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S.chand books class 8 maths solution chapter 1 Rational Numbers exercise 1 C

Exercise 1 C


Question 1

Answer True (T) or False (F)
(i) \dfrac{8}{15}\div\dfrac{7}{8}=\dfrac{7}{8}\div\dfrac{8}{15}

Sol : F
(ii) Addition distributes over multiplication in rational numbers .

Sol : F
(iii) The reciprocal of -8 is \dfrac{1}{8}

Sol : F
(iv) \dfrac{3}{4}\div\dfrac{-8}{7} is a rational numbers .

Sol : T
(v) 0\times \dfrac{4}{5}=\dfrac{4}{5}\times 0 implies that 0 is the multiplicative identity for rational numbers .

Sol : F


Question 2

(i) \dfrac{1}{6}\text{ by }\dfrac{12}{7}

Sol:

=\frac{1}{6} \times \frac{12}{17}

=\frac{12}{102}=\frac{6}{51}=\frac{2}{17}

(ii) \dfrac{-5}{12}\text{ by }\dfrac{9}{10}

Sol:

=\frac{-5}{12} \times \frac{9}{10}

=\frac{-45}{120}

=-\frac{9}{24}=\frac{-3}{8}

(iii) \dfrac{-14}{33}\text{ by }\dfrac{-3}{28}

Sol:

\frac{-14}{33} \times \frac{-3}{28}

=\frac{1}{11 \times 2}=\frac{1}{22}

(iv) \dfrac{-80}{13}\text{ by }\dfrac{-65}{72}

Sol:

\frac{-80}{13} \times \frac{-65}{72}

=\frac{-80}{1} \times \frac{-5}{72}

=-\frac{10}{1} \times-\frac{5}{9}

=\frac{50}{9}

(v) 5\dfrac{1}{2}\text{ by }-1\dfrac{5}{6}

Sol:

5 \frac{1}{2} x-1 \frac{5}{6}

=\frac{11}{2} \times-\frac{11}{6}

=-\frac{121}{12}

=-10 \frac{1}{12}

(vi) -6 by 5\dfrac{2}{3}

Sol :

-6 \times 5 \frac{2}{3}

=-\frac{6}{1} \times \frac{17}{3}

=\frac{-2 \times 17}{1}

=\frac{-34}{1} or-34

 


Question 3

Simplify :

(i) \left(\dfrac{-3}{4}\times \dfrac{-24}{15}\right)+\left(\dfrac{-11}{13}\times \dfrac{78}{55}\right)

Sol :

\left(\frac{-3}{4} \times \frac{-24}{15}\right)+\left(-\frac{11}{13} \times \frac{78}{55}\right)

\left(\frac{-1}{1} \times \frac{-6}{5}\right)+\left(-\frac{1}{1} \times \frac{6}{5}\right)

\left(\frac{6}{5}\right)+\left(-\frac{6}{5}\right)

=0

 


(ii) \left(\dfrac{-4}{5}\times \dfrac{15}{8}\right)+\left(\dfrac{-1}{3}\times \dfrac{-9}{7}\right)-\left(\dfrac{2}{9}\times \dfrac{27}{14}\right)

Sol :

\left(-\frac{4}{5} \times \frac{15}{8}\right)+\left(\frac{-1}{3} \times-\frac{9}{7}\right)-\left(\frac{2}{9} \times \frac{27}{14}\right)

=\left(\frac{4}{1} \times \frac{3}{8}\right)+\left(\frac{-1}{1} \times \frac{-3}{7}\right)-\left(\frac{2}{1} \times \frac{3}{14}\right)

=\left(-\frac{1}{1} \times \frac{3}{2}\right)+\left(\frac{-1}{1} \times \frac{-3}{7}\right)-\left(\frac{1}{1} \times \frac{3}{7}\right)

=-\frac{3}{2}+\frac{3}{7}-\frac{3}{7}

=-\frac{3}{2}

 


Question 4

Fill in the blanks.
(i) \dfrac{-19}{40}\times \dfrac{8}{1}=\dfrac{8}{11}\times \dfrac{-19}{40}

(ii) -25\times\dfrac{-7}{12}=\dfrac{-7}{12}\times (-25)

(iii) \left(\dfrac{6}{11}\times \dfrac{-20}{21}\right)\times \left(\dfrac{-7}{8}\right)=\dfrac{6}{11} \times\left(\dfrac{-20}{21}\times \dfrac{-7}{8}\right)

(iv) \dfrac{-21}{40}\times \left(\dfrac{3}{7}\times \dfrac{17}{-24}\right)=\left(\dfrac{-21}{40}\times \dfrac{17}{-24}\right)\times \dfrac{3}{7}

(v) \dfrac{2}{9}\times \left(\dfrac{-4}{9}\div\dfrac{6}{17}\right)=\dfrac{2}{9}\times \dfrac{-4}{9} \div \dfrac{2}{9} \times \dfrac{6}{7}


Verify the statements given in questions 5 to 7 . Also , name the properties of multiplication illustrated by these statements.

