Divide through everything by <em>b</em> :
Since <em>a/b</em> < <em>c/d</em>, it follows that
Multiply through everything on the right side by <em>b/d</em> to get
and so (<em>a</em> + <em>c</em>)/(<em>b</em> + <em>d</em>) < <em>c/d</em>.
For the other side, you can do something similar and divide through everything by <em>d</em> :
and <em>a/b</em> < <em>c/d</em> tells us that
Then
and so (<em>a</em> + <em>c</em>)/(<em>b</em> + <em>d</em>) > <em>a/b</em>.
Then together we get the desired inequality.