Simplify each of the following by using suitable property. Also, …

Question

Simplify each of the following by using suitable property. Also, name the properties.
(i) $\left[\frac{1}{2} \times \frac{1}{4}\right]+\left[\frac{1}{2} \times 6\right]$
(ii) $\left[\frac{1}{5} \times \frac{2}{15}\right]-\left[\frac{1}{5} \times \frac{2}{5}\right]$
(iii) $\frac{-3}{5} \times\left[\frac{3}{7}+\left(\frac{-5}{6}\right)\right]$

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Answer

(i) $\left[\frac{1}{2} \times \frac{1}{4}\right]+\left[\frac{1}{2} \times 6\right]=\frac{1}{2} \times\left[\frac{1}{4}+6\right]\quad$ [by distributive property over addition]
$=\frac{1}{2} \times\left[\frac{1+24}{4}\right]=\frac{25}{8}$
(ii) $\left[\frac{1}{5} \times \frac{2}{15}\right]-\left[\frac{1}{5} \times \frac{2}{5}\right]=\left[\frac{1}{5} \times \frac{2}{15}\right]+\left[\frac{1}{5} \times\left(\frac{-2}{5}\right)\right]$
$=\frac{1}{5} \times\left[\frac{2}{15}-\frac{2}{5}\right]\quad$ [by distributive property over addition]
$=\frac{1}{5} \times\left[\frac{2-6}{15}\right]=\frac{-4}{75}$
(iii) $\frac{-3}{5} \times\left\{\frac{3}{7}+\left(\frac{-5}{6}\right)\right\}=\frac{-3}{5} \times \frac{3}{7}+\left(\frac{-3}{5}\right) \times\left(\frac{-5}{6}\right)\quad$ [by distributive property]
$=\frac{-9}{35}+\frac{15}{30}=\frac{-54+105}{210}=\frac{51}{210}=\frac{17}{70}$