Question 1075937
It is well known that  (a/b)+(c/d) is not equivalent to (a+c)/(b+d). 
Suppose that a, b, c, and d are all positive. 

Show that (a+c)/(b+d) is in fact between the numbers a/b and c/d
<pre>
Case 1.  {{{a/b<(a+c)/(b+d)}}}

Multiply through by LCD b(b+d)

{{{a(b+d)<b(a+c)}}}
{{{ab+ad<ab+bc}}}
Subtract ab from both sides
{{{ad<bc}}}
Add cd to both sides
{{{ad+cd<bc+cd}}}
Factor out d on left, c on right
{{{d(a+c)<c(b+d)}}}
Divide both sides by d(b+d)
{{{(a+c)/(b+d)<c/d}}}

So {{{a/b<(a+c)/(b+d)<c/d}}}

Case 2.  {{{a/b>(a+c)/(b+d)}}}

Multiply through by LCD b(b+d)

{{{a(b+d)>b(a+c)}}}
{{{ab+ad>ab+bc}}}
Subtract ab from both sides
{{{ad>bc}}}
Add cd to both sides
{{{ad+cd>bc+cd}}}
Factor out d on left, c on right
{{{d(a+c)>c(b+d)}}}
Divide both sides by d(b+d)
{{{(a+c)/(b+d)>c/d}}}

So {{{a/b>(a+c)/(b+d)>c/d}}}

In either case, {{{(a+c)/(b+d)}}} is between {{{a/b}}} and {{{c/d}}}

---------------------------------
  
{{{a/b+c/d}}} is not between the numbers {{{a/b}}} and {{{c/d}}}

because the sum of two positive numbers is greater than
either one.

Edwin</pre>