Fractions — Exercise 6(D)

Simplify:

$\dfrac{3}{7} \times \dfrac{2}{5}$

Answer

Solving,

$\Rightarrow \dfrac{3}{7} \times \dfrac{2}{5} \\[1em] = \dfrac{3 \times 2}{7 \times 5} \\[1em] = \dfrac{6}{35}$

Hence, $\dfrac{3}{7} \times \dfrac{2}{5} = \dfrac{6}{35}$.

Simplify:

$\dfrac{4}{9} \times \dfrac{3}{5}$

Answer

$\dfrac{4}{9} \times \dfrac{3}{5}$

$\Rightarrow \dfrac{4}{9} \times \dfrac{3}{5} = \dfrac{4 \times 3}{9 \times 5} \\[1em] = \dfrac{12}{45} \\[1em] = \dfrac{4}{15}.$

Hence, $\dfrac{4}{9} \times \dfrac{3}{5} = \dfrac{4}{15}$.

Simplify:

$\dfrac{5}{12}$ × 8

Answer

$\dfrac{5}{12} \times 8$

$\Rightarrow \dfrac{5}{12} \times 8 = \dfrac{5 \times 8}{12} \\[1em] = \dfrac{40}{12} \\[1em] = \dfrac{10}{3} \\[1em] = 3\dfrac{1}{3}.$

Hence, $\dfrac{5}{12} \times 8 = 3\dfrac{1}{3}$.

Simplify:

$\dfrac{7}{6} \text{ of } \dfrac{3}{14}$

Answer

$\dfrac{7}{6} \text{ of } \dfrac{3}{14}$

$\Rightarrow \dfrac{7}{6} \text{ of } \dfrac{3}{14} = \dfrac{7}{6} \times \dfrac{3}{14} \\[1em] = \dfrac{7 \times 3}{6 \times 14} \\[1em] = \dfrac{21}{84} \\[1em] = \dfrac{1}{4}.$

Hence, $\dfrac{7}{6} \text{ of } \dfrac{3}{14} = \dfrac{1}{4}$.

Simplify:

$3\dfrac{3}{8} \times 3\dfrac{6}{7}$

Answer

$3\dfrac{3}{8} \times 3\dfrac{6}{7}$

$\Rightarrow 3\dfrac{3}{8} \times 3\dfrac{6}{7} = \dfrac{27}{8} \times \dfrac{27}{7} \\[1em] = \dfrac{27 \times 27}{8 \times 7} \\[1em] = \dfrac{729}{56} \\[1em] = 13\dfrac{1}{56}.$

Hence, $3\dfrac{3}{8} \times 3\dfrac{6}{7} = 13\dfrac{1}{56}$.

Simplify:

$\dfrac{1}{2} \text{ of } \dfrac{1}{3} \times \dfrac{3}{4}$

Answer

$\dfrac{1}{2} \text{ of } \dfrac{1}{3} \times \dfrac{3}{4}$

$\Rightarrow \dfrac{1}{2} \text{ of } \dfrac{1}{3} \times \dfrac{3}{4} = \dfrac{1}{6} \times \dfrac{3}{4} \\[1em] = \dfrac{3}{24} \\[1em] = \dfrac{1}{8}.$

Hence, $\dfrac{1}{2} \text{ of } \dfrac{1}{3} \times \dfrac{3}{4} = \dfrac{1}{8}$.

Simplify:

$\dfrac{3}{7} \times \dfrac{5}{9} \times 4\dfrac{1}{5}$

Answer

$\dfrac{3}{7} \times \dfrac{5}{9} \times 4\dfrac{1}{5}$

$\Rightarrow \dfrac{3}{7} \times \dfrac{5}{9} \times 4\dfrac{1}{5} = \dfrac{3}{7} \times \dfrac{5}{9} \times \dfrac{21}{5} \\[1em] = \dfrac{3 \times 5 \times 21}{7 \times 9 \times 5} \\[1em] = \dfrac{315}{315} \\[1em] = 1.$

Hence, $\dfrac{3}{7} \times \dfrac{5}{9} \times 4\dfrac{1}{5} = 1$.

