Question 474077: if (a)/(b)<(c)/(d) , show that (a)/(b)<(a+c)/(b+d)<(c)/(d)
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Here I will assume a,b,c,d > 0 (otherwise we run into problems when b = 0, or when b+d = 0, etc.).
The statement  , is true if and only if  < b(a+c) \Rightarrow ad < bc) . However, this statement is already true (simply cross-multiply the given inequality), so the first part of the inequality is true.
Similarly, the statement  is true if and only if  < c(b+d) \Rightarrow da < cb) , which is the same statement as in the first inequality, so this part of the inequality also holds, and we are done.
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