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Rational Numbers — MATHS STD 8 — Question

Gujarat BoardEnglish MediumSTD 8MATHSRational Numbers3 Marks
Question
Find the following expression using the appropriate property.
(i) $0 \div\left(\frac{2}{3} \times \frac{9}{16}\right)$
(ii) $\frac{16}{9} \times\left(\frac{-14}{17}\right) \times\left(\frac{-27}{4}\right) \times \frac{51}{49}$
(iii) $\frac{4}{9} \times \frac{19}{20}-\frac{4}{9} \times \frac{1}{20}$
(iv) $\frac{5}{2} \times \frac{1}{10}+\frac{2}{7}-\frac{9}{4} \times \frac{1}{10}$
✓

Answer

(i) $0 \div\left(\frac{2}{3} \times \frac{9}{16}\right)$
Since, $\frac{2}{3}$ and $\frac{9}{16}$ are rational numbers. So, $\frac{2}{3} \times \frac{9}{16}$ will be a rational number.
$[\because$ product of 2 rational numbers is also a rational number]
$\because 0 \div\left(\frac{p}{q}\right)=0, \quad$ [where $\frac{p}{q}$ is a rational number]
and when we divide 0 by any non-zero rational number, we get 0
$\Rightarrow 0 \div\left(\frac{2}{3} \times \frac{9}{16}\right)=0$
(ii) $\frac{16}{9} \times\left(\frac{-14}{17}\right) \times\left(\frac{-27}{4}\right) \times \frac{51}{49}$
$=\frac{16}{9} \times\left[\left(\frac{-14}{17}\right) \times\left(\frac{-27}{4}\right)\right] \times \frac{51}{49}$
$=\frac{16}{9} \times\left[\left(\frac{-27}{4}\right) \times\left(\frac{-14}{17}\right)\right] \times \frac{51}{49}\quad$ [by commutative property]
$=\frac{16}{9} \times\left(\frac{-27}{4}\right) \times\left[\left(\frac{-14}{17}\right) \times \frac{51}{49}\right]\quad$ [by associative property]
$=\frac{16 \times(-27)}{9 \times 4} \times \frac{(-14) \times 51}{17 \times 49}=\frac{4 \times(-3)}{1} \times \frac{(-2) \times 3}{7}=\frac{72}{7}$
(iii) $\frac{4}{9} \times \frac{19}{20}-\frac{4}{9} \times \frac{1}{20}=\frac{4}{9} \times\left(\frac{19}{20}-\frac{1}{20}\right)\quad$ [by distributive property over subtraction of rational numbers]
$=\frac{4}{9} \times\left(\frac{19-1}{20}\right)$
$=\frac{4}{9} \times \frac{18}{20}=\frac{4 \times 18}{9 \times 20}=\frac{2}{5}$
(iv) $\frac{5}{2} \times \frac{1}{10}+\frac{2}{7}-\frac{9}{4} \times \frac{1}{10}$
$=\frac{5}{2} \times \frac{1}{10}+\left[\frac{2}{7}+\left(\frac{-9}{4}\right) \times \frac{1}{10}\right]$
$=\frac{5}{2} \times \frac{1}{10}+\left[\left(\frac{-9}{4}\right) \times \frac{1}{10}+\frac{2}{7}\right]\quad$ [by commutative property for addition of rational numbers]
$=\left[\frac{5}{2} \times \frac{1}{10}+\left(\frac{-9}{4}\right) \times \frac{1}{10}\right]+\frac{2}{7}\quad$ [by associative property]
$=\left(\frac{10-9}{4 \times 10}\right)+\frac{2}{7}$
$=\frac{1}{40}+\frac{2}{7}=\frac{7+40 \times 2}{40 \times 7}=\frac{87}{280}$

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