Simplify:

$1\dfrac{1}{3} \times 1\dfrac{2}{7} \text{ of } 1\dfrac{1}{4}$

Answer

$1\dfrac{1}{3} \times 1\dfrac{2}{7} \text{ of } 1\dfrac{1}{4}$

$\Rightarrow 1\dfrac{1}{3} \times 1\dfrac{2}{7} \text{ of } 1\dfrac{1}{4} = \dfrac{4}{3} \times \dfrac{9}{7} \times \dfrac{5}{4} \\[1em] = \dfrac{4 \times 9 \times 5}{3 \times 7 \times 4} \\[1em] = \dfrac{180}{84} \\[1em] = \dfrac{15}{7} \\[1em] = 2\dfrac{1}{7}.$

Hence, $1\dfrac{1}{3} \times 1\dfrac{2}{7} \text{ of } 1\dfrac{1}{4} = 2\dfrac{1}{7}$.

Simplify:

$\dfrac{2}{3} \div 1\dfrac{1}{5}$

Answer

$\dfrac{2}{3} \div 1\dfrac{1}{5}$

$\Rightarrow \dfrac{2}{3} \div 1\dfrac{1}{5} = \dfrac{2}{3} \div \dfrac{6}{5} \\[1em] = \dfrac{2}{3} \times \dfrac{5}{6} \\[1em] = \dfrac{10}{18} \\[1em] = \dfrac{5}{9}.$

Hence, $\dfrac{2}{3} \div 1\dfrac{1}{5} = \dfrac{5}{9}$.

Simplify:

$4\dfrac{1}{2} \div \dfrac{4}{9}$

Answer

$4\dfrac{1}{2} \div \dfrac{4}{9}$

$\Rightarrow 4\dfrac{1}{2} \div \dfrac{4}{9} = \dfrac{9}{2} \div \dfrac{4}{9} \\[1em] = \dfrac{9}{2} \times \dfrac{9}{4} \\[1em] = \dfrac{81}{8} \\[1em] = 10\dfrac{1}{8}.$

Hence, $4\dfrac{1}{2} \div \dfrac{4}{9} = 10\dfrac{1}{8}$.

Simplify:

1 ÷ $\dfrac{2}{5}$

Answer

1 ÷ $\dfrac{2}{5}$

$\Rightarrow 1 \div \dfrac{2}{5} = 1 \times \dfrac{5}{2} \\[1em] = \dfrac{5}{2} \\[1em] = 2\dfrac{1}{2}.$

Hence, $1 \div \dfrac{2}{5} = 2\dfrac{1}{2}$.

Simplify:

$\dfrac{4}{9} \div \dfrac{4}{9}$

Answer

$\dfrac{4}{9} \div \dfrac{4}{9}$

$\Rightarrow \dfrac{4}{9} \div \dfrac{4}{9} = \dfrac{4}{9} \times \dfrac{9}{4} \\[1em] = 1.$

Hence, $\dfrac{4}{9} \div \dfrac{4}{9} = 1$.

Simplify:

$2\dfrac{1}{3} \div 1\dfrac{3}{4}$

Answer

$2\dfrac{1}{3} \div 1\dfrac{3}{4}$

$\Rightarrow 2\dfrac{1}{3} \div 1\dfrac{3}{4} = \dfrac{7}{3} \div \dfrac{7}{4} \\[1em] = \dfrac{7}{3} \times \dfrac{4}{7} \\[1em] = \dfrac{4}{3} \\[1em] = 1\dfrac{1}{3}.$

Hence, $2\dfrac{1}{3} \div 1\dfrac{3}{4} = 1\dfrac{1}{3}$.

