How can I generalize the result for the n=2 case to a larger set of numbers?

[maquina]

Homework Statement

a,b,c,d in a ordered set K and b and d>0
Show that $$\frac{a+c}{b+d}$$ stay between the minimum and max from $$\frac {a}{b}$$ and $$\frac {c}{d}$$. Generalize for $$a_1,\hdots,a_n,b_1,\hdots,b_n \in K$$ with $$b_1\hdots,b_n >0$$ so $$\frac{a_1+\hdots+a_n}{b_1+\hdots+b_n}$$ is between the max and min elements from $$\frac{a_1}{b_1},\hdots,\frac{a_n}{b_n}$$

I could do it for the the first case but in a way it's impossible to generalize
any ideas?
tks in advance

 

[maquina]

to correct, it`s a ordered field

 

[maquina]

if u consider a/b<c/d
u can do a/b - (a+c)/(b+d)}=(ad-bc)/(b(b+d))>0
and c/d - (a+c)/(b+d)}=(bc-ad)/(b(b+d))>0
and done

but them using samething for generalizing i couldn't make it :(

 

[HallsofIvy]

Looks like proof by induction would work.

 

[Dick]

Let's redo the n=2 case in a way it will be easier to generalize. Write (a1+a2)/(b1+b2)=(b1/(b1+b2))*(a1/b1)+(b2/(b1+b2))*(a2/b2). Notice that the bi/(b1+b2) terms are positive and sum to 1. (This means (a1+a2)/(b1+b2) is in the 'convex hull' of the bi/ai.) If I replace the ai/bi by their minimum and maximum, what do I conclude?