Fractions — Exercise 4(F)

Find:

(i) $\dfrac{1}{8}$ of 40

(ii) $\dfrac{4}{11}$ of $4\dfrac{2}{5}$

(iii) $1\dfrac{3}{5}$ of $6\dfrac{1}{4}$

(iv) $\dfrac{3}{4}$ of ₹ 1

(v) $\dfrac{5}{8}$ of 1 km

(vi) $\dfrac{5}{12}$ of 1 hour

Answer

(i) $\dfrac{1}{8}$ of 40

$\Rightarrow \dfrac{1}{8} \times 40\\[1em] \Rightarrow \dfrac{1 \times 40}{8}\\[1em] \Rightarrow \dfrac{40}{8}\\[1em] \Rightarrow 5$

Hence, $\dfrac{1}{8}$ of 40 = 5.

(ii) $\dfrac{4}{11}$ of $4\dfrac{2}{5}$

$\Rightarrow \dfrac{4}{11} \times 4\dfrac{2}{5}\\[1em] \Rightarrow \dfrac{4}{11} \times \dfrac{22}{5}\\[1em] \Rightarrow \dfrac{4}{5} \times \dfrac{22}{11}\\[1em] \Rightarrow \dfrac{4}{5} \times 2 \\[1em] \Rightarrow \dfrac{8}{5}\\[1em] \Rightarrow 1\dfrac{3}{5}.$

Hence, $\dfrac{4}{11} \text{ of } 4\dfrac{2}{5} = 1\dfrac{3}{5}$.

(iii) $1\dfrac{3}{5}$ of $6\dfrac{1}{4}$

$\Rightarrow 1\dfrac{3}{5} \times 6\dfrac{1}{4}\\[1em] \Rightarrow \dfrac{8}{5} \times \dfrac{25}{4}\\[1em] \Rightarrow \dfrac{8 \times 25}{5 \times 4}\\[1em] \Rightarrow \dfrac{200}{20}\\[1em] \Rightarrow 10.$

Hence, $1\dfrac{3}{5} \text{ of } 6\dfrac{1}{4} = 10$.

(iv) $\dfrac{3}{4}$ of ₹ 1

$\Rightarrow \dfrac{3}{4} \times ₹ 1\\[1em] \Rightarrow \dfrac{3}{4} \times 100 \text{ paise}\\[1em] \Rightarrow \dfrac{3 \times 100}{4}\text{ paise}\\[1em] \Rightarrow \dfrac{300}{4}\text{ paise}\\[1em] \Rightarrow 75\text{ paise}$

Hence, $\dfrac{3}{4}$ of ₹ 1 = 75 paise.

(v) $\dfrac{5}{8}$ of 1 km

$\Rightarrow \dfrac{5}{8} \times 1\text{ km}\\[1em] \Rightarrow \dfrac{5}{8} \times 1000\text{ m}\\[1em] \Rightarrow \dfrac{5 \times 1000}{8}\text{ m}\\[1em] \Rightarrow \dfrac{5000}{8}\text{ m}\\[1em] \Rightarrow 625\text{ m}\\[1em]$

Hence, $\dfrac{5}{8}$ of 1 km = 625 m.

(vi) $\dfrac{5}{12}$ of 1 hour

$\Rightarrow \dfrac{5}{12} \times 1 \text{ hour}\\[1em] \Rightarrow \dfrac{5}{12} \times 60 \text{ min}\\[1em] \Rightarrow \dfrac{5 \times 60}{12} \text{ min}\\[1em] \Rightarrow \dfrac{300}{12} \text{ min}\\[1em] \Rightarrow 25 \text{ min}\\[1em]$

Hence, $\dfrac{5}{12}$ of 1 hour = 25 min.