Simplify:

$\dfrac{1}{4} \text{ of } 2\dfrac{2}{7} \div \dfrac{3}{5}$

Answer

$\dfrac{1}{4} \text{ of } 2\dfrac{2}{7} \div \dfrac{3}{5}$

$\Rightarrow \dfrac{1}{4} \text{ of } 2\dfrac{2}{7} \div \dfrac{3}{5} = \dfrac{1}{4} \times \dfrac{16}{7} \div \dfrac{3}{5} \\[1em] = \dfrac{4}{7} \div \dfrac{3}{5} \\[1em] = \dfrac{4}{7} \times \dfrac{5}{3} \\[1em] = \dfrac{20}{21}.$

Hence, $\dfrac{1}{4} \text{ of } 2\dfrac{2}{7} \div \dfrac{3}{5} = \dfrac{20}{21}$.

Simplify:

$1\dfrac{1}{4} \times \dfrac{1}{2} \div 1\dfrac{1}{3}$

Answer

$1\dfrac{1}{4} \times \dfrac{1}{2} \div 1\dfrac{1}{3}$

$\Rightarrow 1\dfrac{1}{4} \times \dfrac{1}{2} \div 1\dfrac{1}{3} = \dfrac{5}{4} \times \dfrac{1}{2} \div \dfrac{4}{3} \\[1em] = \dfrac{5}{8} \div \dfrac{4}{3} \\[1em] = \dfrac{5}{8} \times \dfrac{3}{4} \\[1em] = \dfrac{15}{32}.$

Hence, $1\dfrac{1}{4} \times \dfrac{1}{2} \div 1\dfrac{1}{3} = \dfrac{15}{32}$.

Simplify:

$6\dfrac{1}{7} \times 0 \times 5\dfrac{3}{8}$

Answer

$6\dfrac{1}{7} \times 0 \times 5\dfrac{3}{8}$

Any number multiplied by 0 is 0.

$\Rightarrow 6\dfrac{1}{7} \times 0 \times 5\dfrac{3}{8} = 0.$

Hence, $6\dfrac{1}{7} \times 0 \times 5\dfrac{3}{8} = 0$.

Simplify:

$\dfrac{3}{4} \times 1\dfrac{1}{3} \div \dfrac{3}{7} \text{ of } 2\dfrac{5}{8}$

Answer

$\dfrac{3}{4} \times 1\dfrac{1}{3} \div \dfrac{3}{7} \text{ of } 2\dfrac{5}{8}$

Solving 'of' first,

$\Rightarrow \dfrac{3}{7} \text{ of } 2\dfrac{5}{8} = \dfrac{3}{7} \times \dfrac{21}{8} = \dfrac{9}{8}\\[1em] \Rightarrow \dfrac{3}{4} \times 1\dfrac{1}{3} \div \dfrac{9}{8} = \dfrac{3}{4} \times \dfrac{4}{3} \div \dfrac{9}{8} \\[1em] = 1 \div \dfrac{9}{8} \\[1em] = 1 \times \dfrac{8}{9} \\[1em] = \dfrac{8}{9}.$

Hence, $\dfrac{3}{4} \times 1\dfrac{1}{3} \div \dfrac{3}{7} \text{ of } 2\dfrac{5}{8} = \dfrac{8}{9}$.

Simplify:

$2\dfrac{1}{4} \div \dfrac{2}{7} \text{ of } 1\dfrac{1}{3} \times \dfrac{2}{3}$

Answer

$2\dfrac{1}{4} \div \dfrac{2}{7} \text{ of } 1\dfrac{1}{3} \times \dfrac{2}{3}$

Solving 'of' first,

$\Rightarrow \dfrac{2}{7} \text{ of } 1\dfrac{1}{3} = \dfrac{2}{7} \times \dfrac{4}{3} = \dfrac{8}{21}\\[1em] \Rightarrow 2\dfrac{1}{4} \div \dfrac{8}{21} \times \dfrac{2}{3} = \dfrac{9}{4} \div \dfrac{8}{21} \times \dfrac{2}{3} \\[1em] = \dfrac{9}{4} \times \dfrac{21}{8} \times \dfrac{2}{3} \\[1em] = \dfrac{9 \times 21 \times 2}{4 \times 8 \times 3} \\[1em] = \dfrac{378}{96} \\[1em] = \dfrac{63}{16} \\[1em] = 3\dfrac{15}{16}.$

Hence, $2\dfrac{1}{4} \div \dfrac{2}{7} \text{ of } 1\dfrac{1}{3} \times \dfrac{2}{3} = 3\dfrac{15}{16}$.

