Question

I have a simple proof question I can not figure out.

Prove:
If a/b < c/d, then a/b < (a+c)/(b+d) < c/d

not sure where to start. :s thanks for any help

Answer 1

so there are no restrictions given on a,b,c, and d?

Answer 2

a/b < c/d

ad < bc

now add ab on both sides,
what do u get?

Answer 3

ofcourse, assuming all a,b,c,d are non-zero

Answer 4

then
ad < bc
add cd on both sides.
combine the 2 results and you'll be done :)

Answer 5

Alright, simple enough.

Let's make the starting equation a little simpler.

\[\frac{ a }{ b } < \frac{c}{d}\]
\[ad<bc \]

So that's Inequality #1. For now.

Let's move on to Inequality #2, which is a bit more ambiguous. You can either go left, or right. It's better if you go both ways and prove both sides. Let's go left first.

\[\frac{ a }{ b } < \frac{a+c}{b+d}\]
Simplifying gives us
\[ab+ad<ab+cb\]
Cancel them ab's out. And voila!

\[ad<bc\]

Compare that to Inequality #1. It's the same, right? You've proven it.

But not fully.

Let's take \[\frac{ a+c }{ b+d } < \frac{ c }{ d }\]
Cross multiply them, you get
\[ad+cd<cb+cd\]

The cd's cancel out, giving you
\[ad<bc\] Compare this to Inequality #1 we obtained earlier.

Pretty much the same, right?
Mission accomplished. Apollo, we are go for launch!

Hope this helped. -swas

Answer 6

Thanks so much for all of your help! :)

Answer 7

Anytime.