Exercise 1D
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)
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
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
Question 4
Using distributivity, find $\frac{8}{15}\times \left(-\frac{7}{18}\right)+\left(\frac{8}{15}\times -\frac{11}{18}\right)$
Sol :
⇒$\frac{8}{15}\left[\frac{-7}{18}+\frac{-11}{18}\right]$
$=\frac{8}{15}\times \frac{-18}{18}=\frac{-8}{15}$
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) $=+\frac{20}{23}$
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
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