Fractions — Exercise 3(D)

Simplify:

$6 + \Bigg\lbrace\dfrac{4}{3} + \left(\dfrac{3}{4} - \dfrac{1}{3}\right)\Bigg\rbrace$

Answer

Simplifying,

$6 + \Bigg\lbrace\dfrac{4}{3} + \left(\dfrac{3}{4} - \dfrac{1}{3}\right)\Bigg\rbrace \\[1em] = 6 + \Bigg\lbrace\dfrac{4}{3} + \left(\dfrac{9 - 4}{12}\right)\Bigg\rbrace \\[1em] = 6 + \Bigg\lbrace\dfrac{4}{3} + \dfrac{5}{12}\Bigg\rbrace \\[1em] = 6 + \Bigg\lbrace\dfrac{16}{12} + \dfrac{5}{12}\Bigg\rbrace \\[1em] = 6 + \dfrac{21}{12} \\[1em] = 6 + \dfrac{7}{4} \\[1em] = \dfrac{24 + 7}{4} = \dfrac{31}{4} = 7\dfrac{3}{4}$

Hence, the required value is $7\dfrac{3}{4}$.

Simplify:

$8 - \Bigg\lbrace\dfrac{3}{2} + \left(\dfrac{3}{5} - \dfrac{1}{2}\right)\Bigg\rbrace$

Answer

Simplifying,

$8 - \Bigg\lbrace\dfrac{3}{2} + \left(\dfrac{3}{5} - \dfrac{1}{2}\right)\Bigg\rbrace \\[1em] = 8 - \Bigg\lbrace\dfrac{3}{2} + \left(\dfrac{6 - 5}{10}\right)\Bigg\rbrace \\[1em] = 8 - \Bigg\lbrace\dfrac{3}{2} + \dfrac{1}{10}\Bigg\rbrace \\[1em] = 8 - \Bigg\lbrace\dfrac{15}{10} + \dfrac{1}{10}\Bigg\rbrace \\[1em] = 8 - \dfrac{16}{10} \\[1em] = 8 - \dfrac{8}{5} \\[1em] = \dfrac{40 - 8}{5} \\[1em] = \dfrac{32}{5} = 6\dfrac{2}{5}$

Hence, the required value is $6\dfrac{2}{5}$.

Simplify:

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

Answer

Simplifying,

$\dfrac{1}{4}\left(\dfrac{1}{4} + \dfrac{1}{3}\right) - \dfrac{2}{5} \\[1em] = \dfrac{1}{4}\left(\dfrac{3 + 4}{12}\right) - \dfrac{2}{5} \\[1em] = \dfrac{1}{4} \times \dfrac{7}{12} - \dfrac{2}{5} \\[1em] = \dfrac{7}{48} - \dfrac{2}{5} \\[1em] = \dfrac{7 \times 5}{240} - \dfrac{2 \times 48}{240} \\[1em] = \dfrac{35 - 96}{240} = -\dfrac{61}{240}$

Hence, the required value is $-\dfrac{61}{240}$.

Simplify:

$2\dfrac{3}{4} - \left[3\dfrac{1}{8} \div \Bigg\lbrace 5 - \left(4\dfrac{2}{3} - \dfrac{11}{12}\right)\Bigg\rbrace\right]$

Answer

Simplifying,

$2\dfrac{3}{4} - \left[3\dfrac{1}{8} \div \Bigg\lbrace 5 - \left(4\dfrac{2}{3} - \dfrac{11}{12}\right)\Bigg\rbrace\right] \\[1em] = 2\dfrac{3}{4} - \left[3\dfrac{1}{8} \div \Bigg\lbrace 5 - \left(\dfrac{14}{3} - \dfrac{11}{12}\right)\Bigg\rbrace\right] \\[1em] = 2\dfrac{3}{4} - \left[3\dfrac{1}{8} \div \Bigg\lbrace 5 - \left(\dfrac{56 - 11}{12}\right)\Bigg\rbrace\right] \\[1em] = 2\dfrac{3}{4} - \left[3\dfrac{1}{8} \div \Bigg\lbrace 5 - \dfrac{45}{12}\Bigg\rbrace\right] \\[1em] = 2\dfrac{3}{4} - \left[3\dfrac{1}{8} \div \Bigg\lbrace 5 - \dfrac{15}{4}\Bigg\rbrace\right] \\[1em] = 2\dfrac{3}{4} - \left[3\dfrac{1}{8} \div \left(\dfrac{20 - 15}{4}\right)\right] \\[1em] = 2\dfrac{3}{4} - \left[3\dfrac{1}{8} \div \dfrac{5}{4}\right] \\[1em] = 2\dfrac{3}{4} - \left[\dfrac{25}{8} \times \dfrac{4}{5}\right] \\[1em] = 2\dfrac{3}{4} - \dfrac{5}{2} \\[1em] = \dfrac{11}{4} - \dfrac{10}{4} \\[1em] = \dfrac{1}{4}$

