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

$$\eqalign{ & {\text{According to the question,}} \cr & \sqrt {\frac{{{{\left( {0.03} \right)}^2} + {{\left( {0.21} \right)}^2} + {{\left( {0.065} \right)}^2}}}{{{{\left( {0.003} \right)}^2} + {{\left( {0.021} \right)}^2} + {{\left( {0.0065} \right)}^2}}}} \cr & = \sqrt {\frac{{0.0009 + 0.0441 + 0.004225}}{{0.000009 + 0.000441 + 0.00004225}}} \cr & = \sqrt {\frac{{0.049225}}{{0.00049225}}} \cr & = \sqrt {100} \cr & = 10 \cr} $$

$$\eqalign{ & {\text{According to the question,}} \cr & {\text{8}}\frac{1}{2} - \left[ {3\frac{1}{4} + \left\{ {1\frac{1}{4} - \frac{1}{2}\left( {1\frac{1}{2} - \frac{1}{3} - \frac{1}{6}} \right)} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \left\{ {\frac{5}{4} - \frac{1}{2}\left( {\frac{3}{2} - \frac{1}{3} - \frac{1}{6}} \right)} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \left\{ {\frac{5}{4} - \frac{1}{2}\left( {\frac{{9 - 2 - 1}}{6}} \right)} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \left\{ {\frac{5}{4} - \frac{1}{2} \times 1} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \left\{ {\frac{{5 - 2}}{4}} \right\}} \right] \cr & \Rightarrow \frac{{17}}{2} - \left[ {\frac{{13}}{4} + \frac{3}{4}} \right] \cr & \Rightarrow \frac{{17}}{2} - \frac{{16}}{4} \cr & \Rightarrow \frac{{34 - 16}}{4} \cr & \Rightarrow \frac{9}{2} \cr & \Rightarrow 4\frac{1}{2} \cr} $$

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

( 2² + 3² + 4² + .......... + 20² )
= 2² ( 1² + 2² + 3² + .......... + 10² )
= 4 * 385
= 1540

$$\eqalign{ & {\text{According to the question,}} \cr & \frac{{\root 3 \of 8 }}{{\sqrt {16} }} \div \sqrt {\frac{{100}}{{49}}} \times \root 3 \of {125} \, \cr & \Rightarrow \frac{2}{4} \div \frac{{10}}{7} \times 5 \cr & \Rightarrow \frac{2}{4} \times \frac{7}{{10}} \times 5 \cr & \Rightarrow \frac{7}{4} \cr & \Rightarrow 1\frac{3}{4} \cr} $$

$$\eqalign{ & {\text{According to the question}} \cr & {\text{ }}\frac{{{{\left( {3.4567} \right)}^2} - {{\left( {3.4533} \right)}^2}}}{{0.0034}} \cr & \left[ {\because {a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)} \right] \cr & \Rightarrow \frac{{\left( {3.4567 + 3.4533} \right)\left( {3.4567 - 3.4533} \right)}}{{0.0034}} \cr & \Rightarrow \frac{{6.91 \times 0.0034}}{{0.0034}} \cr & \Rightarrow 6.91 \cr} $$

$$\eqalign{ & {\text{According to the question,}} \cr & \frac{{{{\left( {0.75} \right)}^3}}}{{1 - 0.75}} + \left[ {0.75 + {{\left( {0.75} \right)}^2} + 1} \right] \cr & \Rightarrow \frac{{{{\left( {0.75} \right)}^3}}}{{1 - 0.75}} + \frac{{{1^3} - {{\left( {0.75} \right)}^3}}}{{1 - 0.75}}\, \cr & \left[ {\because {a^3} - {b^3} = \left( {a - b} \right)\left( {{a^2} + {b^2} + ab} \right)} \right] \cr & \Rightarrow \frac{{{{\left( {0.75} \right)}^3} + 1 - {{\left( {0.75} \right)}^3}}}{{0.25}} \cr & \Rightarrow \frac{1}{{0.25}} \cr & \Rightarrow 4 \cr} $$