Simplify :

$1\dfrac{1}{6} ÷ 1\dfrac{5}{9} \times 3\dfrac{1}{3}$

Answer

Solving,

$\Rightarrow 1\dfrac{1}{6} ÷ 1\dfrac{5}{9} \times 3\dfrac{1}{3}\\[1em] \Rightarrow \dfrac{7}{6} ÷ \dfrac{14}{9} \times \dfrac{10}{3}\\[1em] \Rightarrow \dfrac{7}{6} \times \dfrac{9}{14} \times \dfrac{10}{3}\\[1em] \Rightarrow \dfrac{7}{14} \times \dfrac{9}{3} \times \dfrac{10}{6}\\[1em] \Rightarrow \dfrac{1}{2} \times 3 \times \dfrac{5}{3}\\[1em] \Rightarrow \dfrac{5}{2}\\[1em] \Rightarrow 2\dfrac{1}{2}.$

Hence, $1\dfrac{1}{6} ÷ 1\dfrac{5}{9} \times 3\dfrac{1}{3} = 2\dfrac{1}{2}$.

Simplify :

$1\dfrac{1}{3} ÷ \dfrac{3}{7}$ of $2\dfrac{5}{8} + 1\dfrac{1}{9}$

Answer

Solving,

$\Rightarrow 1\dfrac{1}{3} ÷ \dfrac{3}{7} \text{ of } 2\dfrac{5}{8} + 1\dfrac{1}{9}\\[1em] \Rightarrow \dfrac{4}{3} ÷ \Big(\dfrac{3}{7} \times \dfrac{21}{8}\Big) + \dfrac{10}{9}\\[1em] \Rightarrow \dfrac{4}{3} ÷ \dfrac{3 \times 21}{7 \times 8} + \dfrac{10}{9}\\[1em] \Rightarrow \dfrac{4}{3} ÷ \dfrac{63}{56} + \dfrac{10}{9}\\[1em] \Rightarrow \dfrac{4}{3} ÷ \dfrac{9}{8} + \dfrac{10}{9}\\[1em] \Rightarrow \dfrac{4}{3} \times \dfrac{8}{9} + \dfrac{10}{9}\\[1em] \Rightarrow \dfrac{4 \times 8}{3 \times 9} + \dfrac{10}{9}\\[1em] \Rightarrow \dfrac{32}{27} + \dfrac{10}{9}\\[1em] \Rightarrow \dfrac{32}{27} + \dfrac{10 \times 3}{9 \times 3}\\[1em] \Rightarrow \dfrac{32}{27} + \dfrac{30}{27}\\[1em] \Rightarrow \dfrac{32 + 30}{27}\\[1em] \Rightarrow \dfrac{62}{27}\\[1em] \Rightarrow 2\dfrac{8}{27}$

Hence, $1\dfrac{1}{3} ÷ \dfrac{3}{7}$ of $2\dfrac{5}{8} + 1\dfrac{1}{9} = 2\dfrac{8}{27}$.

Simplify :

$6\dfrac{1}{5} ÷ 3\dfrac{1}{10}$ of $2\dfrac{1}{2} ÷ \dfrac{1}{4}$

Answer

$\Rightarrow 6\dfrac{1}{5} ÷ 3\dfrac{1}{10} \text{ of } 2\dfrac{1}{2} ÷ \dfrac{1}{4}\\[1em] \Rightarrow 6\dfrac{1}{5} ÷ \Big(\dfrac{31}{10} \times \dfrac{5}{2}\Big) ÷ \dfrac{1}{4}\\[1em] \Rightarrow 6\dfrac{1}{5} ÷ \dfrac{31 \times 5}{10 \times 2} ÷ \dfrac{1}{4}\\[1em] \Rightarrow 6\dfrac{1}{5} ÷ \dfrac{155}{20} ÷ \dfrac{1}{4}\\[1em] \Rightarrow \Big(6\dfrac{1}{5} ÷ \dfrac{31}{4}\Big) ÷ \dfrac{1}{4}\\[1em] \Rightarrow \Big(\dfrac{31}{5} ÷ \dfrac{31}{4}\Big) ÷ \dfrac{1}{4}\\[1em] \Rightarrow \Big(\dfrac{31}{5} \times \dfrac{4}{31}\Big) ÷ \dfrac{1}{4}\\[1em] \Rightarrow \dfrac{4}{5} ÷ \dfrac{1}{4} \\[1em] \Rightarrow \dfrac{4}{5} \times 4 \\[1em] \Rightarrow \dfrac{16}{5}\\[1em] \Rightarrow 3\dfrac{1}{5}.$

Hence, $6\dfrac{1}{5} ÷ 3\dfrac{1}{10}$ of $2\dfrac{1}{2} ÷ \dfrac{1}{4} = 3\dfrac{1}{5}$.