Question 5

(i) \dfrac{4}{5}\times \dfrac{7}{9}=\dfrac{7}{9}\times \dfrac{4}{5}

Sol :

\dfrac{4\times 7}{5\times 9}=\dfrac{7\times 4}{9\times 5}

\dfrac{28}{45}=\dfrac{28}{45}

by commutative property


(ii) \dfrac{8}{7}\times \dfrac{9}{-10}=\dfrac{9}{-10}\times \dfrac{8}{7}

Sol :

\dfrac{8\times 9}{7\times -10}=\dfrac{9\times 8}{-10\times 7}

\dfrac{72}{-70}=\dfrac{72}{-70}

by commutative property


Question 6

(i) \left(\dfrac{3}{4}\times \dfrac{1}{2}\right)\times \dfrac{5}{7}=\dfrac{3}{4}\times \left(\dfrac{1}{2}\times \dfrac{5}{7}\right)

Sol :

\left(\dfrac{3}{8}\right)\times \dfrac{5}{7}=\dfrac{3}{4}\times \left( \dfrac{5}{14}\right)

\dfrac{15}{56}=\dfrac{15}{56}

By associative property


(ii) \left(-\dfrac{7}{6}\times \dfrac{-2}{5}\right)\times \dfrac{3}{8}=\dfrac{-7}{6}\times \left(\dfrac{-2}{3}\times \dfrac{3}{8}\right)

Sol :

\left(\dfrac{14}{30}\right)\times \dfrac{3}{8}=\dfrac{-7}{6}\times \left(\dfrac{-6}{24}\right)

\dfrac{42}{240}=\dfrac{42}{240}

By associative property


Question 7

(i) \dfrac{2}{3}\times \left(\dfrac{4}{5}+ \dfrac{7}{8}\right)=\left(\dfrac{2}{3}\times \dfrac{4}{5}\right)+ \left(\dfrac{2}{3}\times \dfrac{7}{8}\right)

Sol :

\frac{2}{3} \times\left(\frac{4}{5}+\frac{7}{8}\right)=\left(\frac{2}{3} \times \frac{4}{5}\right)+\left(\frac{2}{3} \times \frac{7}{8}\right)

LCM OF 5,8 is 40

\frac{2}{3} \times\left(\frac{4 \times 8+7 \times 5}{40}\right)=\left(\frac{8}{15}+\frac{14}{24}\right)

LCM of 15 , 24  is 120

\frac{2}{3} \times\left(\frac{32+35}{40}\right)=\frac{(8 \times 8+14 \times 5)}{120}

\frac{2 \times 67}{120}=\frac{64+70}{120}

\frac{134}{120}=\frac{134}{120}

 


(ii) \dfrac{-6}{15}\times \left(\dfrac{7}{8}+ \dfrac{-5}{12}\right)=\left(\dfrac{-6}{15}\times\dfrac{7}{8}\right)+ \left(\dfrac{-6}{15}\times \dfrac{-5}{12}\right)

Sol :

 


Question 8

Use the distributivity of multiplication of rational numbers over addition to simplify:

(i) \dfrac{5}{8}\times \left(\dfrac{2}{3}+ \dfrac{1}{2}\right)

Sol :


(ii) \dfrac{-2}{3}\times \left(\dfrac{4}{7}+ \dfrac{-11}{14}\right)

Sol :


(iii) \dfrac{-2}{9}\times \left(\dfrac{3}{4}- 36\right)

Sol :


Question 9

Find the multiplication inverse , i.e., the reciprocal of

(i) 8

Sol : \dfrac{1}{8}

 


(ii) -18

Sol : \dfrac{-1}{18}

 


(iii) \dfrac{14}{23}

Sol : \dfrac{23}{14}

 


(iv) \dfrac{-29}{16}

Sol : \dfrac{16}{-29}

 


(v) \dfrac{14}{-9}

Sol : \dfrac{-9}{14}

 


(vi) \dfrac{-8}{-9}

Sol : \dfrac{-9}{-8} or \dfrac{9}{8}

 


(vii) \dfrac{4}{5}\times \dfrac{15}{8}

Sol :

\dfrac{\not{4}}{5} \times \dfrac{15}{\not{8}}

\dfrac{1}{\not{5}} \times \dfrac{\not{15}}{2}

\dfrac{1}{1} \times \dfrac{3}{2}

\dfrac{3}{2}


(viii) 0\times \dfrac{2}{9}

Sol : 0

 


Question 10

Match the mathematical sentence in Column A with the property illustrated by the statement in Column B

Column AColumn B
(i) \dfrac{1}{3}\times 0=0(a) Property of 1
(ii) \dfrac{-2}{5}\times 1=\dfrac{-2}{5}(b) Closure Property
(iii) \dfrac{1}{4}\times 4=1(c) Property of 0
(iv) \dfrac{-6}{7}\times \dfrac{2}{3}=-\dfrac{4}{5}(d) Commutative Property
(v) 0\times -\dfrac{7}{8}=\dfrac{-7}{8}\times 0(e) Multiplicative Inverse

Sol :

(i)→ (c)

(ii)→ (a)

(iii) (e)

(iv)→ (b)

(v)→ (d)


Question 11

Divide:

(i) \dfrac{6}{13}\text{ by }3

Sol :


(ii) \dfrac{5}{6}\text{ by }\dfrac{-10}{21}

(iii) \dfrac{10}{33}\text{ by }\dfrac{2}{-11}

Sol :


(iv) -\dfrac{21}{22}\text{ by }\dfrac{-7}{11}

(v) \dfrac{9}{14}\text{ by }\dfrac{-3}{28}

Sol :


(vi) -15\dfrac{3}{4}\text{ by }-2\dfrac{5}{8}

Sol :


Question 12

Evaluate:

(i) \left(\dfrac{5}{9}\div\dfrac{15}{36}\right)\div \left(-\dfrac{5}{6}\right)

Sol :


(ii) \left(\dfrac{-3}{29}\div\dfrac{9}{87}\right)\div\dfrac{-1}{\phantom{-}7}

Sol :


 

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