Bài 1: 

a. $\frac{x}{2}=\frac{3,6}{1,2}=3$

$x=3.2=6$

b. 

$\frac{8}{2x+1}=\frac{4}{3}$

$2x+1=\frac{8.3}{4}=6$

$2x=6-1=5$

$x=\frac{5}{2}$

c. $\frac{x}{4}=\frac{9}{x}$

$x^2=9.4=36=6^2=(-6)^2$

$\Rightarrow x=\pm 6$

d.

$\frac{x+1}{2}=\frac{32}{x+1}$

$(x+1)^2=32.2=64=8^2=(-8)^2$

$\Rightarrow x+1=8$ hoặc $x+1=-8$

$\Rightarrow x=7$ hoặc $x=-9$

Lời giải:

Gọi diện tích cày được của 3 máy lần lượt là $a,b,c$. Theo bài ra ta có:

$a+b+c=106$

$\frac{a}{3}=\frac{b}{5}; \frac{a}{4}=\frac{c}{7}$

$\Rightarrow \frac{a}{12}=\frac{b}{20}=\frac{c}{21}$

Áp dụng tính chất dãy tỉ số bằng nhau:

$\frac{a}{12}=\frac{b}{20}=\frac{c}{21}=\frac{a+b+c}{12+20+21}=\frac{106}{53}=2$

$\Rightarrow a=12.2=24; b=20.2=40; c=21.2=42$ (ha)

Lời giải:

Gọi khối lượng 3 phần gạo lần lượt là $a,b,c$ (tạ).

Theo bài ra ta có: 

$a+b+c=780$

$2a=3b=4c$

$\Rightarrow \frac{a}{\frac{1}{2}}=\frac{b}{\frac{1}{3}}=\frac{c}{\frac{1}{4}}=\frac{a+b+c}{\frac{1}{2}+\frac{1}{3}+\frac{1}{4}}=\frac{780}{\frac{13}{12}}=720$

$\Rightarrow a=720:2=360; bb=720:3=240; c=720:4=180$ (tạ)

Ta có :

\(3x=4y=5z\)

\(\Rightarrow\dfrac{x}{\dfrac{1}{3}}=\dfrac{2y}{\dfrac{1}{2}}=\dfrac{3z}{\dfrac{3}{5}}=\dfrac{x+2y-3z}{\dfrac{1}{3}+\dfrac{1}{2}-\dfrac{3}{5}}=\dfrac{28}{\dfrac{7}{30}}=\dfrac{28.30}{7}=120\) \(\left(x+2y-3z=28\right)\)

\(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{1}{3}.120=40\\y=\dfrac{1}{4}.120=30\\z=\dfrac{1}{5}.120=24\end{matrix}\right.\)

\(A=\dfrac{4}{3}-\left(\dfrac{5}{3}-\dfrac{3}{4}+\dfrac{5}{18}\right)+\left(\dfrac{9}{4}-\dfrac{8}{3}\right)\\ A=\dfrac{4}{3}-\dfrac{5}{3}+\dfrac{3}{4}-\dfrac{5}{18}+\dfrac{9}{4}-\dfrac{8}{3}\\ A=\left(\dfrac{4}{3}-\dfrac{5}{3}-\dfrac{8}{3}\right)+\left(\dfrac{3}{4}+\dfrac{9}{4}\right)-\dfrac{5}{18}\\ A=-3+3-\dfrac{5}{18}\\ A=-\dfrac{5}{18.}\)

Lời giải:
Ta thấy:

$\widehat{yBA}+\widehat{BAx}=124^0+56^0=180^0$. Mà 2 góc này ở vị trí trong cùng phía nên $By\parallel Ax$ (đpcm)

\(\dfrac{-4}{7}:x=\dfrac{-2}{5}\)

\(\Rightarrow x=\dfrac{-4}{7}:\dfrac{-2}{5}\)

\(\Rightarrow x=\dfrac{10}{7}\)

\(\dfrac{-4}{7}:x=\dfrac{-2}{5}\)

\(x=\dfrac{-4}{7}.\dfrac{-5}{2}\)

\(x=\dfrac{10}{7}\)

\(\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{20}}{\dfrac{19}{1}+\dfrac{18}{2}+\dfrac{17}{3}+....+\dfrac{1}{19}}\)

\(=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{20}}{1+\left(\dfrac{18}{2}+1\right)+\left(\dfrac{17}{3}+1\right)+\left(\dfrac{1}{19}+1\right)}\)

\(=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{20}}{1+\dfrac{20}{2}+\dfrac{20}{3}+...+\dfrac{20}{19}}\)

\(=\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{20}}{20.\left(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{19}+\dfrac{1}{20}\right)}\)

\(=\dfrac{1}{20}\)