Approximate Ratio: Steps to Find Reasonable Solutions

[onako]

The following is an approximation problem. Given a ratio:
$$D =\frac{ \sum a_ib_i}{\sum b_ib_i}$$

I wonder what steps to follow to give a reasonable approximation.
This is an intuition
$$D =\frac{ \sum a_i}{\sum b_i}$$.
Clearly, given that all b_i terms are the same, the approximation is the correct solution.
But, this is not the case. The question is, under which conditions (assumptions on a_i or b_i) is the
above approximation accurate.
I guess, the lower the standard deviation of the b_i's, the more accurate the approximation. But, suppose the
b_i's are [1,2,3,4,5,6,7...]. What condition needs be meet for the accurate approximation.
In a sense, the ratio I'm trying to approximate is the weighted average, and in the approximation I'm discarding
the weights. If this interpretation makes it easier to further interpret, please use it.

Also, as for the conditions, I thought of
$$a_i>a_k, => b_i>b_k$$
How is this condition affecting the accuracy of the ratio approximation. Is the approximation more accurate with
this assumption.
Thanks
(topic in Number theory and Calculus subforum)

 

[fresh_42]

There is no reasonable way to compare the two quantities, other than all $b_i$ are almost equal.