Simplify :

$3\dfrac{2}{3} - \dfrac{3}{11}$ of $2\dfrac{3}{4} ÷ 1\dfrac{1}{4} \times 1\dfrac{2}{3} + \dfrac{1}{3}$

Answer

$\Rightarrow 3\dfrac{2}{3} - \dfrac{3}{11} \text{ of } 2\dfrac{3}{4} ÷ 1\dfrac{1}{4} \times 1\dfrac{2}{3} + \dfrac{1}{3}\\[1em] \Rightarrow 3\dfrac{2}{3} - \dfrac{3}{11} \times 2\dfrac{3}{4} ÷ 1\dfrac{1}{4} \times 1\dfrac{2}{3} + \dfrac{1}{3}\\[1em] \Rightarrow 3\dfrac{2}{3} - \dfrac{3}{11} \times \dfrac{11}{4} ÷ 1\dfrac{1}{4} \times 1\dfrac{2}{3} + \dfrac{1}{3}\\[1em] \Rightarrow 3\dfrac{2}{3} - \dfrac{3 \times 11}{11 \times 4} ÷ 1\dfrac{1}{4} \times 1\dfrac{2}{3} + \dfrac{1}{3}\\[1em] \Rightarrow 3\dfrac{2}{3} - \dfrac{3 \times 11}{11 \times 4} ÷ 1\dfrac{1}{4} \times 1\dfrac{2}{3} + \dfrac{1}{3}\\[1em] \Rightarrow 3\dfrac{2}{3} - \dfrac{3}{4} ÷ 1\dfrac{1}{4} \times 1\dfrac{2}{3} + \dfrac{1}{3}\\[1em] \Rightarrow 3\dfrac{2}{3} - \dfrac{3}{4} ÷ \dfrac{5}{4} \times 1\dfrac{2}{3} + \dfrac{1}{3}\\[1em] \Rightarrow 3\dfrac{2}{3} - \dfrac{3}{4} \times \dfrac{4}{5} \times \dfrac{5}{3} + \dfrac{1}{3}\\[1em] \Rightarrow 3\dfrac{2}{3} - \dfrac{3 \times 4 \times 5}{4 \times 5 \times 3} + \dfrac{1}{3}\\[1em] \Rightarrow 3\dfrac{2}{3} - 1 + \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{11}{3} + \dfrac{1}{3} - 1\\[1em] \Rightarrow \dfrac{11 + 1}{3} - 1\\[1em] \Rightarrow \dfrac{12}{3} - 1\\[1em] \Rightarrow 4 - 1\\[1em] \Rightarrow 3$

Hence, $3\dfrac{2}{3} - \dfrac{3}{11}$ of $2\dfrac{3}{4} ÷ 1\dfrac{1}{4} \times 1\dfrac{2}{3} + \dfrac{1}{3} = 3$.

Simplify :

$2\dfrac{2}{7}$ of $15\dfrac{3}{4} \times 2\dfrac{1}{4} ÷ \dfrac{4}{7}$ of $2\dfrac{5}{8}$

Answer

$\Rightarrow \Big(2\dfrac{2}{7} \text{ of } 15\dfrac{3}{4}\Big) \times 2\dfrac{1}{4} ÷ \Big(\dfrac{4}{7} \text{ of } 2\dfrac{5}{8}\Big)\\[1em] \Rightarrow \dfrac{16}{7} \times \dfrac{63}{4} \times 2\dfrac{1}{4} ÷ \dfrac{4}{7} \times \dfrac{21}{8}\\[1em] \Rightarrow \dfrac{16 \times 63}{7 \times 4} \times 2\dfrac{1}{4} ÷ \dfrac{4 \times 21}{7 \times 8} \\[1em] \Rightarrow 4 \times 9 \times 2\dfrac{1}{4} ÷ \dfrac{3}{2} \\[1em] \Rightarrow 36 \times 2\dfrac{1}{4} ÷ \dfrac{3}{2} \\[1em] \Rightarrow 36 \times 2\dfrac{1}{4} \times \dfrac{2}{3} \\[1em] \Rightarrow 36 \times \dfrac{9}{4} \times \dfrac{2}{3} \\[1em] \Rightarrow 36 \times \dfrac{3}{2} \\[1em] \Rightarrow 18 \times 3 \\[1em] \Rightarrow 54.$

Hence, $2\dfrac{2}{7}$ of $15\dfrac{3}{4} \times 2\dfrac{1}{4} ÷ \dfrac{4}{7}$ of $2\dfrac{5}{8} = 54$.