$$\eqalign{ & x = \frac{1}{{2 + \frac{1}{2}}} = \frac{2}{5} \cr & \Rightarrow \frac{1}{x} = \frac{5}{2} \cr} $$

$$\eqalign{ & {\text{According to the question,}} \cr & \frac{{\frac{1}{2} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6}}}{{\frac{2}{5} - \frac{5}{9} + \frac{3}{5} - \frac{7}{{18}}}} \cr & \Rightarrow \frac{{\frac{{30 - 15 + 12 - 10}}{{60}}}}{{\frac{{36 - 50 + 54 - 35}}{{90}}}} \cr & \Rightarrow \frac{{17}}{{60}} \times \frac{{90}}{5} \cr & \Rightarrow 5\frac{1}{{10}} \cr} $$

$$\eqalign{ & {\text{According to the question,}} \cr & \frac{{\sqrt {72} - \sqrt {18} }}{{\sqrt {12} }} \cr & \Rightarrow \sqrt {\frac{{72}}{{12}}} - \sqrt {\frac{{18}}{{12}}} \cr & \Rightarrow \sqrt 6 - \sqrt {\frac{6}{4}} \cr & \Rightarrow \frac{{\sqrt 6 }}{2} \cr} $$

$$\eqalign{ & {\text{According to the question,}} \cr & \sqrt {4 + \sqrt {44 + \sqrt {10000} } } \cr & \Rightarrow \sqrt {4 + \sqrt {44 + 100} } \cr & \Rightarrow \sqrt {4 + \sqrt {144} } \cr & \Rightarrow \sqrt {4 + 12} \cr & \Rightarrow \sqrt {16} \cr & \Rightarrow 4 \cr} $$

$$\eqalign{ & {\text{Let,}} \cr & 45 - \left[ {28 - \left\{ {37 - \left( {15 - x} \right)} \right\}} \right] = 58 \cr & {\text{then,}} \cr & {\text{45}} - \left[ {28 - \left\{ {37 - 15 + x} \right\}} \right] = 58 \cr & \Rightarrow 45 - \left[ {28 - \left\{ {22 + x} \right\}} \right] = 58 \cr & \Rightarrow 45 - \left[ {28 - 22 - x} \right] = 58 \cr & \Rightarrow 45 - \left[ {6 - x} \right] = 58 \cr & \Rightarrow 45 - 6 + x = 58 \cr & \Rightarrow 39 + x = 58 \cr & \Rightarrow x = 58 - 39 \cr & \Rightarrow x = 19 \cr} $$

Let the missing number is x.
then, $$\frac{{354750}}{{4096 + x}}$$   = 55
⇒ 55( 4096 + x ) = 354750
⇒ 55x = 354750 - 225280
⇒ 55x = 129470
⇒ x = 2354

$$\eqalign{ & {\text{According to question}} \cr & \frac{{\frac{5}{3} \times \frac{7}{{51}}{\text{of}}\frac{17}{5} - \frac{1}{3}}}{{\frac{2}{9} \times \frac{5}{7}{\text{of}}\frac{{28}}{5} - \frac{2}{3}}}\, \cr & \Rightarrow \frac{{\frac{5}{3} \times \frac{7}{{51}} \times \frac{17}{5} - \frac{1}{3}}}{{\frac{2}{9} \times \frac{5}{7} \times \frac{{28}}{5} - \frac{2}{3}}}\, \cr & \Rightarrow \frac{{\frac{5}{3} \times \frac{7}{{15}} - \frac{1}{3}}}{{\frac{2}{9} \times 4 - \frac{2}{3}}} \cr & \, \Rightarrow \frac{{\frac{7}{9} - \frac{1}{3}}}{{\frac{8}{9} - \frac{2}{3}}} \cr & \, \Rightarrow \frac{{\frac{{7 - 3}}{9}}}{{\frac{{8 - 6}}{9}}} \cr & \Rightarrow \frac{4}{2} \cr & \Rightarrow 2 \cr} $$

