Then (a+c)/(b+d) = (a+e)/(b+f) = (c+e)/(d+f) = (a+c+e)/(b+d+f) = a/b = c/d = e/f =

Prove:

Take (a+c+e)/(b+d+f)=a/b=c/d=e/f=k as an example.

a/b=c/d=e/f=k

∴a=bk,c=dk,e=fk

∴a+c+e=bk+dk+fk=(b+d+f)k

∴(a+c+e)/(b+d+f)=(b+d+f)k/b+d+f=k=a/b=c/d=e/f

The same is true of subtraction:

If a/b=c/d=e/f=k

Then (a-c)/(b-d) = (a-e)/(b-f) = (c-e)/(d-f) = (a-c-e)/(b-d-f) = a/b = c/d = e/f =

The same is true when combined, for example, (a-c+e)/(b-d+f) = a/b = c/d = e/f = k.

In fact, as long as the denominator is aligned, it is equal.