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Solve the following using the suitable property.$\frac{-16}{7} \times (\frac{-8}{9} + \frac{-7}{6}) = (\frac{-16}{7} \times \frac{-8}{9}) + (\frac{-16}{7} \times \frac{-7}{6})$
Given :
The given expression is $\frac{-16}{7} \times (\frac{-8}{9} + \frac{-7}{6}) = (\frac{-16}{7} \times \frac{-8}{9}) + (\frac{-16}{7} \times \frac{-7}{6})$.
To do :
We have to solve the given expressions using a suitable property.
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:
$a (b+c) = a\times b + a\times c$
$a (b-c) = a\times b - a\times c$
$\frac{-16}{7} \times (\frac{-8}{9} + \frac{-7}{6}) = (\frac{-16}{7} \times \frac{-8}{9}) + (\frac{-16}{7} \times \frac{-7}{6})$ (Distributive property)
LHS :
$\frac{-16}{7} \times (\frac{-8}{9} + \frac{-7}{6})= \frac{-16}{7} \times (\frac{-8\times 6}{9 \times 6} + \frac{-7 \times 9}{6 \times 9})$
$ = \frac{-16}{7} \times (\frac{-48}{54} + \frac{-63}{54})$
$= \frac{-16}{7} \times (\frac{-48-63}{54})$
$= \frac{-16}{7} \times \frac{-111}{54} $
$ = \frac{8}{7} \times \frac{111}{27}$
$ = \frac{8}{7} \times \frac{37}{9}$
$= \frac{296}{63}$.
RHS:
$\frac{-16}{7} \times (\frac{-8\times 6}{9 \times 6} + \frac{-7 \times 9}{6 \times 9}) = \frac{(16\times8)}{(7\times9)} + \frac{(8\times1)}{(1\times3)}$
$ = \frac{128}{63} + \frac{8}{3}$
$= \frac{(128+8\times21)}{63}$
$ = \frac{(128+168)}{63}$
$= \frac{296}{63}$.
Therefore, LHS $=$ RHS.
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