Suppose their paths cross after x minutes
Then,
11 + 57x = 51 - 63x
⇒ 57x + 63x = 51 - 11
⇒ 120x = 40
⇒ x = $$\frac{1}{3}$$
Number of floors covered by David in $$\frac{1}{3}$$ min.
$$\eqalign{ & = {\frac{1}{3} \times 57} \cr & = 19 \cr} $$
So, their paths cross at (11 + 19) i.e., 30^th floor

$$\eqalign{ & {\text{According to question,}} \cr & \frac{{\frac{5}{3} \times \frac{7}{{51}}{\text{ of }}\frac{17}{5} - \frac{1}{3}}}{{\frac{2}{9} \times \frac{5}{7}{\text{ of }}\frac{{28}}{5} - \frac{2}{3}}}\, \cr & \Rightarrow \frac{{\frac{5}{3} \times \frac{7}{{51}} \times \frac{17}{5} - \frac{1}{3}}}{{\frac{2}{9} \times \frac{5}{7} \times \frac{{28}}{5} - \frac{2}{3}}}\, \cr & \Rightarrow \frac{{\frac{5}{3} \times \frac{7}{{15}} - \frac{1}{3}}}{{\frac{2}{9} \times 4 - \frac{2}{3}}} \cr & \, \Rightarrow \frac{{\frac{7}{9} - \frac{1}{3}}}{{\frac{8}{9} - \frac{2}{3}}} \cr & \, \Rightarrow \frac{{\frac{{7 - 3}}{9}}}{{\frac{{8 - 6}}{9}}} \cr & \Rightarrow \frac{4}{2} \cr & \Rightarrow 2 \cr} $$

1 - [5 - {2 + (- 5 + 6 - 2) 2}]
= 1 - [5 - {2 + (- 1) 2}]
= 1 - [5 - {2 - 2}]
= 1 - [5 - 0]
= 1 - 5
= -4

$$\eqalign{ & {\text{Given expression,}} \cr & \frac{7}{4} - \frac{6}{5} + \frac{{13}}{8} \cr & = \frac{{70 - 48 + 65}}{{40}} \cr & = \frac{{87}}{{40}} \cr & = 2\frac{7}{{40}} \cr} $$

$$\eqalign{ & \frac{{113 \times 4 - x \times 2}}{{13 \times 9 - 5 \times 7}} = 5 \cr & \Rightarrow \frac{{452 - 2x}}{{117 - 35}} = 5 \cr & \Rightarrow \frac{{452 - 2x}}{{82}} = 5 \cr & \Rightarrow 452 - 2x = 410 \cr & \Rightarrow 2x = 452 - 410 = 42 \cr & \Rightarrow x = 21 \cr} $$

$$\eqalign{ & {\text{Let,}} \cr & {\text{853 + }}x \div 17 = 1000 \cr & {\text{Then,}} \cr & {\text{853 + }}\frac{x}{{17}} = 1000 \cr & \Leftrightarrow \frac{x}{{17}} = 1000 - 853 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 147 \cr & \Leftrightarrow x = 147 \times 17 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 2499 \cr} $$

$$\eqalign{ & {\text{Let,}} \cr & \left( {x - 968} \right) \div 79 \times 4 = 512 \cr & {\text{Then,}} \cr & \frac{{x - 968}}{{79}} \times 4 = 512 \cr & \Rightarrow x - 968 = \frac{{512 \times 79}}{4} \cr & \Rightarrow x - 968 = 10112 \cr & \Rightarrow x = 10112 + 968 \cr & \,\,\,\,\,\,\,\,\,\,\,\,\, = 11080 \cr} $$

