SOLUTION: 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

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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, while (a/b)+(c/d) is not
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
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
Case 1.  

Multiply through by LCD b(b+d)



Subtract ab from both sides

Add cd to both sides

Factor out d on left, c on right

Divide both sides by d(b+d)


So 

Case 2.  

Multiply through by LCD b(b+d)



Subtract ab from both sides

Add cd to both sides

Factor out d on left, c on right

Divide both sides by d(b+d)


So 

In either case,  is between  and 

---------------------------------
  
 is not between the numbers  and 

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

Edwin
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