Tìm x $\in \mathbb{N}$, biết:

$\begin{array}{l}a)\frac{11}{8}+\frac{-5}{8}<x<\frac{-3}{4}+\frac{15}{4}\\ \\ b)\frac{31}{23}+\frac{-7}{32}+\frac{-8}{23}<x<\frac{-8}{45}+5+\frac{-37}{45}\\ \\ c)\frac{-7}{2}+\frac{1}{10}+\frac{-8}{5}<x<\frac{1}{12}+\frac{-5}{6}+\frac{-5}{4}\end{array}$

Tính nhanh:

$\begin{array}{l}a)\frac{-3}{7}+\frac{5}{13}+\frac{3}{7};\begin{array}{cccc}& & & \end{array}\begin{array}{cc}& \end{array}b)\frac{-5}{21}+\frac{-2}{21}+\frac{8}{24};\\ \\ c)\frac{-5}{11}+\left(\frac{-6}{11}+2\right);\begin{array}{ccc}& & \end{array}\begin{array}{cc}& \end{array}d)\left(\frac{-1}{32}+\frac{1}{2}\right)+\frac{-15}{32}\end{array}$

Chứng minh rằng:

$\begin{array}{l}a)\frac{1}{101}+\frac{1}{102}+\frac{1}{103}+\mathrm{...}+\frac{1}{150}>\frac{1}{3}\\ \\ b)\frac{1}{201}+\frac{1}{202}+\frac{1}{203}+\mathrm{...}+\frac{1}{400}<1\\ \\ c)\frac{3}{5}<\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+\mathrm{...}+\frac{1}{60}<\frac{4}{5}\end{array}$

Tìm x $\in \mathbb{N}$, biết:

$\begin{array}{l}a)\frac{5}{6}+\frac{1}{6}\le x\le \frac{13}{4}+\frac{14}{8}\\ \\ b)\frac{-5}{6}+\frac{8}{3}+\frac{29}{-6}\le x\le \frac{-1}{2}+2+\frac{5}{2}\\ \\ c)\frac{79}{15}+\frac{7}{5}+\frac{-8}{3}\le x\le \frac{10}{3}+\frac{15}{4}+\frac{23}{12}\end{array}$

Tính các tổng sau một cách hợp lí:

$\begin{array}{l}a)\frac{4}{7} +\frac{4}{28}+\frac{9}{21}; \\ \\ b)\frac{4}{7}+\frac{-9}{21}+\frac{-1}{2}+\frac{7}{14} \\ \\ c)\frac{-27}{10}+\frac{1}{4}+\frac{13}{2}+\frac{5}{4}+\frac{-3}{10};\\ \\ d)\frac{8}{5}+\frac{-11}{4}+\frac{-7}{20}+\frac{7}{5}+\frac{-9}{4}\\ \\ e)\frac{7}{5}+\frac{-11}{3}+\frac{-1}{15}+\frac{11}{2}+\frac{1}{6}+(-2)\end{array}$

Tính nhanh:

$\begin{array}{l}a)\frac{2}{3}+\frac{5}{7}+\frac{-2}{3};\begin{array}{cccc}& & & \end{array}\begin{array}{cc}& \end{array}b)\frac{-1}{4}+\frac{5}{8}+\frac{-3}{8};\\ \\ c)\frac{-6}{13}+\left(\frac{-7}{13}+2\right);\begin{array}{ccc}& & \end{array}\begin{array}{cc}& \end{array}d)\left(\frac{-5}{27}+\frac{1}{3}\right)+\frac{-4}{27}\end{array}$

Tính các tổng sau một cách hợp lí

$\begin{array}{l}a)\frac{1}{3}+\frac{-2}{6}+\frac{3}{9};\\ \\ b)\frac{-4}{12}+\frac{18}{45}+\frac{-6}{9}+\frac{24}{30};\\ \\ c)\frac{7}{23}+\frac{-10}{18}+\frac{-4}{9}+\frac{16}{23}+\frac{-5}{8};\\ \\ d)\frac{1}{3}+\frac{-3}{4}+\frac{3}{5}+\frac{-1}{36}+\frac{1}{15}+\frac{-2}{9}\\ \\ e)\frac{-1}{2}+\frac{3}{5}+\frac{-1}{9}+\frac{-7}{18}+\frac{4}{35}+\frac{2}{7}\end{array}$