How do expressions of the form $\frac{a}{b}+\frac{c}{d}+......$ compare to $\frac{a+c+....}{b+d+...}$, for positive $a,b,c,d....$?

Question

Is the question too general to answer? I'm talking about when the former expression will be greater (or smaller) than the latter?

Here's an extension:

Compare expressions $$p\frac{a}{b}+q\frac{c}{d}+r\frac{e}{f}......$$

And

$$(p+q+r+.....)(\frac{a+c+e+....}{b+d+f+...})$$

For positive $a,b,c,d...$ and $p,q,r,s.....$. This is of course assuming that the first sequence of variables doesn't have variables having the same names has those in the second sequence (say, we start naming them by greek letters once we reach $p$).

Answer 1

If the variables are all positive, $\frac{a}{b}+\frac{c}{d}+… \gt \frac{a+c+…}{b+d+…}$. Break up the right side as $\frac a{b+d+\ldots}+\frac c{b+d+\ldots}+\ldots$ and notice that each term on the right has a matching one on the left, but the one on the right has greater denominator.

Your extension can go either way, because the product has many cross terms.