Simplify :

$1 ÷ \dfrac{4}{7} - \dfrac{1}{3}$ of $3\dfrac{3}{4} + \dfrac{1}{2} ÷ 3$

Answer

$\Rightarrow 1 ÷ \dfrac{4}{7} - \Big(\dfrac{1}{3} \text{ of } 3\dfrac{3}{4}\Big) + \dfrac{1}{2} ÷ 3\\[1em] \Rightarrow 1 ÷ \dfrac{4}{7} - \Big(\dfrac{1}{3} \times 3\dfrac{3}{4}\Big) + \dfrac{1}{2} ÷ 3\\[1em] \Rightarrow 1 ÷ \dfrac{4}{7} - \dfrac{1}{3} \times \dfrac{15}{4} + \dfrac{1}{2} ÷ 3\\[1em] \Rightarrow 1 ÷ \dfrac{4}{7} - \dfrac{1 \times 15}{3 \times 4} + \dfrac{1}{2} ÷ 3\\[1em] \Rightarrow 1 ÷ \dfrac{4}{7} - \dfrac{15}{12} + \dfrac{1}{2} ÷ 3\\[1em] \Rightarrow 1 ÷ \dfrac{4}{7} - \dfrac{5}{4} + \dfrac{1}{2} ÷ 3\\[1em] \Rightarrow 1 \times \dfrac{7}{4} - \dfrac{5}{4} + \dfrac{1}{2} \times \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{7}{4} - \dfrac{5}{4} + \dfrac{1 \times 1}{2 \times 3}\\[1em] \Rightarrow \dfrac{7}{4} - \dfrac{5}{4} + \dfrac{1}{6}\\[1em] \Rightarrow \dfrac{7 \times 3}{4 \times 3} - \dfrac{5 \times 3}{4 \times 3} + \dfrac{1 \times 2}{6 \times 2}\\[1em] \Rightarrow \dfrac{21}{12} - \dfrac{15}{12} + \dfrac{2}{12}\\[1em] \Rightarrow \dfrac{21 - 15 + 2}{12}\\[1em] \Rightarrow \dfrac{21 - 13}{12}\\[1em] \Rightarrow \dfrac{8}{12}\\[1em] \Rightarrow \dfrac{2}{3}$

Hence, $1 ÷ \dfrac{4}{7} - \dfrac{1}{3}$ of $3\dfrac{3}{4} + \dfrac{1}{2} ÷ 3 = \dfrac{2}{3}$.

Simplify :

$\Big(\dfrac{2}{3} + \dfrac{4}{9}\Big)$ of $\dfrac{3}{5} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3}$

Answer

$\Rightarrow \Big(\dfrac{2}{3} + \dfrac{4}{9}\Big) \text{ of } \dfrac{3}{5} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \Big(\dfrac{2 \times 3}{3 \times 3} + \dfrac{4}{9}\Big) \text{ of } \dfrac{3}{5} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \Big(\dfrac{6}{9} + \dfrac{4}{9}\Big) \text{ of } \dfrac{3}{5} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \Big(\dfrac{6 + 4}{9}\Big) \text{ of } \dfrac{3}{5} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{10}{9} \times \dfrac{3}{5} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{10 \times 3}{9 \times 5} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{30}{45} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{2}{3} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{2}{3} ÷ \dfrac{5}{3} \times \dfrac{5}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{2}{3} \times \dfrac{3}{5} \times \dfrac{5}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{2 \times 3 \times 5}{3 \times 5 \times 4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{2 \times \cancel{3} \times \cancel{5}}{\cancel{3} \times \cancel{5} \times 4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{2}{4} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{1}{2} - \dfrac{1}{3}\\[1em] \Rightarrow \dfrac{1 \times 3}{2 \times 3} - \dfrac{1 \times 2}{3 \times 2}\\[1em] \Rightarrow \dfrac{3}{6} - \dfrac{2}{6}\\[1em] \Rightarrow \dfrac{3 - 2}{6}\\[1em] \Rightarrow \dfrac{1}{6}$