Hence, the required value is $\dfrac{1}{4}$.

Simplify:

$12\dfrac{1}{2} - \left[8\dfrac{1}{2} + \Bigg\lbrace 9 - \left(5 - \overline{3 - 2}\right)\Bigg\rbrace\right]$

Answer

Simplifying,

$12\dfrac{1}{2} - \left[8\dfrac{1}{2} + \Bigg\lbrace 9 - \left(5 - \overline{3 - 2}\right)\Bigg\rbrace\right] \\[1em] = 12\dfrac{1}{2} - \left[8\dfrac{1}{2} + \Bigg\lbrace 9 - \left(5 - 1\right)\Bigg\rbrace\right] \\[1em] = 12\dfrac{1}{2} - \left[8\dfrac{1}{2} + \Bigg\lbrace 9 - 4\Bigg\rbrace\right] \\[1em] = 12\dfrac{1}{2} - \left[8\dfrac{1}{2} + 5\right] \\[1em] = 12\dfrac{1}{2} - 13\dfrac{1}{2} \\[1em] = \dfrac{25}{2} - \dfrac{27}{2} \\[1em] = -\dfrac{2}{2} = -1$

Hence, the required value is -1.

Simplify:

$1\dfrac{1}{5} \div \Bigg\lbrace 2\dfrac{1}{3} - \left(5 + \overline{2 - 3}\right)\Bigg\rbrace - 3\dfrac{1}{2}$

Answer

Simplifying,

$1\dfrac{1}{5} \div \Bigg\lbrace 2\dfrac{1}{3} - \left(5 + \overline{2 - 3}\right)\Bigg\rbrace - 3\dfrac{1}{2} \\[1em] = 1\dfrac{1}{5} \div \Bigg\lbrace 2\dfrac{1}{3} - \left(5 + (-1)\right)\Bigg\rbrace - 3\dfrac{1}{2} \\[1em] = 1\dfrac{1}{5} \div \Bigg\lbrace 2\dfrac{1}{3} - 4\Bigg\rbrace - 3\dfrac{1}{2} \\[1em] = 1\dfrac{1}{5} \div \left(\dfrac{7}{3} - \dfrac{12}{3}\right) - 3\dfrac{1}{2} \\[1em] = 1\dfrac{1}{5} \div \left(-\dfrac{5}{3}\right) - 3\dfrac{1}{2} \\[1em] = \dfrac{6}{5} \times \left(-\dfrac{3}{5}\right) - 3\dfrac{1}{2} \\[1em] = -\dfrac{18}{25} - \dfrac{7}{2} \\[1em] = -\dfrac{36}{50} - \dfrac{175}{50} \\[1em] = -\dfrac{211}{50} = -4\dfrac{11}{50}$

Hence, the required value is $-4\dfrac{11}{50}$.

Simplify:

$\left(\dfrac{1}{2} + \dfrac{2}{3}\right) \div \left(\dfrac{3}{4} - \dfrac{2}{9}\right)$

Answer

Simplifying,

$\left(\dfrac{1}{2} + \dfrac{2}{3}\right) \div \left(\dfrac{3}{4} - \dfrac{2}{9}\right) \\[1em] = \left(\dfrac{3 + 4}{6}\right) \div \left(\dfrac{27 - 8}{36}\right) \\[1em] = \dfrac{7}{6} \div \dfrac{19}{36} \\[1em] = \dfrac{7}{6} \times \dfrac{36}{19} \\[1em] = \dfrac{7 \times 36}{6 \times 19} = \dfrac{252}{114} \\[1em] = \dfrac{42}{19} = 2\dfrac{4}{19}$

Hence, the required value is $2\dfrac{4}{19}$.

