Lời giải:

$B=\frac{10^{11}+10}{10^{12}+10}$

Đặt $10^{11}-1=a; 10^{12}-1=b$ thì $0< a< b$. Khi đó:

$A-B=\frac{a}{b}-\frac{a+11}{b+11}=\frac{11(a-b)}{b(b+11)}<0$

$\Rightarrow A< B$

Dễ vãi

Lời giải:

a) Xét hiệu \(\frac{a+n}{b+n}-\frac{a}{b}=\frac{(a+n).b-a(b+n)}{b(b+n)}=\frac{n(b-a)}{b(b+n)}\)

Nếu $b>a$ thì $\frac{a+n}{b+n}-\frac{a}{b}>0\Rightarrow \frac{a+n}{b+n}>\frac{a}{b}$

Nếu $b<a$ thì $\frac{a+n}{b+n}-\frac{a}{b}<0\Rightarrow \frac{a+n}{b+n}<\frac{a}{b}$

Nếu $b=a$ thì $\frac{a+n}{b+n}-\frac{a}{b}=0\Rightarrow \frac{a+n}{b+n}=\frac{a}{b}$

b) Rõ ràng $10^{11}-1< 10^{12}-1$. 

Đặt $10^{11}-1=a; 10^{12}-1=b; 11=n$ thì: $a< b$; $A=\frac{a}{b}$ và $B=\frac{10^{11}+10}{10^{12}+10}=\frac{a+n}{b+n}$

Áp dụng kết quả phần a:

$b>a\Rightarrow \frac{a+n}{b+n}>\frac{a}{b}$ hay $B>A$

Lời giải:
a.

\(\frac{n+1}{n+2}=\frac{n+1}{n+2}+1-1=\frac{2n+3}{n+2}-1\)

\(> \frac{2n+3}{n+3}-1=\frac{(n+3)+n}{n+3}-1=\frac{n}{n+3}\)

b.

\(10A=\frac{10^{12}-10}{10^{12}-1}=\frac{(10^{12}-1)-9}{10^{12}-1}=1-\frac{9}{10^{12}-1}<1\)

\(10B=\frac{10^{11}+10}{10^{11}+1}=\frac{(10^{11}+1)+9}{10^{11}+1}=1+\frac{9}{10^{11}+1}>1\)

$\Rightarrow 10A< 10B\Rightarrow A< B$

a. 19/10 > 10/11

b. 11/10 = 12/11

c. 9/10 = 10/11

a)\(\dfrac{19}{10}>\dfrac{10}{11}\)

b)\(\dfrac{11}{10}=\dfrac{12}{11}\)

c)\(\dfrac{9}{10}< \dfrac{10}{11}\)

a) \(< \)

b) \(>\)

c) \(< \)

d) \(>\)

e) \(< \)

g) \(>\)

h) \(>\)

k) \(>\)

ta có: \(A=\dfrac{10.10^{10}-1}{10.10^{11}-1}=\dfrac{10^{10}-1}{10^{11}-1}\)

so sánh: \(A=\dfrac{10^{10}-1}{10^{11}-1}\)và \(B=\dfrac{10^{10}+1}{10^{11}+1}\)

\(\Rightarrow A< B\)

10A=\(\dfrac{10^{12}-10}{10^{12}-1}=1-\dfrac{9}{10^{12}-1}\)

10B =\(\dfrac{10^{11}+10}{10^{11}+1}=1+\dfrac{9}{10^{11}+1}\)

\(\dfrac{9}{10^{12}-1}< \dfrac{9}{10^{12}+1}\) =>\(1-\dfrac{9}{10^{12}-1}< 1+\dfrac{9}{10^{11}+1}\)

=> 10A < 10B

=> A<B

Vậy A < B

\(A=\dfrac{10^{11}-1}{10^{12}-1};B=\dfrac{10^{10}+1}{10^{11}+1}\)

\(A=\dfrac{10.10^{10}-1}{10.10^{11}-1}\)và \(B=\dfrac{10^{10}+1}{10^{11}+1}\)

Vậy A<B

\(A=\dfrac{2021^{10}-2021+2020}{2021^9-1}\\ =\dfrac{2021\left(2021^9-1\right)+2020}{2021^9-1}\\ =2021+\dfrac{2020}{2021^9-1}\\ B=\dfrac{2021^{11}-1}{2021^{10}-1}=2021+\dfrac{2020}{2021^{10}-1}\)

Ta có:

 \(2021^9-1< 2021^{10}-1\\ \Rightarrow\dfrac{2020}{2021^9-1}>\dfrac{2020}{2021^{10}-1}\)

Do đó A > B.

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 \(C< D\left(đpcm\right)\)