S.chand publication New Learning Composite mathematics solution of class 8 Chapter 1 Rational numbers Exercise 1D

 Exercise 1D

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Q1 | Ex-1D |Class 8 |Rational Numbers | S.Chand | New Learning | Composite maths | myhelper

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Question 1

State the property used by the following statements.

(a) $-\frac{7}{8}\times \frac{11}{15}=\frac{11}{15}\times -\frac{7}{8}$

Sol :

$=\frac{-7}{8}\times \frac{11}{15}=\frac{-77}{120}$ , $\frac{11}{15}\times \frac{-7}{8}=\frac{-77}{120}$

$\frac{-7}{8}\times \frac{11}{15}=\frac{11}{15}\times\frac{-7}{8}$ (commutative property)

(b) $\frac{6}{7}\times \frac{7}{6}=1$

Sol :

(Multiplicative inverse)

(c) $-\frac{1}{8}+\frac{1}{8}=0$

Sol :

(Additive inverse)

(d) $-\frac{21}{29}+\frac{6}{19}=\frac{6}{19}+\left(\frac{-21}{29}\right)$

Sol :

⇒$\frac{-21}{29}+\frac{6}{19}=\frac{-399+126}{551}=\frac{-273}{551}$

⇒$\frac{6}{19}+\left(-\frac{21}{29}\right)=\frac{126-399}{551}=-\frac{273}{551}$

∴$-\frac{21}{29}+\frac{6}{19}=\frac{6}{19}+\left(-\frac{21}{29}\right)$

 (commutative property)

(e) $-\frac{23}{70}\times 1=-\frac{23}{70}$

Sol :

⇒$\frac{-23}{70}\times 1=\frac{-23}{70}$ (multiplicative identity)

(f) $\frac{5}{8}\left(\frac{1}{2}-\frac{1}{3}\right)=\frac{5}{8}\times \frac{1}{2}-\frac{5}{8}\times \frac{1}{3}$ 

Sol :

⇒$\frac{5}{8}\left(\frac{1}{2}-\frac{1}{3}\right)=\frac{5}{8}\left(\frac{3-2}{6}\right)$ $=\frac{5}{8}\times  \frac{1}{6}=\frac{5}{48}$

⇒$\frac{5}{8}\times \frac{1}{2}-\frac{5}{8}\times \frac{1}{3}=\frac{5}{16}-\frac{5}{24}=\frac{15-10}{48}$ $=\frac{5}{48}$

 (distributive property)

(g) $\left(\frac{1}{2}\times \frac{1}{3}\right)\times \frac{1}{5}=\frac{1}{2}\times \left(\frac{1}{3}\times \frac{1}{5}\right)$

Sol :

⇒$\left(\frac{1}{2}\times \frac{1}{3}\right)\times \frac{1}{5}=\frac{1}{6}\times \frac{1}{5}=\frac{1}{30}$

⇒$\frac{1}{2}\left(\frac{1}{3}\times \frac{1}{5})\right)=\frac{1}{2}\times \frac{1}{15}=\frac{1}{30}$

(Associative property)

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Q2 | Ex-1D |Class 8 |Rational Numbers | S.Chand | New Learning | Composite maths | myhelper

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Question 2

Verify the following and state the property used.

(a) $\frac{17}{135}\times \frac{15}{-51}=\frac{15}{-51}\times \frac{17}{135}$

Sol :

⇒$\frac{17}{135}\times \frac{15}{-51}=\frac{1}{-27}$

⇒$\frac{15}{-51}\times \frac{17}{135}=-\frac{1}{27}$

(commutative)

(b) $\frac{1}{9}\left(\frac{18}{5}+\frac{3}{20}\right)=\frac{1}{9}\times \frac{18}{5}+\frac{1}{9}\times \left(\frac{-3}{20}\right)$

Sol :

⇒$\frac{1}{9}\left(\frac{18}{5}+\frac{-3}{20}\right)=\frac{1}{9}\left(\frac{72-3}{20}\right)=\frac{1}{9}\times \frac{69}{20}=\frac{23}{60}$

⇒$\frac{1}{9}\times \frac{18}{5}+\frac{1}{9}\times \left(\frac{-3}{20}\right)=\frac{2}{5}-\frac{1}{60}$ $=\frac{24-1}{60}=\frac{23}{60}$

distributive 

(c) $\left(-\frac{1}{2}+\frac{3}{7}\right)+\left(-\frac{4}{3}\right)=-\frac{1}{2}+\left[\frac{3}{7}+\left(-\frac{4}{3}\right)\right]$

Sol :

⇒$\left(-\frac{1}{2}+\frac{3}{7}\right)+\left(-\frac{4}{3}\right)=\frac{-1}{2}+\frac{3}{7}-\frac{4}{3}$