Simplify:

$\dfrac{6}{5} \text{ of } \left(3\dfrac{1}{3} - 2\dfrac{1}{2}\right) \div \left(2\dfrac{5}{21} - 2\right)$

Answer

Simplifying,

$\dfrac{6}{5} \text{ of } \left(3\dfrac{1}{3} - 2\dfrac{1}{2}\right) \div \left(2\dfrac{5}{21} - 2\right) \\[1em] = \dfrac{6}{5} \text{ of } \left(\dfrac{10}{3} - \dfrac{5}{2}\right) \div \left(\dfrac{47}{21} - 2\right) \\[1em] = \dfrac{6}{5} \text{ of } \left(\dfrac{20 - 15}{6}\right) \div \left(\dfrac{47 - 42}{21}\right) \\[1em] = \dfrac{6}{5} \text{ of } \dfrac{5}{6} \div \dfrac{5}{21} \\[1em] = \dfrac{6}{5} \times \dfrac{5}{6} \div \dfrac{5}{21} \\[1em] = 1 \div \dfrac{5}{21} \\[1em] = 1 \times \dfrac{21}{5} = \dfrac{21}{5} = 4\dfrac{1}{5}$

Hence, the required value is $4\dfrac{1}{5}$.

Simplify:

$10\dfrac{1}{8} \text{ of } \dfrac{4}{5} \div \dfrac{35}{36} \text{ of } \dfrac{20}{49}$

Answer

Simplifying,

$10\dfrac{1}{8} \text{ of } \dfrac{4}{5} \div \dfrac{35}{36} \text{ of } \dfrac{20}{49} \\[1em] = \left(\dfrac{81}{8} \times \dfrac{4}{5}\right) \div \left(\dfrac{35}{36} \times \dfrac{20}{49}\right) \\[1em] = \dfrac{81}{10} \div \dfrac{25}{63} \\[1em] = \dfrac{81}{10} \times \dfrac{63}{25} \\[1em] = \dfrac{81 \times 63}{10 \times 25} = \dfrac{5103}{250} \\[1em] = 20\dfrac{103}{250}$

Hence, the required value is $20\dfrac{103}{250}$.

Simplify:

$5\dfrac{3}{4} - \dfrac{3}{7} \times 15\dfrac{3}{4} + 2\dfrac{2}{35} \div 1\dfrac{11}{25}$

Answer

Simplifying,

$5\dfrac{3}{4} - \dfrac{3}{7} \times 15\dfrac{3}{4} + 2\dfrac{2}{35} \div 1\dfrac{11}{25} \\[1em] = 5\dfrac{3}{4} - \dfrac{3}{7} \times \dfrac{63}{4} + \dfrac{72}{35} \div \dfrac{36}{25} \\[1em] = 5\dfrac{3}{4} - \dfrac{3}{7} \times \dfrac{63}{4} + \dfrac{72}{35} \times \dfrac{25}{36} \\[1em] = 5\dfrac{3}{4} - \dfrac{27}{4} + \dfrac{10}{7} \\[1em] = \dfrac{23}{4} - \dfrac{27}{4} + \dfrac{10}{7} \\[1em] = -\dfrac{4}{4} + \dfrac{10}{7} \\[1em] = -1 + \dfrac{10}{7} \\[1em] = \dfrac{-7 + 10}{7} = \dfrac{3}{7}$

Hence, the required value is $\dfrac{3}{7}$.

Simplify:

$\dfrac{3}{4} \text{ of } 7\dfrac{3}{7} - 5\dfrac{3}{5} \div 3\dfrac{4}{15}$

Answer

Simplifying,

$\dfrac{3}{4} \text{ of } 7\dfrac{3}{7} - 5\dfrac{3}{5} \div 3\dfrac{4}{15} \\[1em] = \dfrac{3}{4} \times \dfrac{52}{7} - \dfrac{28}{5} \div \dfrac{49}{15} \\[1em] = \dfrac{3}{4} \times \dfrac{52}{7} - \dfrac{28}{5} \times \dfrac{15}{49} \\[1em] = \dfrac{39}{7} - \dfrac{12}{7} \\[1em] = \dfrac{27}{7} = 3\dfrac{6}{7}$

Hence, the required value is $3\dfrac{6}{7}$.