How to select best k fractions out of n fractions (k<=n) so as to have (numerator sum / denominator sum) maximum?
For a list of fractions, define the ratio as (sum of numerator) / (sum of denominator). 

Question: Can we find a list of k fractions with the ratio ≥ r, for some given r?

Let the sum of numerators be X, and the sum of denominators be Y, then we require X / Y ≥ r or X - rY ≥ 0. For a single fraction x / y, we can calculate x - ry. We pick the k fractions where x - ry is largest. If the sum X - rY ≥ 0, then we have a list with ratio ≥ r, otherwise there is none.

We have obvious lower and upper bounds for r (the k-largest and the largest value of any fractions). With binary search we can easily find a list with maximum ratio. 

Responding to some comments: What I presented is the basic idea. There is plenty of room for improvement. 

For example "we pick the k fractions where x - ry is largest" - that can easily be done without sorting the array, much faster than n log n. Start like Quicksort, but only process subarrays that you need to determine the k largest values. 

And it's not just binary search, because each time you actually find a list of fractions and can update the lower or upper bound for r or both. (For example if the lower and upper bound were 6.0 ≤ r ≤ 10.0, you look for a list with ratio 8.0 and only find one with ratio 7.6, then in the next search you know 7.6 ≤ r &lt; 8.0 is possible). 

Since the lower bound for r is one that was actually achieved, you can also check which is the smallest eps for which you would have put a different fraction in the top k list of fractions.