$=\frac{-21+18-56}{42}=\frac{-59}{42}$

⇒$-\frac{1}{2}+\left[\frac{3}{7}+\left(-\frac{4}{3}\right)\right]=\frac{-1}{2}+\frac{3}{7}-\frac{4}{3}$

$=\frac{-21+18-56}{42}=\frac{-59}{42}$

(Associative property)

(d) $\left(\frac{5}{3}\times -\frac{4}{5}\right)\times \frac{3}{5}=\frac{5}{3}\times \left[\left(-\frac{4}{5}\times \frac{3}{5}\right)\right]$

Sol :

⇒$\left(\frac{5}{3}\times -\frac{4}{5}\right)\times \frac{3}{5}=\frac{-20}{15}\times \frac{3}{5}=-\frac{4}{5}$

⇒$\frac{5}{3}\times \left(-\frac{4}{5}\times \frac{3}{5}\right)=\frac{5}{3}\times \frac{-12}{25}=\frac{-4}{5}$

(Associative property)

(e) $-\frac{19}{20}\times 1=1\times \left(-\frac{19}{20}\right)=-\frac{19}{20}$

Sol :

(Commutative property)

also, 1 is multiplicative identity

(f) $-\frac{17}{24}\times \frac{24}{-17}=1$

Sol :

(Multiplicative inverse)

(g) $-\frac{2}{3}+0=0+\left(-\frac{2}{3}\right)=-\frac{2}{3}$

Sol :

0 is additive identity

(h) $\frac{1}{7}+0=0+\frac{1}{7}=\frac{1}{7}$

Sol :

0 is additive identity

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Q3 | Ex-1D |Class 8 |Rational Numbers | S.Chand | New Learning | Composite maths | myhelper

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Question 3

Taking, a = $\frac{4}{7}$,b = $-\frac{5}{2}$ and c =$\frac{4}{3}$, show that

a÷(b÷c)≠(a÷b)÷c, that is not associative for rational numbers.

Sol :

a÷(b÷c)$=\frac{4}{7}\div \left(-\frac{5}{2}\div \frac{4}{3}\right)$ 

$=\frac{4}{7}\div \left(-\frac{5}{2}\times \frac{3}{4}\right)=\frac{4}{7}\times \frac{8}{-15}=\frac{32}{-105}$

(a÷b)÷c$=\left[\frac{4}{7}\div \left(-\frac{5}{2}\right)\right]\div \frac{4}{3}$

$=\left(\frac{4}{7}\times \frac{2}{-5}\right)\div \frac{4}{3}=\frac{6}{-35}\times \frac{3}{4}=\frac{-6}{35}$

∴a÷(b÷c)≠(a÷b)÷c

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Q4 | Ex-1D |Class 8 |Rational Numbers | S.Chand | New Learning | Composite maths | myhelper

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Question 4

Using distributivity, find $\frac{8}{15}\times \left(-\frac{7}{18}\right)+\left(\frac{8}{15}\times -\frac{11}{18}\right)$

Sol :

⇒$\frac{8}{15}\left[\frac{-7}{18}+\frac{-11}{18}\right]$

⇒$=\frac{8}{15}\left[\frac{-7-11}{18}\right]$

$=\frac{8}{15}\times \frac{-18}{18}=\frac{-8}{15}$

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Q5 | Ex-1D |Class 8 |Rational Numbers | S.Chand | New Learning | Composite maths | myhelper

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Question 5

Find the additive inverse of each of the following.

(a) $\frac{5}{16}$ 

(b) $\frac{-15}{-16}$ 

(c) $-\frac{8}{19}$ 

(d) $\frac{20}{-23}$

Sol :

(a) $=-\frac{5}{16}$

(b) $=-\frac{15}{16}$

(c) $=\frac{8}{19}$

(d) $=+\frac{20}{23}$

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Q6 | Ex-1D |Class 8 |Rational Numbers | S.Chand | New Learning | Composite maths | myhelper

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Question 6

Write the multiplicative inverse of each of the following.

(a) -7

(b) 10

(c) $\frac{17}{41}$

(d) $\frac{28}{-59}$

(e) $\frac{-29}{-36}$

(f) $\left(\frac{1}{3}-\frac{1}{4}\right)\times (-2)$

(g) $\frac{5}{8}\div \frac{15}{16} \times \left(-\frac{3}{2}\right)$

(h) 16 ÷ (-32)

Sol :

(a) $=-\frac{1}{7}$

(b) $=\frac{1}{10}$

(c) $=\frac{41}{17}$

(d) $=\frac{-59}{28}$

(e) $=\frac{36}{29}$

(f) $=\left(\frac{4-3}{120}\right)\times (-2)$

$=\frac{1}{12}\times -2=-\frac{1}{6}=-6$

(g) $=\frac{5}{8}\times \frac{16}{15} \times \left(-\frac{3}{2}\right)$

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

(h) $=16\times \frac{1}{-32}$=-2