# CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

###### CAT Questions | Algebra | Inequalities - Maximum possible value

The question is about maximum possible value. Our task is to maximize the product of three integers given their range and their sum. Inequalities are crucial to understand many topics that are tested in the CAT exam. Having a good foundation in this subject can help us tackle questions in Coordinate Geometry, Functions, and most importantly in Algebra. A range of CAT questions can be asked based on this simple concept.

Question 14: If a, b, c are integers such that – 50 < a, b, c < 50 and a + b + c = 30, what is the maximum possible value of abc?

1. 1000
2. 5000
3. 4410
4. 4560

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##### Method of solving this CAT Question from Algebra - Inequalities : When 'a' + 'b' is given to be a constant, 'ab' has its maximum value when they are closest to each other, or if 'a' = 'b'. What happens when 'a' and 'b' are distinct?

abc will be maximum when it is positive. So, a, b, c can all be positive or two of the three can be negative and one positive.

When all are positive, max product is when the numbers are 10, 10 and 10.

When two are negative and one positive, the best–case scenario would be when two negative numbers are as low as possible (magnitudes as high as possible) so that the product can be high. Now, in order for the product to be maximum, the positive number should be as high as possible. So, let the positive number be 49. Then the sum of the two negative numbers should be –19. The best– case scenario would be when numbers are 49, –9, –10.

Product would be 4410.

The question is "what is the maximum possible value of abc?"

##### Hence the answer is "4410"

Choice C is the correct answer.

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