Hence, $\Big(\dfrac{2}{3} + \dfrac{4}{9}\Big)$ of $\dfrac{3}{5} ÷ 1\dfrac{2}{3} \times 1\dfrac{1}{4} - \dfrac{1}{3} = \dfrac{1}{6}$

Simplify :

$\Big(\dfrac{14}{15} ÷ 1\dfrac{1}{6} + \dfrac{7}{10}\Big) \times \dfrac{3}{4}$

Answer

$\Rightarrow \Big(\dfrac{14}{15} ÷ 1\dfrac{1}{6} + \dfrac{7}{10}\Big) \times \dfrac{3}{4}\\[1em] \Rightarrow \Big(\dfrac{14}{15} ÷ \dfrac{7}{6} + \dfrac{7}{10}\Big) \times \dfrac{3}{4}\\[1em] \Rightarrow \Big(\dfrac{14}{15} \times \dfrac{6}{7} + \dfrac{7}{10}\Big) \times \dfrac{3}{4}\\[1em] \Rightarrow \Big(\dfrac{14 \times 6}{15 \times 7} + \dfrac{7}{10}\Big) \times \dfrac{3}{4}\\[1em] \Rightarrow \Big(\dfrac{84}{105} + \dfrac{7}{10}\Big) \times \dfrac{3}{4}\\[1em] \Rightarrow \Big(\dfrac{4}{5} + \dfrac{7}{10}\Big) \times \dfrac{3}{4}\\[1em] \Rightarrow \Big(\dfrac{4 \times 2}{5 \times 2} + \dfrac{7}{10}\Big) \times \dfrac{3}{4}\\[1em] \Rightarrow \Big(\dfrac{8}{10} + \dfrac{7}{10}\Big) \times \dfrac{3}{4}\\[1em] \Rightarrow \Big(\dfrac{8 + 7}{10}\Big) \times \dfrac{3}{4}\\[1em] \Rightarrow \dfrac{15}{10} \times \dfrac{3}{4}\\[1em] \Rightarrow \dfrac{15 \times 3}{10 \times 4}\\[1em] \Rightarrow \dfrac{45}{40}\\[1em] \Rightarrow \dfrac{9}{8}\\[1em] \Rightarrow 1\dfrac{1}{8}$

Hence, $\Big(\dfrac{14}{15} ÷ 1\dfrac{1}{6} + \dfrac{7}{10}\Big) \times \dfrac{3}{4} = 1\dfrac{1}{8}$

Simplify :

$\dfrac{1}{3}\Big(2\dfrac{1}{2} + 3\dfrac{1}{3}\Big) ÷ \dfrac{2}{9}\Big(3\dfrac{1}{8} - 1\dfrac{1}{12}\Big)$

Answer

$\Rightarrow \dfrac{1}{3}\Big(2\dfrac{1}{2} + 3\dfrac{1}{3}\Big) ÷ \dfrac{2}{9}\Big(3\dfrac{1}{8} - 1\dfrac{1}{12}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{5}{2} + \dfrac{10}{3}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{25}{8} - \dfrac{13}{12}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{5 \times 3}{2 \times 3} + \dfrac{10 \times 2}{3 \times 2}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{25 \times 3}{8 \times 3} - \dfrac{13 \times 2}{12 \times 2}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{15}{6} + \dfrac{20}{6}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{75}{24} - \dfrac{26}{24}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{15 + 20}{6}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{75 - 26}{24}\Big)\\[1em] \Rightarrow \dfrac{1}{3}\Big(\dfrac{35}{6}\Big) ÷ \dfrac{2}{9}\Big(\dfrac{49}{24}\Big)\\[1em] \Rightarrow \dfrac{35 \times 1}{6 \times 3} ÷ \dfrac{49 \times 2}{24 \times 9}\\[1em] \Rightarrow \dfrac{35}{18} ÷ \dfrac{98}{216}\\[1em] \Rightarrow \dfrac{35}{18} \times \dfrac{216}{98}\\[1em] \Rightarrow \dfrac{35}{98} \times \dfrac{216}{18} \\[1em] \Rightarrow \dfrac{5}{14} \times 12 \\[1em] \Rightarrow \dfrac{60}{14}\\[1em] \Rightarrow \dfrac{30}{7} \\[1em] \Rightarrow 4\dfrac{2}{7}$

