CAT Quantitative Aptitude Questions | CAT Algebra - Inequalities questions

CAT Questions | Algebra | Inequalities - Natural Numbers

The question is about three distinct natural numbers. Given the sum of three distinct natural numbers we need to maximize the value of their product. 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 6: The sum of three distinct natural numbers is 25. What is the maximum value of their product?

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Explanatory Answer

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?

Let the three natural numbers be a, b and c. We know that a + b + c = 25.
Let us substitute some values and see where this is headed
a = 1, b = 2 c = 22, Product = 44
a = 2, b = 3 c = 20, product = 120
a = 5, b = 6 c = 14, product = 420

The numbers should be as close to each other as possible. (Just trial and error and one can figure this out). This property is an extension of the AM-GM Inequality.

25 divided by 3 is approximately equal to 8. So, we should choose a, b, and c close enough to 8.
8 + 8 + 9 = 25. But the numbers need to be distinct. 8 * 7 * 10 come closest. The maximum product would be 560.

The question is "The sum of three distinct natural numbers is 25. What is the maximum value of their product?"

Hence the answer is "The maximum product is 560."

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