Question

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

Answer 1

$\left(i\right)$ $\frac{-2}{3}\times \frac{3}{5}-\frac{3}{5}\times \frac{1}{6}+\frac{5}{2}$

Using the commutative property, we can write the given expression as:

$\frac{-3}{5}\times \frac{2}{3}-\frac{3}{5}\times \frac{1}{6}+\frac{5}{2}$

Now, using the distributive property of multiplication over addition,

$\left(\begin{array}{c}\frac{-3}{5}\end{array}\right)\left(\begin{array}{c}\frac{2}{3}+\frac{1}{6}\end{array}\right)+\frac{5}{2}$

$\Rightarrow \left(\frac{-3}{5}\right)\left(\frac{5}{6}\right)+\frac{5}{2}=\frac{-1}{2}+\frac{5}{2}$

$\Rightarrow \frac{4}{2}=2$

Hence, using appropriate properties, $\frac{-2}{3}\times \frac{3}{5}-\frac{3}{5}\times \frac{1}{6}+\frac{5}{2}=2$

$\left(ii\right)$ $\frac{2}{5}\times \left(\begin{array}{c}\frac{-3}{7}\end{array}\right)-\frac{1}{6}\times \frac{3}{2}+\frac{1}{14}\times \frac{2}{5}$

Using the commutative property, we can write the same expression as:

$\frac{2}{5}\times \left(\begin{array}{c}\frac{-3}{7}\end{array}\right)+\frac{1}{14}\times \frac{2}{5}-\frac{1}{6}\times \frac{3}{2}$

Now, using the distributive property of multiplication over addition, we get:

$\frac{2}{5}\times \left(\begin{array}{c}\frac{-3}{7}+\frac{1}{14}\end{array}\right)-\frac{1}{4}$

$\Rightarrow \frac{2}{5}\times \left(\begin{array}{c}\frac{-5}{14}\end{array}\right)-\frac{1}{4}$

$\Rightarrow -\frac{1}{7}-\frac{1}{4}=\frac{-11}{28}$

Hence, using appropriate properties, $\frac{2}{5}\times \left(\begin{array}{c}\frac{-3}{7}\end{array}\right)-\frac{1}{6}\times \frac{3}{2}+\frac{1}{14}\times \frac{2}{5}=\frac{-11}{28}$