What is the sum of the maximum and minimum of two real numbers?

[Jameson]

This problem is geared at younger high school students but is a good introduction to proofs and laying out mathematical arguments.

For real numbers $a,b$ what is $\max \left({a, b}\right) + \min \left({a, b}\right)$? Show all steps in your solution.

Hint:

Spoiler

There are 3 cases to consider.

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[Jameson]

Congratulations to the following members for their correct solutions:

1) eddybob123
2) kaliprasad
3) MarkFL

This problem was on one hand very easy but it focuses on making strong mathematical arguments for general situations, which is a great skill to develop at a young age. So if you're in middle school, high school or just starting to get into math I suggest you try this problem before looking at the solution.

Solution (from eddybob123):

Spoiler

Let us consider three cases:

Case 1 $a>b$: It follows that $\max(a,b)=a$ and $\min(a,b)=b$, and their sum is $a+b$.

Case 2 $a<b$: This is the opposite of the first case. We have $\max(a,b)=b$ and $\min(a,b)=a$, and their sum is also $a+b$.

Case 3 $a=b$: Since $a=b$, it does not matter what values we use for $\max(a,b)$ and $\min(a,b)$. Their sum can be represented in either of the forms $2a$, $a+b$, or $2b$.

In all three cases, the sum of the maximum and the minimum of two real numbers is simply the sum of the numbers.