question_answer

Using appropriates find

(a) \[-\frac{2}{3}\times \frac{3}{5}+\frac{5}{2}-\frac{3}{5}\times \frac{1}{6}\]

(b) \[\frac{2}{5}\times \left( -\frac{3}{7} \right)-\frac{1}{6}\times \frac{3}{2}+\frac{1}{14}\times \frac{2}{5}\]

Answer:

(a) \[-\frac{2}{3}\times \frac{3}{5}+\frac{5}{2}-\frac{3}{5}\times \frac{1}{6}=-\frac{2}{3}\times \frac{3}{5}-\frac{3}{5}\times \frac{1}{6}+\frac{5}{2}\]

\[=\frac{3}{5}\left( -\frac{2}{3}-\frac{1}{6} \right)+\frac{5}{2}\]  (by commutativity)

\[=\frac{3}{5}\left( \frac{-4-1}{6} \right)+\frac{5}{2}\]

\[=\frac{3}{5}\times \frac{-5}{6}+\frac{5}{2}\]

\[=\frac{-1}{2}+\frac{5}{2}=2\]

(b) \[\frac{2}{5}\times \left( \frac{-3}{7} \right)-\frac{1}{6}\times \frac{3}{2}+\frac{1}{14}\times \frac{2}{5}=\frac{2}{5}\times \left( \frac{-3}{7} \right)+\frac{1}{14}\times \frac{2}{5}-\frac{1}{6}\times \frac{3}{2}\] (by associativity)

\[=\frac{2}{5}\times \left( \frac{-3}{7}+\frac{1}{14} \right)-\frac{1}{4}\] (by distributivity)

\[=\frac{2}{5}\left( \frac{-6+1}{14} \right)-\frac{1}{4}\]

\[=\frac{2}{5}\times \frac{-5}{14}-\frac{1}{4}\]

\[=-\frac{1}{7}-\frac{1}{4}=\frac{-4-7}{28}=\frac{-11}{28}\]