Simplify:

$5 - \left(\dfrac{8}{11} - 3\dfrac{3}{11}\right)$

Answer

Using BODMAS we simplify bracket first.

$5 - \left(\dfrac{8}{11} - 3\dfrac{3}{11}\right)$

$\Rightarrow 5 - \left(\dfrac{8}{11} - \dfrac{36}{11}\right) = 5 - \left(\dfrac{8 - 36}{11}\right) \\[1em] = 5 - \left(\dfrac{-28}{11}\right) \\[1em] = 5 + \dfrac{28}{11} \\[1em] = \dfrac{55}{11} + \dfrac{28}{11} \\[1em] = \dfrac{83}{11} \\[1em] = 7\dfrac{6}{11}.$

Hence, $5 - \left(\dfrac{8}{11} - 3\dfrac{3}{11}\right) = 7\dfrac{6}{11}$.

Simplify:

$\dfrac{1}{2} \div \left(\dfrac{7}{8} - \dfrac{3}{5}\right)$

Answer

Using BODMAS we simplify bracket first.

$\dfrac{1}{2} \div \left(\dfrac{7}{8} - \dfrac{3}{5}\right)$

$\Rightarrow \dfrac{1}{2} \div \left(\dfrac{7}{8} - \dfrac{3}{5}\right) = \dfrac{1}{2} \div \left(\dfrac{35}{40} - \dfrac{24}{40}\right) \\[1em] = \dfrac{1}{2} \div \dfrac{11}{40} \\[1em] = \dfrac{1}{2} \times \dfrac{40}{11} \\[1em] = \dfrac{20}{11} \\[1em] = 1\dfrac{9}{11}.$

Hence, $\dfrac{1}{2} \div \left(\dfrac{7}{8} - \dfrac{3}{5}\right) = 1\dfrac{9}{11}$.

Simplify:

$2\dfrac{1}{3} \div \left(5\dfrac{1}{2} + 3\dfrac{3}{4}\right)$

Answer

Using BODMAS we simplify bracket first.

$2\dfrac{1}{3} \div \left(5\dfrac{1}{2} + 3\dfrac{3}{4}\right)$

$\Rightarrow 2\dfrac{1}{3} \div \left(\dfrac{11}{2} + \dfrac{15}{4}\right) = \dfrac{7}{3} \div \left(\dfrac{22}{4} + \dfrac{15}{4}\right) \\[1em] = \dfrac{7}{3} \div \dfrac{37}{4} \\[1em] = \dfrac{7}{3} \times \dfrac{4}{37} \\[1em] = \dfrac{28}{111}.$

Hence, $2\dfrac{1}{3} \div \left(5\dfrac{1}{2} + 3\dfrac{3}{4}\right) = \dfrac{28}{111}$.

Simplify:

$\left(3\dfrac{7}{8} - 3\dfrac{3}{5}\right) \div \dfrac{1}{2}$

Answer

Using BODMAS we simplify bracket first.

$\left(3\dfrac{7}{8} - 3\dfrac{3}{5}\right) \div \dfrac{1}{2}$

$\Rightarrow \left(\dfrac{31}{8} - \dfrac{18}{5}\right) \div \dfrac{1}{2} = \left(\dfrac{155}{40} - \dfrac{144}{40}\right) \div \dfrac{1}{2} \\[1em] = \dfrac{11}{40} \div \dfrac{1}{2} \\[1em] = \dfrac{11}{40} \times 2 \\[1em] = \dfrac{22}{40} \\[1em] = \dfrac{11}{20}.$

Hence, $\left(3\dfrac{7}{8} - 3\dfrac{3}{5}\right) \div \dfrac{1}{2} = \dfrac{11}{20}$.

