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If $\dfrac{c}{d} > \dfrac{a}{b}$, then $\dfrac{c}{d} > \dfrac{a+c}{b+d} > \dfrac{a}{b}$

This is intuitively clear to me but I don't know how to prove this. I tried rewriting all the fractions to the same for but I still couldn't figure it out because I can't really say anything about the size of $a,b,c,d$ on their own.

What tactic should one use for these types of problems?

Loerades
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1 Answers1

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The tactic is to cross multiply. From $\dfrac{c}{d}>\dfrac{a}{b}$ you know that $bc>ad$.

For $\dfrac{c}{d}>\dfrac{a+c}{b+d}$ you need to show that $bc+cd>ad+cd$, which reduces to $bc>ad$. But you know this is true.

For $\dfrac{a+c}{b+d}>\dfrac{a}{b}$ you can apply a similar argument.

pi66
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