Division of a fraction - class-VIII

$\dfrac{(3\frac{2}{3})^2-(2\frac{1}{2})^2}{(4\frac{3}{2})^2-(3\frac{1}{3})^2} \div \dfrac{3\frac{2}{3}-2\frac{1}{2}}{4\frac{3}{2}-3\frac{1}{3}}$

$\left[\dfrac{170}{3} +\dfrac{6}{7}\right] \div \left[\dfrac{2}{7} \times \dfrac{11}{2}\right]$

Difference of  $\dfrac{1}{5}$  and $\dfrac{2}{7}$ is  

Given exp =$\displaystyle =\frac{2+4}{1\times 4}\div \frac{\frac{1}{2}\div \frac{1}{4}}{\frac{1}{2}+\frac{1}{4}}$
$\displaystyle =\frac{6}{2}\div\frac{\frac{1}{2}\times \frac{4}{1}}{\frac{3}{4}}=3\div \frac{2}{\frac{3}{4}} $
$\displaystyle =3\div \frac{8}{3}=3\times \frac{3}{8}=\frac{9}{8}=1\frac{1}{8}$

Given expression

(Using BODMAS)
$\displaystyle \frac{\frac{1}{3}\div \frac{1}{3}\times \frac{1}{3}}{\frac{1}{3}\div \frac{1}{3}of\frac{1}{3}}-\frac{1}{9}=\frac{\frac{1}{3}\times \frac{3}{1}\times \frac{1}{3}}{\frac{1}{3}\div \frac{1}{9}}-\frac{1}{9}$
$\displaystyle =\frac{1\times \frac{1}{3}}{\frac{1}{3}\times \frac{9}{1}}-\frac{1}{9}=\frac{1}{3\times 3}-\frac{1}{9}=\frac{1}{9}-\frac{1}{9}=0$

$Let\quad fraction\quad x\quad then\ x\times x\div \dfrac { 1 }{ x } =18\dfrac { 26 }{ 27 } =\dfrac { 512 }{ 27 }$

$\displaystyle \frac{2}{\displaystyle1+\frac{1}{\displaystyle1-\frac{1}{\displaystyle2}}}\times\frac{3}{\displaystyle\frac{5}{\displaystyle6}of\frac{3}{\displaystyle2}\div 1\frac{1}{\displaystyle4}}$
$=\displaystyle \frac{2}{\displaystyle1+\frac{1}{\displaystyle\frac{1}{\displaystyle2}}}\times\frac{3}{\displaystyle\frac{15}{\displaystyle12}\div \frac{5}{\displaystyle4}}$

Given exp.
$\displaystyle =\left [ \frac{13}{4}\div \left { \frac{5}{4}-\frac{1}{2}\left ( \frac{5}{2}-\frac{3, , ,2}{2} \right ) \right } \right ]\div \left ( \frac{1}{2}of\frac{13}{3} \right )$
$\displaystyle =\left [ \frac{13}{4}\div \left { \frac{5}{4}-\frac{1}{2}\left ( \frac{5}{2}-\frac{1}{12} \right ) \right } \right ]\div \frac{13}{6} $
$=\displaystyle \left [ \frac{13}{4}\div \left { \frac{5}{4}-\frac{1}{2}\times \frac{30-1}{12} \right } \right ]\div \frac{13}{6}$
$\displaystyle =\left [ \frac{13}{4}\div \left { \frac{5}{4}-\frac{29}{24} \right } \right ]\div \frac{13}{6}$
$\displaystyle =\left [ \frac{13}{4}\div \frac{30-29}{24} \right ]\div \frac{13}{6}$
$\displaystyle =\left ( \frac{13}{4} \div \frac{1}{24}\right )\div \frac{13}{4}=\frac{13}{4}\times 24\times \frac{6}{13}=36$

15 of $\displaystyle \frac{1}{5}, =, 15, \times, \displaystyle \frac{1}{5}, =, 3$

That is the role of the decimal point. The decimal point separates the place values that are whole values on the left from the place values that are fractional parts on the right.

$\dfrac{6\dfrac{1}{3}}{9}$ is complex fraction.

Given that 

$\dfrac{4}{5} \div 2 = \dfrac{\dfrac{4}{5}}{2} = \dfrac{4}{10}$

Every number has a reciprocal except 0 $(\dfrac{1}{0}$ is undefined$)$. The reciprocal is shown as $\dfrac{1}{x}$. 

sum of $\dfrac{65}{12}$  and $\dfrac{12}{7} $

Given, $ 8 \times \displaystyle \frac {\frac {5}{24}}{\frac {7}{12}} $

$ I=\cfrac{3}{4}\times \cfrac{6}{5}=\cfrac{9}{10}$

$\displaystyle 1\frac{1}{3}\div 1\frac{1}{2}=\frac{4}{3}\div \frac{3}{2}\div \frac{10}{9}$
$\displaystyle =\frac{4}{3}\times \frac{2}{3}\div \frac{10}{9}=\frac{8}{9}\times \frac{9}{10}=\frac{4}{5}$
$\displaystyle \therefore \frac{1}{5}$ should be added to $\displaystyle \frac{4}{5}$ to make it an integer

$\frac{a}{b}=\frac{\frac{7}{4}}{\frac{5}{3}}$
$=\frac{21}{20}$

$\frac{b}{a}=\frac{\frac{5}{3}}{\frac{7}{4}}$
$=\frac{20}{21}$

Now check all the options