Question

Cho các số hữu tỉ \(\dfrac{a}{b}\)và\(\dfrac{c}{d}\) với mẫu dương, trong đó \(\dfrac{a}{b}\)<\(\dfrac{c}{d}\). Chứng minh rằng:

A) ad<bc

B) \(\dfrac{a}{b}\)<\(\dfrac{a+c}{b+d}\)< \(\dfrac{c}{d}\)

Answer 1

a) \(\dfrac{a}{b}< \dfrac{c}{d}\Rightarrow ad< bc\)

b) Tham khảo:https://olm.vn/hoi-dap/tim-kiem?q=cho+c%C3%A1c+s%E1%BB%91+h%E1%BB%AFu+t%E1%BB%89+a/b+v%C3%A0+c/d+v%E1%BB%9Bi+m%E1%BA%ABu+d%C6%B0%C6%A1ng+,+trong+%C4%91%C3%B3+a/b+%3Cc/d+.+c/m+r%E1%BA%B1ng+a)+a.d+%3Cb.c+b)+a/b+%3C+(a+c)/(b+d)%3Cc/d+&id=174343

Answer 2

a) Ta có: \(\left\{{}\begin{matrix}\dfrac{a}{b}< \dfrac{c}{d}\\b,d>0\end{matrix}\right.\)

\(\Rightarrow\dfrac{a}{b}.bd< \dfrac{c}{d}.bd\Rightarrow ad< bc\)

b) Ta có: \(ad< bc\Rightarrow ad+ab< bc+ab\)

\(\Rightarrow a\left(b+d\right)< b\left(a+c\right)\Rightarrow\dfrac{a}{b}< \dfrac{a+c}{b+d}\left(1\right)\)(do \(b,d>0\))

\(bc>ad\Rightarrow bc+cd>ad+cd\)

\(\Rightarrow c\left(b+d\right)>d\left(a+c\right)\Rightarrow\dfrac{c}{d}>\dfrac{a+c}{b+d}\left(2\right)\)

\(\left(1\right),\left(2\right)\Rightarrow\dfrac{a}{b}< \dfrac{a+c}{b+d}< \dfrac{c}{d}\)