Hence, $\dfrac{1}{3}\Big(2\dfrac{1}{2} + 3\dfrac{1}{3}\Big) ÷ \dfrac{2}{9}\Big(3\dfrac{1}{8} - 1\dfrac{1}{12}\Big) = 4\dfrac{2}{7}$.

Simplify :

$\Big(\dfrac{1}{4} - \dfrac{1}{9}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{1}{4} ÷ \dfrac{1}{3}\Big)$

Answer

$\Rightarrow \Big(\dfrac{1}{4} - \dfrac{1}{9}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{1}{4} ÷ \dfrac{1}{3}\Big)\\[1em] \Rightarrow \Big(\dfrac{1 \times 9}{4 \times 9} - \dfrac{1 \times 4}{9 \times 4}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{1}{4} \times \dfrac{3}{1}\Big)\\[1em] \Rightarrow \Big(\dfrac{9}{36} - \dfrac{4}{36}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{1 \times 3}{4 \times 1}\Big)\\[1em] \Rightarrow \Big(\dfrac{9 - 4}{36}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{1 \times 2}{2\times 2} + \dfrac{3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{2}{4} + \dfrac{3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{2 + 3}{4}\Big)\\[1em] \Rightarrow \Big(\dfrac{5}{36}\Big) ÷ \Big(\dfrac{5}{4}\Big)\\[1em] \Rightarrow \dfrac{5}{36} \times \dfrac{4}{5}\\[1em] \Rightarrow \dfrac{5 \times 4}{36 \times 5}\\[1em] \Rightarrow \dfrac{4}{36}\\[1em] \Rightarrow \dfrac{1}{9}.$

Hence, $\Big(\dfrac{1}{4} - \dfrac{1}{9}\Big) ÷ \Big(\dfrac{1}{2} + \dfrac{1}{4} ÷ \dfrac{1}{3}\Big) = \dfrac{1}{9}$.

Simplify :

$3\dfrac{7}{8} - \left\lbrace1\dfrac{3}{8} \div \Big(2\dfrac{4}{5} - 1\dfrac{7}{10}\Big)\right\rbrace$

Answer

$\Rightarrow 3\dfrac{7}{8} - \left\lbrace1\dfrac{3}{8} \div \Big(2\dfrac{4}{5} - 1\dfrac{7}{10}\Big)\right\rbrace\\[1em] \Rightarrow 3\dfrac{7}{8} - \left\lbrace1\dfrac{3}{8} \div \Big(\dfrac{14}{5} - \dfrac{17}{10}\Big)\right\rbrace\\[1em] \Rightarrow 3\dfrac{7}{8} - \left\lbrace1\dfrac{3}{8} \div \Big(\dfrac{14 \times 2}{5 \times 2} - \dfrac{17}{10}\Big)\right\rbrace\\[1em] \Rightarrow 3\dfrac{7}{8} - \left\lbrace1\dfrac{3}{8} \div \Big(\dfrac{28}{10} - \dfrac{17}{10}\Big)\right\rbrace\\[1em] \Rightarrow 3\dfrac{7}{8} - \left\lbrace1\dfrac{3}{8} \div \Big(\dfrac{28 - 17}{10}\Big)\right\rbrace\\[1em] \Rightarrow 3\dfrac{7}{8} - \left\lbrace1\dfrac{3}{8} \div \Big(\dfrac{11}{10}\Big)\right\rbrace\\[1em] \Rightarrow 3\dfrac{7}{8} - \left\lbrace\dfrac{11}{8} \div \dfrac{11}{10}\right\rbrace\\[1em] \Rightarrow 3\dfrac{7}{8} - \left\lbrace\dfrac{11}{8} \times \dfrac{10}{11}\right\rbrace\\[1em] \Rightarrow 3\dfrac{7}{8} - \dfrac{10}{8}\\[1em] \Rightarrow \dfrac{31}{8} - \dfrac{10}{8}\\[1em] \Rightarrow \dfrac{31 - 10}{8}\\[1em] \Rightarrow \dfrac{21}{8}\\[1em] \Rightarrow 2\dfrac{5}{8}$

Hence, $3\dfrac{7}{8} - \left\lbrace1\dfrac{3}{8} \div \Big(2\dfrac{4}{5} - 1\dfrac{7}{10}\Big)\right\rbrace = 2\dfrac{5}{8}$.

