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Given that ${a\over b} < {c\over d}$ show that $${a\over b} < {a+c\over b+d} < {c\over d}$$

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closed as off-topic by Theoretical Economist, GNUSupporter 8964民主女神 地下教會, José Carlos Santos, YuiTo Cheng, Arnaud D. May 14 at 8:22

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$${a+c\over b+d} < {c\over d}\iff d(a+c)<c(b+d)\iff da<cb\iff {a\over b} < {c\over d} $$

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The second part $$\frac{a+c}{b+d}-\frac{a}{b}=\frac{bc-ad}{(b+d)b}>0$$ if $$bc>ad$$ if $$\frac{c}{d}>\frac{a}{b}$$

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