A prize of $27,000 is to be divided among three people in the ratio 3:5:7. What is the largest share? This is not homework; I was just reviewing some old math flash cards and I came across this one I couldn't solve. I'm not interested in the solution so much as the reasoning.
Thanks

Lets $a$ amount that get first person $b$ amount of second and $c$ amount of third person, from conditions in question follow system
$$a+b+c=27000$$
$$a:b:c=3:5:7$$ 
from second equation follow $a:3=b:5=c:7=k$ and $a=3k,b=5k,c=7k$ if these values put in first equation of system above we get $$3k+5k+7k=27000, 15k=27000, k=27000/15=1800$$ $k$ is coefficient of proportionality, clearly
$$a=3k=5400$$
$$b=5k=9000$$
$$c=7k=12600$$
This method can be generalized.
