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} $.

Given:

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

(ii) \( \frac{2}{5} \times\left(-\frac{3}{7}\right)-\frac{1}{6} \times \frac{3}{2}+\frac{1}{14} \times \frac{2}{5} \).

To do:

We have to use appropriate properties and find the values of the given expressions.

Solution:

Distributive Property:

The distributive property of multiplication states that when a factor is multiplied by the sum or difference of two terms, it is essential to multiply each of the two numbers by the factor, and finally perform the addition or subtraction operation.

This property is symbolically stated as:

$a (b+c) = a\times b + a\times c$

$a (b-c) = a\times b - a\times c$

Therefore,

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

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

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

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

$= \frac{-1}{1}\times\frac{1}{2} + \frac{5}{2}$

$= \frac{-1}{2} + \frac{5}{2}$

$= \frac{5-1}{2}$

$= \frac{4}{2}$

$= 2$. 

(ii) $\frac{2}{5} \times(-\frac{3}{7})-\frac{1}{6} \times \frac{3}{2}+\frac{1}{14} \times \frac{2}{5}=\frac{2}{5} \times (-\frac{3}{7})-\frac{1}{6} \times \frac{3}{2}+\frac{2}{5} \times \frac{1}{14}$

$=\frac{2}{5} \times (-\frac{3}{7})+\frac{2}{5} \times \frac{1}{14}-\frac{1}{6} \times \frac{3}{2}$

$=\frac{2}{5} \times [(-\frac{3}{7})+\frac{1}{14}]-\frac{1}{6} \times \frac{3}{2}$

$=\frac{2}{5} \times(\frac{-6+1}{14})-(\frac{1}{6} \times \frac{3}{2})$

$=\frac{2}{5} \times \frac{-5}{14}-\frac{1}{6} \times \frac{3}{2}$

$=\frac{-1}{7}-\frac{1}{4}$

$=\frac{-4-7}{28}$

$=\frac{-11}{28}$

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Updated on: 10-Oct-2022

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