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

Question

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

KarishmaB · Accepted Answer

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.

Bunuel · Answer

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

rishabh3057 · Answer

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.

leeto · Answer

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 ) ?

Bunuel · Answer

Unless it is explicitly stated otherwise, different variables CAN represent the same number.

leeto · Answer

Many thanks, your answer get rid of a lot of doubts.

KarishmaB · Answer

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.

leeto · Answer

I like you analogy with quadratic equation, thank you for this idea.

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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=

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