# Weighted division?

Let's say I have 10,000 dollars I want divided among 10 people. With simple division each person gets $1,000. Easy enough. Now suppose each person has a score on a test from 0 to 200. Now I want to divide the money among all the people but weighted by their score on the test, such that people who scored higher will get more money. How would I do that? -$W=\frac{Individual~ test~ score}{sum~ of~ all~ test~ scores}$– Simon Hayward Dec 7 '12 at 23:07 ## 1 Answer Suppose that the test scores are$x_1,x_2,\dots,x_{10}$. Let$t$be the sum of the test scores; then the first person’s share of the total is$\frac{x_1}t$, the second’s is$\frac{x_2}t$, and so on. These ten fractions add up to$1$, so just give person$k$(for$k=1,2,\dots,10$) $$10000\cdot\frac{x_k}t\text{ dollars}\;.$$ - Damn you have to get in quick to get ahead of Brian M. Scott. – Simon Hayward Dec 7 '12 at 23:09 Is there a way to tweak the weighting such that even with a large disparity in scores there will not be as large a disparity in the money amount? – User Dec 11 '12 at 7:40 @User: I used a linear weighting of the test scores, but other weightings are possible. You could, for example, replace$x_k$by$y_k=\log x_k$and apply the same algorithm to the numbers$y_k$; this would greatly flatten the top end, since multiplying a score by$10$(if you’re using logs base$10$) would only add$1$to the modified score. Or you could replace$x_k$by$\sqrt{x_k}$, which would have a similar but less dramatic effect. You might try replacing$x_k$by$y_k=x_k^\alpha$for different values of$\alpha$between$0$and$1$and choosing an$\alpha\$ that gives ... – Brian M. Scott Dec 11 '12 at 7:59
... satisfactory results. – Brian M. Scott Dec 11 '12 at 8:02