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

Exercise 1 D


Question 1

Find a rational number between a and b, if

(i) a=\dfrac{1}{6} , b=\dfrac{1}{5}

Sol :


(ii) a=\dfrac{1}{8} , b=\dfrac{7}{12}

Sol :


(iii) a=\dfrac{-3}{4} , b=\dfrac{-2}{3}

Sol :

(iv) a=-\dfrac{4}{9} , b=\dfrac{11}{6}

Sol :


Question 2

Find the two rational numbers between -2 and 2 .

Sol :

Two rational numbers are 0 , 1


Question 3

Find the three rational numbers between -\dfrac{7}{2}\text{ and }-2

Sol :

Let q1 , q2 and q3 are three required rational numbers

q1=\dfrac{1}{2}\left(\dfrac{-7}{2}+\dfrac{-2}{1}\right)

=\dfrac{1}{2}\left(\dfrac{-7\times 1 -2\times 2}{2}\right)

=\dfrac{1}{2}\left(\dfrac{-7-4}{2}\right)

=\dfrac{1}{2}\times \dfrac{-11}{2}=\dfrac{-11}{4}

 

q2=\dfrac{1}{2}\left(\dfrac{-11}{4}+\dfrac{-2}{1}\right)

=\dfrac{1}{2}\left(\dfrac{-11\times 1-2\times 4}{4}\right)

=\dfrac{1}{2}\left(\dfrac{-11-8}{4}\right)

=\dfrac{1}{2}\times \dfrac{-19}{4}=\dfrac{-19}{8}

 

q3=\dfrac{1}{2}\left(\dfrac{-19}{8}+\dfrac{-2}{1}\right)

=\dfrac{1}{2}\left(\dfrac{-19\times 1-2\times 8}{8}\right)

=\dfrac{1}{2}\left(\dfrac{-19-16}{8}\right)

=\dfrac{1}{2}\times \dfrac{-35}{8}=\dfrac{-35}{16}

 


Question 4

Find four rational numbers between -1 and -\dfrac{1}{2} .

Sol :

Let q1 , q2 , q3 and q4 are four required rational numbers

q1=\dfrac{1}{2}\left(\dfrac{-1}{1}+\dfrac{-1}{2}\right)

=\dfrac{1}{2}\left(\dfrac{-1\times 2-1\times 1}{2}\right)

=\dfrac{1}{2}\left(\dfrac{-2-1}{2}\right)

=\dfrac{1}{2}\times \dfrac{-3}{2}=\dfrac{-3}{4}

 

q2=\dfrac{1}{2}\left(\dfrac{-3}{4}+\dfrac{-1}{2}\right)

L.C.M of 4  and 2 is 4

=\dfrac{1}{2}\left(\dfrac{-3\times 1-1\times 2}{4}\right)

=\dfrac{1}{2}\left(\dfrac{-3-2}{4}\right)

=\dfrac{1}{2}\times \dfrac{-5}{4}=\dfrac{-5}{8}

 

q3=\dfrac{1}{2}\left(\dfrac{-5}{8}+\dfrac{-1}{2}\right)

L.C.M of 8  and 2 is 8

=\dfrac{1}{2}\left(\dfrac{-5\times 1-1\times 4}{8}\right)

=\dfrac{1}{2}\left(\dfrac{-5-4}{8}\right)

=\dfrac{1}{2}\times \dfrac{-9}{8}=\dfrac{-9}{16}

 

q4=\dfrac{1}{2}\left(\dfrac{-9}{16}+\dfrac{-1}{2}\right)

L.C.M of 16 and 2 is 16

=\dfrac{1}{2}\left(\dfrac{-9\times 1-1\times 8}{16}\right)

=\dfrac{1}{2}\left(\dfrac{-9-8}{8}\right)

=\dfrac{1}{2}\times \dfrac{-17}{8}=\dfrac{-17}{32}

 


Question 5

Find six rational numbers between -1 and 0

Sol :

Let q1 , q2 , q3 , q4 , q5 and q6 are six required rational numbers

q1=\dfrac{1}{2}\left(-1+0\right)

=\dfrac{-1}{2}

 

q2=\dfrac{1}{2}\left(\dfrac{-1}{2}+0\right)

=\dfrac{1}{2}\left(\dfrac{-1+0}{2}\right)

=\dfrac{-1}{4}

 

q3=\dfrac{1}{2}\left(\dfrac{-1}{4}+0\right)

=\dfrac{1}{2}\left(\dfrac{-1+0}{4}\right)

=\dfrac{-1}{8}

 

q4=\dfrac{1}{2}\left(\dfrac{-1}{8}+0\right)

=\dfrac{1}{2}\left(\dfrac{-1+0}{8}\right)

=\dfrac{-1}{16}

 

q5=\dfrac{1}{2}\left(\dfrac{-1}{16}+0\right)

=\dfrac{1}{2}\left(\dfrac{-1+0}{16}\right)

=\dfrac{-1}{32}

 

q6=\dfrac{1}{2}\left(\dfrac{-1}{32}+0\right)

=\dfrac{1}{2}\left(\dfrac{-1+0}{32}\right)

=\dfrac{-1}{64}

 


Question 6

Insert 10 rational numbers between \dfrac{-1}{2}\text{ and }\dfrac{7}{8} .

Sol :

Taking equivalent rationals \dfrac{-1}{2}\times \dfrac{4}{4}=\dfrac{-4}{8} for \dfrac{-1}{2}

So , 10 rational numbers are \dfrac{-3}{8},\dfrac{-2}{8},\dfrac{-1}{8},\dfrac{-0}{8},\dfrac{1}{8} , \dfrac{2}{8},\dfrac{3}{8},\dfrac{4}{8},\dfrac{5}{8},\dfrac{6}{8}

 


Question 7

Insert 100 rational numbers between \dfrac{-3}{13}\text{ and }\dfrac{9}{13} .

Sol :

Taking equivalent rationals \dfrac{-3}{13}\times \dfrac{10}{10}=\dfrac{-30}{130} and \dfrac{9}{13}\times \dfrac{10}{10}=\dfrac{90}{130}

So , ten rational numbers are \dfrac{-29}{130},\dfrac{-28}{130}...\dfrac{-1}{130},0,\dfrac{1}{130},...\dfrac{70}{130}

 


Question 8

Find (i) Six (ii) Sixty

(iii) Six hundred rational numbers between \dfrac{-5}{8}\text{ and }\dfrac{3}{8}

Sol :

(i)

six rational numbers are \dfrac{-4}{8},\dfrac{-3}{8},\dfrac{-2}{8},\dfrac{-1}{8},0,\dfrac{1}{8}

(ii)

Taking equivalent rationals \dfrac{-5}{8}\times \dfrac{10}{10}=\dfrac{-50}{80} and  \dfrac{3}{8}\times \dfrac{10}{10}=\dfrac{30}{80}

Sixty rational numbers are

\dfrac{-49}{80},\dfrac{-48}{80},..0..\dfrac{10}{80}

 

(iii)

Taking equivalent rationals \dfrac{-5}{8}\times \dfrac{100}{100}=\dfrac{-500}{800} and  \dfrac{3}{8}\times \dfrac{100}{100}=\dfrac{300}{800}

Six hundred rational numbers are

\dfrac{-499}{800},\dfrac{-498}{80},..0..\dfrac{99}{800},\dfrac{100}{80}


 

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