Question

abcd都是正数，且a／b<c／b，求证a／b<a+c／b+d<c／d

Answer 1

因为a,b,c,d是正数,a/b<c/d,即有bc>ad,所以bc+ab>ad+ab,提公因式为b(a+c)>a(b+d),
两边除b(b+d)为,(a+c)/(b+d)>a/b①,
同理两边同时加上cd,即有bc+cd>ad+cd,提公因式为,c(b+d)>d(a+c)两边除d(b+d)为,
c/d>(a+c)/(b+d)②,
⑵综合①②得证a/b <(a+c)/(b+d)< c/d

Answer 2

a/b<c/d,b>0,d>0, 故ad<bc
所以ab+ad<ab+bc, ad+cd<bc+cd
即a(b+d)<b(a+c), d(a+c)<c(b+d)
b+d>0,
故a/b<(a+c)/(b+d), (a+c)/(b+d)<c/d

Answer 3

由于abcd都是正数,
a／b<c／d → ad＜bc → ad+ab＜bc+ab → a(a+d)＜b(a+c) → a／b<(a+c)／(b+d)
a／b<c／d → ad＜bc → ad+dc＜bc+dc → d(a+c)＜c(b+d) → (a+c)／(b+d)<c／d