let a,b,c and d be positive integers such that a/b < c/d. Show that a/b < a+c/b+d < c/d

Question

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

Answer 1

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

Answer 2

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