question_answer 1)

                Using appropriate properties find:                 (i) \[-\frac{2}{3}\times \frac{3}{5}\,+\frac{5}{2}\,-\frac{3}{5}\times \frac{1}{6}\]                                 (ii) \[\frac{2}{5}\times \,\left( -\frac{3}{7} \right)\,-\frac{1}{6}\times \frac{3}{2}\,+\frac{1}{14}\,\times \frac{2}{5}\]                

Answer:

                (i) \[-\frac{2}{3}\times \frac{3}{5}\,+\frac{5}{2}\,-\frac{3}{5}\times \frac{1}{6}\] \[-\frac{2}{3}\times \frac{3}{5}\,+\frac{5}{2}\,-\frac{3}{5}\times \frac{1}{6}\] \[=\frac{3}{5}\times \frac{-2}{3}+\frac{5}{2}-\frac{3}{5}\times \frac{1}{6}\]                          | by commutativity \[=\frac{3}{5}\times \frac{-2}{3}-\frac{3}{5}\times \,\frac{1}{6}+\frac{5}{2}\]                       | by associativity \[=\frac{3}{5}\times \,\left( \frac{-2}{3}-\frac{1}{6} \right)\,+\frac{5}{2}\]                             | by distributivity \[=\frac{3}{5}\times \left( \frac{-4-1}{6} \right)+\frac{5}{2}\] \[=\frac{3}{5}\times \,\left( \frac{-5}{6} \right)+\frac{5}{2}\] \[=\frac{-1}{2}\,\,+\frac{5}{2}\,=\frac{(-1)+5}{2}\,=\frac{4}{2}=2\] (ii) \[\frac{2}{5}\times \,\left( -\frac{3}{7} \right)\,-\frac{1}{6}\times \frac{3}{2}\,+\frac{1}{14}\,\times \frac{2}{5}\] \[\frac{2}{5}\times \,\left( -\frac{3}{7} \right)\,-\frac{1}{6}\times \frac{3}{2}\,+\frac{1}{14}\,\times \frac{2}{5}\] \[=\frac{2}{5}\times \,\left( -\frac{3}{7} \right)\,-\frac{1}{6}\times \frac{3}{2}\,+\frac{2}{5}\times \,\frac{1}{14}\]  | by commutativity    \[=\frac{2}{5}\times \,\left( -\frac{3}{7} \right)\,+\frac{2}{5}\,\times \frac{1}{14}\,-\frac{1}{6}\,\times \frac{3}{2}\]       | by associativity \[=\frac{2}{5}\times \left\{ \left( -\frac{3}{7} \right)+\frac{1}{14} \right\}-\frac{1}{6}\times \frac{3}{2}\] I by distributivity \[=\frac{2}{5}\times \,\left\{ \frac{(-6)+1}{14} \right\}\,-\frac{1}{6}\times \frac{3}{2}\] \[=\frac{2}{5}\times \left\{ \frac{-5}{14} \right\}-\frac{1}{6}\times \frac{3}{2}\,=\frac{-1}{7}-\frac{1}{4}\] \[=\frac{-4-7}{28}\,=\frac{-11}{28}\].