Use the distributivity of multiplication of rational numbers over their addition to simplify:
(i) $ \frac{3}{5} \times\left(\frac{35}{24}+\frac{10}{1}\right) $
(ii) $ \frac{-5}{4} \times\left(\frac{8}{5}+\frac{16}{5}\right) $
(iii) $ \frac{2}{7} \times\left(\frac{7}{16}-\frac{21}{4}\right) $
(iv) $ \frac{3}{4} \times\left(\frac{8}{9}-40\right) $

To do:

We have to use the distributivity of multiplication of rational numbers over their addition to simplify the given expressions.

Solution:

According to distributive property of multiplication,

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

Therefore,

(i) $\frac{3}{5} \times(\frac{35}{24}+\frac{10}{1})=\frac{3}{5} \times \frac{35}{24}+\frac{3}{5} \times \frac{10}{1}$

$=\frac{3 \times 35}{5 \times 24}+\frac{3 \times 10}{5 \times 1}$

$=\frac{1 \times 7}{1 \times 8}+\frac{3 \times 2}{1 \times 1}$

$=\frac{7}{8}+6$

$=\frac{7+6\times8}{8}$

$=\frac{7+48}{8}$

$=\frac{55}{8}$

(ii) $\frac{-5}{4} \times(\frac{8}{5}+\frac{16}{5})=\frac{-5}{4} \times \frac{8}{5}+\frac{-5}{4} \times \frac{16}{5}$

$=\frac{-5 \times 8}{4 \times 5}+\frac{-5 \times 16}{4 \times 5}$

$=\frac{-1 \times 2}{1 \times 1}+\frac{-1 \times 4}{1 \times 1}$

$=-2-4$

$=-6$

(iii) $\frac{2}{7} \times(\frac{7}{16}-\frac{21}{4})=\frac{2}{7} \times \frac{7}{16}-\frac{2}{7} \times \frac{21}{4}$

$=\frac{2 \times 7}{7 \times 16}-\frac{2 \times 21}{7 \times 4}$

$=\frac{1}{8}-\frac{3}{2}$

$=\frac{1-12}{8}$

$=\frac{-11}{8}$

(iv) $\frac{3}{4} \times(\frac{8}{9}-40)=\frac{3}{4} \times \frac{8}{9} - \frac{3}{4} \times \frac{40}{1}$

$=\frac{3 \times 8}{4 \times 9}-\frac{3 \times 40}{4 \times 1}$

$=\frac{2}{3}-\frac{30}{1}$

$=\frac{2-90}{3}$

$=\frac{-88}{3}$

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Updated on: 10-Oct-2022

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