Simplify:

$\dfrac{4}{7} \div \left(\dfrac{1}{3} \times 2\dfrac{4}{5}\right)$

Answer

Using BODMAS we simplify bracket first.

$\dfrac{4}{7} \div \left(\dfrac{1}{3} \times 2\dfrac{4}{5}\right)$

$\Rightarrow \dfrac{4}{7} \div \left(\dfrac{1}{3} \times \dfrac{14}{5}\right) = \dfrac{4}{7} \div \dfrac{14}{15} \\[1em] = \dfrac{4}{7} \times \dfrac{15}{14} \\[1em] = \dfrac{60}{98} \\[1em] = \dfrac{30}{49}.$

Hence, $\dfrac{4}{7} \div \left(\dfrac{1}{3} \times 2\dfrac{4}{5}\right) = \dfrac{30}{49}$.

Simplify:

$\left(\dfrac{1}{2} + \dfrac{1}{3}\right) \div \left(\dfrac{1}{4} - \dfrac{1}{6}\right)$

Answer

$\left(\dfrac{1}{2} + \dfrac{1}{3}\right) \div \left(\dfrac{1}{4} - \dfrac{1}{6}\right)$

Using BODMAS we simplify bracket first.

$\Rightarrow \left(\dfrac{3}{6} + \dfrac{2}{6}\right) \div \left(\dfrac{3}{12} - \dfrac{2}{12}\right) = \dfrac{5}{6} \div \dfrac{1}{12} \\[1em] = \dfrac{5}{6} \times 12 \\[1em] = 10.$

Hence, $\left(\dfrac{1}{2} + \dfrac{1}{3}\right) \div \left(\dfrac{1}{4} - \dfrac{1}{6}\right) = 10$.

Simplify:

$\left(\dfrac{24}{35} \div \dfrac{6}{7} + \dfrac{5}{9}\right) \times \dfrac{3}{4}$

Answer

$\left(\dfrac{24}{35} \div \dfrac{6}{7} + \dfrac{5}{9}\right) \times \dfrac{3}{4}$

Using BODMAS, we solve the division inside the bracket first,

$\Rightarrow \dfrac{24}{35} \div \dfrac{6}{7} = \dfrac{24}{35} \times \dfrac{7}{6} = \dfrac{4}{5}.\\[1em] \Rightarrow \left(\dfrac{4}{5} + \dfrac{5}{9}\right) \times \dfrac{3}{4} = \left(\dfrac{36}{45} + \dfrac{25}{45}\right) \times \dfrac{3}{4} \\[1em] = \dfrac{61}{45} \times \dfrac{3}{4} \\[1em] = \dfrac{183}{180} \\[1em] = \dfrac{61}{60} \\[1em] = 1\dfrac{1}{60}.$

Hence, $\left(\dfrac{24}{35} \div \dfrac{6}{7} + \dfrac{5}{9}\right) \times \dfrac{3}{4} = 1\dfrac{1}{60}$.

Simplify:

$\dfrac{3}{4} \text{ of } 6\dfrac{1}{8} - \dfrac{2}{3} \text{ of } 2\dfrac{1}{4}$

Answer

$\dfrac{3}{4} \text{ of } 6\dfrac{1}{8} - \dfrac{2}{3} \text{ of } 2\dfrac{1}{4}$

Using BODMAS, we solve 'of' first,

$\Rightarrow \dfrac{3}{4} \text{ of } 6\dfrac{1}{8} = \dfrac{3}{4} \times \dfrac{49}{8} = \dfrac{147}{32}.\\[1em] \Rightarrow \dfrac{2}{3} \text{ of } 2\dfrac{1}{4} = \dfrac{2}{3} \times \dfrac{9}{4} = \dfrac{3}{2}.\\[1em] \Rightarrow \dfrac{147}{32} - \dfrac{3}{2} = \dfrac{147}{32} - \dfrac{48}{32} \\[1em] = \dfrac{99}{32} \\[1em] = 3\dfrac{3}{32}.$

Hence, $\dfrac{3}{4} \text{ of } 6\dfrac{1}{8} - \dfrac{2}{3} \text{ of } 2\dfrac{1}{4} = 3\dfrac{3}{32}$.

