# CAT Practice : Inequalities

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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?

## Natural Numbers

Q.6: The sum of three distinct natural numbers is 25. What is the maximum value of their product?

The maximum product is 560.

## Detailed Solution

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 / 3
∼ 8.
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.
Correct Answer : 560

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## More questions from Inequalities

Inequalities are crucial to understand many topics that are tested in the CAT. Having a good foundation in this subject will make us tackling questions in Coordinate Geometry, Functions, and most importantly in Algebra much more comfortable.