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

Algebra ->  Numeric Fractions Calculators, Lesson and Practice -> 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       Log On


   

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) About Me  (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.  a%2Fb%3C%28a%2Bc%29%2F%28b%2Bd%29

Multiply through by LCD b(b+d)

a%28b%2Bd%29%3Cb%28a%2Bc%29
ab%2Bad%3Cab%2Bbc
Subtract ab from both sides
ad%3Cbc
Add cd to both sides
ad%2Bcd%3Cbc%2Bcd
Factor out d on left, c on right
d%28a%2Bc%29%3Cc%28b%2Bd%29
Divide both sides by d(b+d)
%28a%2Bc%29%2F%28b%2Bd%29%3Cc%2Fd

So a%2Fb%3C%28a%2Bc%29%2F%28b%2Bd%29%3Cc%2Fd

Case 2.  a%2Fb%3E%28a%2Bc%29%2F%28b%2Bd%29

Multiply through by LCD b(b+d)

a%28b%2Bd%29%3Eb%28a%2Bc%29
ab%2Bad%3Eab%2Bbc
Subtract ab from both sides
ad%3Ebc
Add cd to both sides
ad%2Bcd%3Ebc%2Bcd
Factor out d on left, c on right
d%28a%2Bc%29%3Ec%28b%2Bd%29
Divide both sides by d(b+d)
%28a%2Bc%29%2F%28b%2Bd%29%3Ec%2Fd

So a%2Fb%3E%28a%2Bc%29%2F%28b%2Bd%29%3Ec%2Fd

In either case, %28a%2Bc%29%2F%28b%2Bd%29 is between a%2Fb and c%2Fd

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a%2Fb%2Bc%2Fd is not between the numbers a%2Fb and c%2Fd

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

Edwin