Simplification

$${\text{we know that}}$$
$$\frac{5}{{1\frac{7}{8}{\text{of 1}}\frac{1}{3}}} \times \frac{{2\frac{1}{{10}}}}{{3\frac{1}{2}}}{\text{ of 1}}\frac{1}{4}$$
$$ ⇒ \frac{5}{{\frac{{15}}{8} \times {\text{ }}\frac{4}{3}}} \times {\text{ }}\frac{{21}}{{10}} \times \frac{2}{7} \times \frac{5}{4}$$
$$ ⇒ 2 \times \frac{3}{4}$$
$$ ⇒ \frac{3}{2}$$
$$ ⇒ 1\frac{1}{2}$$

$$\frac{{225}}{{836}} \times \frac{{152}}{{245}} \div 1\frac{{43}}{{77}}$$
$$ = \frac{{225}}{{836}} \times \frac{{152}}{{245}} \div \frac{{120}}{{77}}$$
$$ = \frac{{225}}{{836}} \times \frac{{152}}{{245}} \times \frac{{77}}{{120}}$$
$$ = \frac{3}{{28}}$$

$$\frac{3}{2} \times \frac{{11}}{5} \div \left( {\frac{{25}}{{44}} \times \frac{{11}}{5}} \right) \div \frac{{33}}{{15}}$$
$$ = \frac{3}{2} \times \frac{{11}}{5} \div \frac{5}{4} \div \frac{{33}}{{15}}$$
$$ = \frac{3}{2} \times \frac{{11}}{5} \times \frac{4}{5} \times \frac{{15}}{{33}}$$
$$ = \frac{6}{5}$$
$$ = 1\frac{1}{5}$$

$${\text{Given expression,}}$$
$$\frac{{81}}{8}\,of\,\frac{{12}}{{15}} \div \frac{{35}}{{36}}\,of\,\frac{{20}}{{49}}$$
$$ = \frac{{81}}{{10}} \div \frac{{25}}{{63}}$$
$$ = \frac{{81}}{{10}} \times \frac{{63}}{{25}}$$
$$ = \frac{{5103}}{{250}}$$
$$ = 20\frac{{103}}{{250}}\,$$

$${\text{we know that,}}$$
$$\sqrt {\frac{{{{\left( {0.03} \right)}^2} + {{\left( {0.21} \right)}^2} + {{\left( {0.065} \right)}^2}}}{{{{\left( {0.003} \right)}^2} + {{\left( {0.021} \right)}^2} + {{\left( {0.0065} \right)}^2}}}} $$
$$ = \sqrt {\frac{{0.0009 + 0.0441 + 0.004225}}{{0.000009 + 0.000441 + 0.00004225}}} $$
$$ = \sqrt {\frac{{0.049225}}{{0.00049225}}} $$
$$ = \sqrt {100} $$
$$ = 10$$

we know that,
$$\frac{{25 - 5\left[ {2 + 3\left\{ {2 - 2\left( {5 - 3} \right) + 5} \right\} - 10} \right]}}{4}$$
$$ = \frac{{25 - 5\left[ {2 + 3\left\{ {2 - 2 × 2 + 5} \right\} - 10} \right]}}{4}$$
$$ = \frac{{25 - 5\left[ {2 + 3\left\{ {2 - 4 + 5} \right\} - 10} \right]}}{4}$$
$$ = \frac{{25 - 5\left[ {2 + 3 × 3 - 10} \right]}}{4}$$
$$ = \frac{{25 - 5\left[ {11 - 10} \right]}}{4}$$
$$ = 25 - \frac{5}{4}$$
$$ = \frac{{100 - 5}}{4}$$
$$ = \frac{{95}}{4}$$
$$ = 23.75$$

$${\text{Given Expression,}}$$
$$ {\frac{3}{7} \times 455 + \frac{5}{8} \times 456} $$
$$ = \left( {3 \times 65 + 5 \times 57} \right)$$
$$ = 195 + 285$$
$$ = 480$$

$$ {\frac{{41}}{6} \times \frac{{16}}{3} + \frac{{53}}{3} \times \frac{9}{2}} $$
$$ = {\frac{{328}}{9} + \frac{{159}}{2}} $$
$$ = \frac{{656 + 1431}}{{18}}$$
$$ = \frac{{2087}}{{18}}$$
$$ = 115\frac{{17}}{{18}}$$

$${\text{Given expression}}$$
$$\frac{{24}}{5} \div \frac{{32}}{5}$$
$$ = \frac{{24}}{5} \times \frac{5}{{32}}$$
$$ = \frac{3}{4}$$

$$\eqalign{ &×= \frac{1}{{2 + \frac{1}{2}}} = \frac{2}{5}$$
$$ ⇒ \frac{1}{x} = \frac{5}{2}$$