\(A=\dfrac{10^{11}+1}{10^{12}-1}\)
\(\Rightarrow10A=\dfrac{10^{11}+1}{10^{12}-1}.10\)
\(\Rightarrow10A=\dfrac{10\left(10^{11}+1\right)}{10^{12}-1}\)
\(\Rightarrow10A=\dfrac{10^{12}-10}{10^{12}-1}\)
\(B=\dfrac{10^{10}+1}{10^{11}+1}\)
\(\Rightarrow10B=\dfrac{10^{10}+1}{10^{11}+1}.10\)
\(\Rightarrow10B=\dfrac{\left(10^{10}+1\right).10}{10^{11}+1}\)
\(\Rightarrow10B=\dfrac{10^{11}+10}{10^{11}+1}\)
Ta thấy:
\(10^{12}-1>10^{12}-10>0\Rightarrow10A< 1\)
\(0< 10^{11}+1< 10^{11}+10\Rightarrow10B>1\)
Mà \(10A< 1;10B>1\)
\(\Rightarrow B>A\).
Bạn tham khảo cách giải này ạ:
Nếu có 1 phân số \(\dfrac{a}{b}\) < 1 thì a/b < a+n/b+n.
Tương tự ta có: A < (1011 -1)+11/(1012-1)+10
A < 1011+10/1012+10
A < 10(1010+1)/10(1011+1)
A < 10(1010+1)/10(1011+1)
A < 1010+1/1011+1
Vậy A< B ( đcpcm )