24 Mar 2016, 04:21
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 ) ?


Unless it is explicitly stated otherwise, different variables CAN represent the same number.
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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

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New post 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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New post 28 Mar 2016, 22:21
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.
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Re: a, b, c and d are four positive real numbers such that abcd=  [#permalink]

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

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