Ta có: \(10C=\dfrac{10^{12}-10}{10^{12}-1}=1-\dfrac{9}{10^{12}-1}\)
\(10D=\dfrac{10^{11}+10}{10^{11}+1}=1+\dfrac{9}{10^{11}+1}\)
\(\dfrac{9}{10^{12}-1}< \dfrac{9}{10^{12}+1}\Rightarrow1-\dfrac{9}{10^{12}-1}< 1+\dfrac{9}{10^{11}+1}\Rightarrow10A< 10B\)
\(\Rightarrow A< B\)
Vậy A < B
10C=\(\dfrac{10.\left(10^{11}-1\right)}{10^{12}-1}\)=\(\dfrac{10^{12}-10}{10^{12}-1}\)=\(\dfrac{10^{12}-1-9}{1012-1}\)=1-\(\dfrac{9}{10^{12}-1}\)<1 (1)
10D=\(\dfrac{10.\left(10^{10}+1\right)}{10^{11}+1}\)=\(\dfrac{10^{11}+10}{10^{11}+1}\)=\(\dfrac{10^{11}+1+9}{10^{11}+1}\)=1+\(\dfrac{9}{10^{11}+1}\)>1 (2)
(1),(2)=> 1-\(\dfrac{9}{10^{12}-1}\)<1+\(\dfrac{9}{10^{11}+1}\)
hay 10C<10D
=>C<D
\(C=\dfrac{10^{11}-1}{10^{12}-1}>1\\ \Rightarrow C=\dfrac{10^{11}-1}{10^{12}-1}< \dfrac{10^{11}-1+11}{10^{12}-1+11}=\dfrac{10^{11}+10}{10^{12}+10}=\dfrac{10\cdot\left(10^{10}+1\right)}{10\cdot\left(10^{11}+1\right)}=\dfrac{10^{10}+1}{10^{11}+1}=D\)
Vậy \(C< D\left(đpcm\right)\)
