Finding the smaller number of two given the ratio between sum, difference and product How would you find the smaller of two numbers given the ratio between their sum, difference and product? I've been struggling with this one for a while. 
For example: the ratio between the sum, difference and product is 6:4:15 and the numbers are positive integers 

Just looking at the sum and difference, if the unknown numbers are $a,b$, then there is some real $k$ such that $$a+b=6k$$
and $$a-b=4k$$
We can then solve this to get $a=5k, b=k$.  Now, the product is $5k^2$, and using the ratio relationship, $$5k^2=15k$$
from which we conclude $k=3$ and hence $a=15, b=3$.  A second, less interesting, solution, is $k=0$, which gives $a=b=0$.