Simplify:

$\dfrac{7}{30} \text{ of } \left(\dfrac{1}{3} + \dfrac{7}{15}\right) \div \left(\dfrac{5}{6} - \dfrac{3}{5}\right)$

Answer

$\dfrac{7}{30} \text{ of } \left(\dfrac{1}{3} + \dfrac{7}{15}\right) \div \left(\dfrac{5}{6} - \dfrac{3}{5}\right)$

Using BODMAS, we simplify the brackets first,

$\Rightarrow \dfrac{1}{3} + \dfrac{7}{15} = \dfrac{5}{15} + \dfrac{7}{15} = \dfrac{12}{15} = \dfrac{4}{5}.\\[1em] \Rightarrow \dfrac{5}{6} - \dfrac{3}{5} = \dfrac{25}{30} - \dfrac{18}{30} = \dfrac{7}{30}.\\[1em] \Rightarrow \dfrac{7}{30} \text{ of } \dfrac{4}{5} \div \dfrac{7}{30} = \dfrac{7}{30} \times \dfrac{4}{5} \div \dfrac{7}{30} \\[1em] = \dfrac{28}{150} \div \dfrac{7}{30} \\[1em] = \dfrac{28}{150} \times \dfrac{30}{7} \\[1em] = \dfrac{4}{5}.$

Hence, $\dfrac{7}{30} \text{ of } \left(\dfrac{1}{3} + \dfrac{7}{15}\right) \div \left(\dfrac{5}{6} - \dfrac{3}{5}\right) = \dfrac{4}{5}$.

Simplify:

$2\dfrac{1}{2} - 3\dfrac{1}{2} \times 1\dfrac{3}{4} + 2\dfrac{1}{2}$

Answer

$2\dfrac{1}{2} - 3\dfrac{1}{2} \times 1\dfrac{3}{4} + 2\dfrac{1}{2}$

Using BODMAS, we solve the multiplication first,

$\Rightarrow 3\dfrac{1}{2} \times 1\dfrac{3}{4} = \dfrac{7}{2} \times \dfrac{7}{4} = \dfrac{49}{8}.\\[1em] \Rightarrow 2\dfrac{1}{2} - \dfrac{49}{8} + 2\dfrac{1}{2} = \dfrac{5}{2} - \dfrac{49}{8} + \dfrac{5}{2} \\[1em] = \dfrac{20}{8} - \dfrac{49}{8} + \dfrac{20}{8} \\[1em] = \dfrac{-9}{8} \\[1em] = -1\dfrac{1}{8}.$

Hence, $2\dfrac{1}{2} - 3\dfrac{1}{2} \times 1\dfrac{3}{4} + 2\dfrac{1}{2} = -1\dfrac{1}{8}$.

Simplify:

$4\dfrac{5}{7}\left(3\dfrac{1}{8} \div \dfrac{11}{12}\right)$

Answer

$4\dfrac{5}{7}\left(3\dfrac{1}{8} \div \dfrac{11}{12}\right)$

Using BODMAS, we simplify the bracket first,

$\Rightarrow 3\dfrac{1}{8} \div \dfrac{11}{12} = \dfrac{25}{8} \times \dfrac{12}{11} = \dfrac{300}{88} = \dfrac{75}{22}.\\[1em] \Rightarrow 4\dfrac{5}{7} \times \dfrac{75}{22} = \dfrac{33}{7} \times \dfrac{75}{22} \\[1em] = \dfrac{33 \times 75}{7 \times 22} \\[1em] = \dfrac{2475}{154} \\[1em] = \dfrac{225}{14} \\[1em] = 16\dfrac{1}{14}.$

Hence, $4\dfrac{5}{7}\left(3\dfrac{1}{8} \div \dfrac{11}{12}\right) = 16\dfrac{1}{14}$.

