Why is $\frac{\sum_{i=1}x_iy_i}{\sum_{i=1}x_i^2}\ne \frac{\sum_{i=1}y_i}{\sum_{i=1}x_i}$? I know they are not equal, but I am unable to come up with a convincing explanation as to why they are not equal. Cancelling the $x_i$'s when they are part of the summand is prohibited; Can anyone provide a decent argument as to why this is the case?
Thanks.
 A: Let the $y_i$'s be all $1$, and let's write the summation explicitly, like so:
$${x_1 + x_2 + \cdots +x_n \over x_1^2 + x_2^2 + \dots +x_n^2} \neq {1 + 1 + \dots +1 \over x_1 + x_2 + \dots +x_n} = \frac{n}{x_1 + x_2 + \dots +x_n}$$
Does it seem convincing here that the $x_i$'s can't cancel? It's just not a valid logical step.
For a specific counterexample, take $x_i = 1,2,3,4$. Then we have
$${1+2+3+4 \over 1 + 4 + 9 + 16} = \frac{10}{30} = \frac13 \neq \frac25 = \frac4{10} = \frac4{1 + 2 + 3 + 4}$$
Therefore we cannot cancel the $x_i$'s.
A: Consider the simpler case when $n = 2$ (the upper bound for the summation index). Assume that $\dfrac {ab + cd} {a^2 + c^2} = \dfrac {b + d} {a + c}$. After cross-multiplication and canceling of similar terms, you may grup the remaining ones as $(a-c) (ad - bc) = 0$. This shows that, in order for that equality to be true, either $a = c$ (i.e. $x_1 = \dots = x_n$, which means that you may simplify a common factor), or your numbers have to verify a very special relationship ($ad - bc = 0$), so the equality cannot hold in general, for arbitrary values.
A: Maybe this answer of mine is not rigorous enough but I will give it a try because it gives some kind of alternative view on questions of this sort so I will add it maybe just to have a variety of views on the same question. 
Suppose that we view $\sum_{i=1}^{n}x_iy_i$ as a function of $2n$ variables $x_1,...,x_n,y_1,...,y_n$. Then, in the same spirit, the sum $\sum_{i=1}^{n}x_i$ could be viewed as the function of $n$ variables $x_1,...x_n$. The sum $\sum_{i=1}^{n} {x_i}^2$  as a function of also the $n$ variables $x_1,...,x_n$ and sum $\sum_{i=1}^{n} y_i$ as a function of $n$ variables $y_1,...y_n$.
Let us write now $a(x_1,...x_n,y_1,...y_n)=\sum_{i=1}^{n}x_iy_i$ and $b(x_1,...x_n)=\sum_{i=1}^{n}x_i$ and $c(x_1,...x_n)=\sum_{i=1}^{n} {x_i}^2$ and $d(y_1,...y_n)=\sum_{i=1}^{n} y_i$.
Now, if we suppose that your equality is true then we would have 
$a(x_1,...x_n,y_1,...y_n)=d(y_1,...y_n) \dfrac {c(x_1,...x_n)}{b(x_1,...x_n)}$
So we would have that the function of $2n$ variables is equal to the product of two functions, one that depends on the first $n$ variables and the second that depends on the remaining $n$ variables, and, intuitively, this will, I guess, happen rarely, and in this case it is only always true when $n=1$.