Simplify :

$3 \div \Big[3\times \left\lbrace3 - \Big(3 - \dfrac{1}{4}\Big)\right\rbrace \Big]$

Answer

$\Rightarrow 3 \div \Big[3\times \left\lbrace3 - \Big(3 - \dfrac{1}{4}\Big)\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace3 - \Big(\dfrac{3}{1} - \dfrac{1}{4}\Big)\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace3 - \Big(\dfrac{3 \times 4}{1 \times 4} - \dfrac{1}{4}\Big)\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace3 - \Big(\dfrac{12}{4} - \dfrac{1}{4}\Big)\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace3 - \Big(\dfrac{12 - 1}{4}\Big)\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace3 - \Big(\dfrac{11}{4}\Big)\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace\dfrac{3}{1} - \dfrac{11}{4}\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace\dfrac{3 \times 4}{1 \times 4} - \dfrac{11}{4}\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace\dfrac{12}{4} - \dfrac{11}{4}\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace\dfrac{12 - 11}{4}\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[3\times \left\lbrace\dfrac{1}{4}\right\rbrace \Big]\\[1em] \Rightarrow 3 \div \Big[\dfrac{3 \times 1}{4} \Big]\\[1em] \Rightarrow 3 \div \dfrac{3}{4}\\[1em] \Rightarrow 3 \times \dfrac{4}{3}\\[1em] \Rightarrow \dfrac{4 \times 3}{3}\\[1em] \Rightarrow 4$

Hence, $3 \div \Big[3\times \left\lbrace3 - \Big(3 - \dfrac{1}{4}\Big)\right\rbrace \Big] = 4$.

Simplify :

$5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3}{4} - \dfrac{1}{2} \times \Big(\dfrac{7}{10} - \dfrac{3}{5}\Big)\right\rbrace\Big]$

Answer

$\Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3}{4} - \dfrac{1}{2} \times \Big(\dfrac{7}{10} - \dfrac{3}{5}\Big)\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3}{4} - \dfrac{1}{2} \times \Big(\dfrac{7}{10} - \dfrac{3 \times 2}{5 \times 2}\Big)\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3}{4} - \dfrac{1}{2} \times \Big(\dfrac{7}{10} - \dfrac{6}{10}\Big)\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3}{4} - \dfrac{1}{2} \times \Big(\dfrac{7 - 6}{10}\Big)\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3}{4} - \dfrac{1}{2} \times \dfrac{1}{10}\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3}{4} - \dfrac{1 \times 1}{2 \times 10}\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3}{4} - \dfrac{1}{20}\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3 \times 5}{4 \times 5} - \dfrac{1}{20}\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{15}{20} - \dfrac{1}{20}\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{15 - 1}{20}\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{14}{20}\right\rbrace\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[\dfrac{7}{3} \div \dfrac{14}{20}\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[\dfrac{7}{3} \times \dfrac{20}{14}\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[\dfrac{7 \times 20}{3 \times 14}\Big]\\[1em] \Rightarrow 5\dfrac{1}{3} - \Big[\dfrac{140}{42}\Big]\\[1em] \Rightarrow \dfrac{16}{3} - \dfrac{10}{3}\\[1em] \Rightarrow \dfrac{16 - 10}{3}\\[1em] \Rightarrow \dfrac{6}{3}\\[1em] \Rightarrow 2$

Hence, $5\dfrac{1}{3} - \Big[2\dfrac{1}{3} \div \left\lbrace\dfrac{3}{4} - \dfrac{1}{2} \times \Big(\dfrac{7}{10} - \dfrac{3}{5}\Big)\right\rbrace\Big] = 2$.