Simplify:

$\dfrac{2}{5} \text{ of } \left(\dfrac{1}{7} - \dfrac{1}{12}\right) \text{ of } 1\dfrac{2}{5}$

Answer

$\dfrac{2}{5} \text{ of } \left(\dfrac{1}{7} - \dfrac{1}{12}\right) \text{ of } 1\dfrac{2}{5}$

Using BODMAS, we simplify the bracket first,

$\Rightarrow \dfrac{1}{7} - \dfrac{1}{12} = \dfrac{12}{84} - \dfrac{7}{84} = \dfrac{5}{84}.\\[1em] \Rightarrow \dfrac{2}{5} \text{ of } \dfrac{5}{84} \text{ of } 1\dfrac{2}{5} = \dfrac{2}{5} \times \dfrac{5}{84} \times \dfrac{7}{5} \\[1em] = \dfrac{2 \times 5 \times 7}{5 \times 84 \times 5} \\[1em] = \dfrac{70}{2100} \\[1em] = \dfrac{1}{30}.$

Hence, $\dfrac{2}{5} \text{ of } \left(\dfrac{1}{7} - \dfrac{1}{12}\right) \text{ of } 1\dfrac{2}{5} = \dfrac{1}{30}$.

Simplify:

$\left(\dfrac{1}{2} - \dfrac{1}{3}\right)\left(\dfrac{3}{4} - \dfrac{4}{5}\right) \div \left(\dfrac{1}{2} - \dfrac{2}{5} + \dfrac{1}{7}\right)$

Answer

$\left(\dfrac{1}{2} - \dfrac{1}{3}\right)\left(\dfrac{3}{4} - \dfrac{4}{5}\right) \div \left(\dfrac{1}{2} - \dfrac{2}{5} + \dfrac{1}{7}\right)$

Using BODMAS, we simplify the brackets first,

$\Rightarrow \dfrac{1}{2} - \dfrac{1}{3} = \dfrac{3}{6} - \dfrac{2}{6} = \dfrac{1}{6}.\\[1em] \Rightarrow \dfrac{3}{4} - \dfrac{4}{5} = \dfrac{15}{20} - \dfrac{16}{20} = \dfrac{-1}{20}.\\[1em] \Rightarrow \dfrac{1}{2} - \dfrac{2}{5} + \dfrac{1}{7} = \dfrac{35}{70} - \dfrac{28}{70} + \dfrac{10}{70} = \dfrac{17}{70}.\\[1em] \Rightarrow \dfrac{1}{6} \times \dfrac{-1}{20} \div \dfrac{17}{70} = \dfrac{-1}{120} \times \dfrac{70}{17} \\[1em] = \dfrac{-70}{2040} \\[1em] = \dfrac{-7}{204}.$

Hence, $\left(\dfrac{1}{2} - \dfrac{1}{3}\right)\left(\dfrac{3}{4} - \dfrac{4}{5}\right) \div \left(\dfrac{1}{2} - \dfrac{2}{5} + \dfrac{1}{7}\right) = \dfrac{-7}{204}$.

Simplify:

$\dfrac{5}{6} - \dfrac{3}{5}\left(\dfrac{1}{3} + \dfrac{2}{11}\right)$

Answer

$\dfrac{5}{6} - \dfrac{3}{5}\left(\dfrac{1}{3} + \dfrac{2}{11}\right)$

Using BODMAS, we simplify the bracket first,

$\Rightarrow \dfrac{1}{3} + \dfrac{2}{11} = \dfrac{11}{33} + \dfrac{6}{33} = \dfrac{17}{33}.\\[1em] \Rightarrow \dfrac{5}{6} - \dfrac{3}{5} \times \dfrac{17}{33} = \dfrac{5}{6} - \dfrac{51}{165} \\[1em] = \dfrac{5}{6} - \dfrac{17}{55} \\[1em] = \dfrac{275}{330} - \dfrac{102}{330} \\[1em] = \dfrac{173}{330}.$

Hence, $\dfrac{5}{6} - \dfrac{3}{5}\left(\dfrac{1}{3} + \dfrac{2}{11}\right) = \dfrac{173}{330}$.