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Using appropriate properties, find$\left( i \right){\text{ }} - \dfrac{2}{3} \times \dfrac{3}{5} + \dfrac{5}{2} - \dfrac{3}{5} \times \dfrac{1}{6} \\ \left( {ii} \right){\text{ }}\dfrac{2}{5} \times \left( { - \dfrac{3}{7}} \right) - \dfrac{1}{6} \times \dfrac{3}{2} + \dfrac{1}{{14}} \times \dfrac{2}{5} \\$

Last updated date: 17th Sep 2024
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$\left( i \right){\text{ }} - \dfrac{2}{3} \times \dfrac{3}{5} + \dfrac{5}{2} - \dfrac{3}{5} \times \dfrac{1}{6} \\ = - \dfrac{2}{3} \times \dfrac{3}{5} - \dfrac{3}{5} \times \dfrac{1}{6} + \dfrac{5}{2} \\$
Taking $\dfrac{3}{5}$common, we get
$= \dfrac{3}{5}\left( { - \dfrac{2}{3} - \dfrac{1}{6}} \right) + \dfrac{5}{2} \\ = \dfrac{3}{5}\left( {\dfrac{{ - 2 \times 2 - 1}}{6}} \right) + \dfrac{5}{2} \\ = \dfrac{3}{5}\left( {\dfrac{{ - 5}}{6}} \right) + \dfrac{5}{2} \\ = \dfrac{{ - 1}}{2} + \dfrac{5}{2} \\ = \dfrac{4}{2} \\ = 2 \\$
$\left( {ii} \right){\text{ }}\dfrac{2}{5} \times \left( { - \dfrac{3}{7}} \right) - \dfrac{1}{6} \times \dfrac{3}{2} + \dfrac{1}{{14}} \times \dfrac{2}{5} \\ = \dfrac{2}{5} \times \left( { - \dfrac{3}{7}} \right) + \dfrac{1}{{14}} \times \dfrac{2}{5} - \dfrac{1}{6} \times \dfrac{3}{2} \\$
Taking $\dfrac{2}{5}$common from above equation, we get
$= \dfrac{2}{5} \times \left( { - \dfrac{3}{7} + \dfrac{1}{{14}}} \right) - \dfrac{1}{6} \times \dfrac{3}{2} \\ = \dfrac{2}{5} \times \left( {\dfrac{{ - 3 \times 2 + 1}}{{14}}} \right) - \dfrac{1}{6} \times \dfrac{3}{2} \\ = \dfrac{2}{5} \times \left( {\dfrac{{ - 5}}{{14}}} \right) - \dfrac{1}{6} \times \dfrac{3}{2} \\ = \dfrac{{ - 1}}{7} - \dfrac{1}{2} \times \dfrac{1}{2} \\ = \dfrac{{ - 1}}{7} - \dfrac{1}{4} \\ = \dfrac{{ - 4 - 7}}{{7 \times 4}} \\ = \dfrac{{ - 11}}{{28}} \\$