TITLE: Why everytime the final number comes the same?
QUESTION [22 upvotes]: I have come across an interesting puzzle.
Write $20$ numbers. Erase any two number say $x$ and $y$ and and replace with 
$\text{Number}_{new} = xy/(x + y)$
OR
$\text{Number}_{new}= x + y + xy$

Where some possible choices of  are:
You are to repeat 19 times, until you get a single number.
The final result will be independent of the choices of pairs made. 
Why is the result the same every time ?
Example:
Suppose I take $4$ numbers $\{2 , 4 , 5 , 7\}$

Now if I take $2$ and $4$ and get new number. New set will be  $\{\frac86, 5 , 7\}$
Now if I take 5 and 7. New set will be  $\{\frac86 , \frac{35}{12}\}$ And the final
value will be  $\{\frac{140}{153}\}$
Now if I take $2$ and $5$ and get new number. New set will be  $\{\frac{10}7, 4 ,
    7\}$. Now if I take $4$ and $7$. New set will be  $\{\frac{10}7 , \frac{28}{11}\}$ And
the final value will be  $\{\frac{140}{153}\}$ again

How is this possible?
Thanks in advance.

REPLY [3 votes]: Let us find some characteristic of the set of numbers that does not change when you group two of them the way you describe.
The formula $\frac{xy}{x+y}$ suggests the following characteristic: $$\chi(a,b,c,\ldots)=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\ldots$$ Indeed, if we take two numbers $m$ and $n$ and group them, then $$\chi(a,b,\ldots,m,\ldots,n,\ldots)=\frac{1}{a}+\ldots+\frac{1}{m}+\ldots+\frac{1}{n}+\ldots=\frac{1}{a}+\ldots+\frac{m+n}{mn}+\ldots=\chi(a,b,\ldots,\frac{mn}{m+n},\ldots)$$
Therefore, when you group any two numbers in the set, its characteristic does not change, and what you are going to get at the end (regardless of the sequence of pairs you choose) is one number $x$ such that $$\chi(a,b,c,\ldots)=\frac{1}{a}+\frac{1}{b}+\ldots=\chi(x)=\frac{1}{x}$$