Intermediate Ratio: Proof & Name

[Helios]

Given positive integers a, b, c, d

and for fractions a/b and c/d,

it seems that ( a + c )/( b + d ) is between a/b and c/d.

There's likely an easy proof of this. I'd like to know if there's a formal name for ( a + c )/( b + d ) or the operation that brings it about.

 

[HallsofIvy]

Proof by contradiction:

Assuming that a/b< c/d, then ad< bc.

Suppose (a+ c)/(b+ d) were not between a/b and c/d. The we must have either (a+c)/(b+d)< a/b or (a+c)/(b+d)< c/d.

In the first case, if [itex](a+c)/(b+d)\le a/b[/itex] then [itex](a+c)b= ab+ bc\le a(b+d)= ab+ ad[/itex] or [itex]bc\le ad[/itex] contradicting the inequality above.

In the second case, if [itex]c/d\le (a+c)/(b+d)[/itex] then [/itex]c(b+d)= bc+ cd\le d(a+c)= ad+ dc[/itex] or [itex]bc\le ad[/itex], again a contradiction.