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

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 $$p$$).

• It is too general. Why don't you play around yourself for a while with just two summands? Do some numerical tests. Fix some of the variables and see what happens as the other(s) vary. – Ethan Bolker May 5 at 14:12
• Hint: math.stackexchange.com/questions/751155/… Can you extend the result by induction from a pre-sorted list of fractions ? – zwim May 5 at 14:13

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.