According to the question,
$$\eqalign{ & \sqrt {1.3} + \sqrt {1300} + \sqrt {0.013} \cr & \Rightarrow \sqrt {\frac{{130}}{{100}} }+ \sqrt {1300} + \sqrt {\frac{{130}}{{10000}}} \cr & \Rightarrow \frac{{11.4}}{{10}} + 36.05 + \frac{{11.4}}{{100}} \cr & \Rightarrow 1.14 + 36.05 + 0.114 \cr & \Rightarrow 37.304 \cr} $$

$$\eqalign{ & 1888 \div 32 \div 8 \cr & = \frac{{1888}}{{32}} \div 8 \cr & = 59 \div 8 \cr & = \frac{{59}}{8} \cr & = 7.375 \cr} $$

$$\eqalign{ & \frac{{4848}}{{24}} \times 11 - 222 \cr & = 202 \times 11 - 222 \cr & = 2222 - 222 \cr & = 2000 \cr} $$

$$\eqalign{ & 1 + \frac{1}{{1 + \frac{2}{{2 + \frac{3}{{1 + \frac{4}{5}}}}}}} \cr & = 1 + \frac{1}{{1 + \frac{2}{{2 + \frac{3}{{\frac{9}{5}}}}}}} \cr & = 1 + \frac{1}{{1 + \frac{2}{{2 + \frac{5}{3}}}}} \cr & = 1 + \frac{1}{{1 + \frac{2}{{\frac{{11}}{3}}}}} \cr & = 1 + \frac{1}{{1 + \frac{6}{{11}}}} \cr & = 1 + \frac{1}{{\frac{{17}}{{11}}}} \cr & = 1 + \frac{{11}}{{17}} \cr & = 1\frac{{11}}{{17}} \cr} $$

$$\eqalign{ & 1 + \frac{2}{{1 + \frac{3}{{1 + \frac{4}{5}}}}} \cr & = 1 + \frac{2}{{1 + \frac{3}{{\frac{9}{5}}}}} \cr & = 1 + \frac{2}{{1 + \frac{5}{3}}} \cr & = 1 + \frac{2}{{\frac{8}{3}}} \cr & = 1 + \frac{3}{4} \cr & = \frac{7}{4} \cr} $$

Suppose the man works overtime for x hours.
Now, working hours in 4 weeks = (5 * 8 * 4) = 160.
∴ 160 * 2.40 + x * 3.20 = 432
⇒ 3.20x = 432 - 384 = 48
⇒ x = 15.
Hence, total hours of work = (160 + 15) = 175.

$$\eqalign{ & {\text{Let}}\,{\text{the}}\,{\text{total}}\,{\text{number}}\,{\text{of}}\,{\text{shots}}\,{\text{be}}\,x. \cr & {\text{Then,}}\,{\text{Shots}}\,{\text{fired}}\,{\text{by}}\,{\text{A}} = \frac{5}{8}x \cr & {\text{Shots}}\,{\text{fired}}\,{\text{by}}\,{\text{B}} = \frac{3}{8}x \cr & {\text{Killing}}\,{\text{shots}}\,{\text{by}}\,{\text{A}} \cr & = \frac{1}{3}{\text{of}}\frac{5}{8}x = \frac{5}{{24}}x \cr & {\text{Shots}}\,{\text{missed}}\,{\text{by}}\,{\text{B}} \cr & = \frac{1}{2}{\text{of}}\,\frac{3}{8}x = \frac{3}{{16}}x \cr & \therefore \frac{{3x}}{{16}} = 27\,{\text{or}}\,x = {\frac{{27 \times 16}}{3}} = 144 \cr & {\text{Birds}}\,{\text{killed}}\,{\text{by}}\,{\text{A}} \cr & = \frac{{5x}}{{24}} = {\frac{5}{{24}} \times 144} = 30 \cr} $$

$$\eqalign{ & {\text{According to question,}} \cr & \Rightarrow \sqrt {\frac{8}{3}} \cr & \Rightarrow \sqrt {\frac{{8 \times 3}}{{3 \times 3}}} \cr & \Rightarrow \sqrt {\frac{{24}}{9}} \cr & \Rightarrow \frac{{4.898}}{3} \cr & \Rightarrow 1.633 \cr} $$