Question

(1) Verify that :
$\sf (i) - \dfrac{1}{2} + \bigg [ \dfrac{-4}{3} +\dfrac{3}{7} \bigg] \: and \bigg [ \dfrac{-1}{2} + \dfrac{3}{7} \bigg] + \left ( \dfrac{-4}{3}\right ) are \: the \: same$
$\sf (ii) \dfrac{2}{3} \times \bigg [ \dfrac{-6}{7} +\dfrac{4}{5} \bigg] \: = \bigg [ \dfrac{2}{3} \times \dfrac{4}{5} \bigg] \times \left ( \dfrac{-6}{7}\right )$
(2) Find :
$\sf \dfrac{5}{22} + \dfrac{3}{7} + \left ( \dfrac{-8}{21}\right ) + \left ( \dfrac{-6}{11} \right)$
(3) Find :
$\sf \left ( \dfrac{-14}{9}\right ) \times \dfrac{3}{5} \times \left ( \dfrac{-4}{7}\right ) \times \dfrac{15}{16}$+ \left ( \dfrac{-6}{11} \right)[/tex]
(4) Find :
$\sf \left ( \dfrac{-14}{9}\right ) \times \dfrac{3}{5} \times \left( \dfrac{-4}{7}\right) \times \dfrac{15}{16}$
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↦ Class - 8
↦ Lesson - Rational Numbers

Answer 1

1 Verify that :

$\\\sf (i) - \dfrac{1}{2} + \bigg [ \dfrac{-4}{3} +\dfrac{3}{7} \bigg] \: and \bigg [ \dfrac{-1}{2} + \dfrac{3}{7} \bigg] + \left ( \dfrac{-4}{3}\right ) are \: the \: same$

Solution

$\\\bf LHS=- \dfrac{1}{2} + \bigg [ \dfrac{-4}{3} +\dfrac{3}{7} \bigg] \:$

$\bf{RHS= \bigg [ \dfrac{-1}{2} + \dfrac{3}{7} \bigg] + \left ( \dfrac{-4}{3}\right )}$

On Solving LHS we get,

$\\\sf~~~~~:\implies LHS= - \dfrac{1}{2} + \bigg [ \dfrac{-4}{3} +\dfrac{3}{7} \bigg] \:$

$\sf~~~~~:\implies LHS= - \dfrac{1}{2} + \bigg [ \dfrac{-28 + 9}{21} \bigg] \:$

$\sf~~~~~:\implies LHS= - \dfrac{1}{2} + \ \dfrac{-19}{21} \:$

$\sf~~~~~:\implies LHS= \dfrac{ - 21 - 38}{42} \:$

$\sf~~~~~:\implies LHS= \dfrac{ - 59 \: }{ \: \: 42 \: } \:$

On Solving RHS we get

$\sf{~~~~~:\implies RHS= \bigg [ \dfrac{-1}{2} + \dfrac{3}{7} \bigg] + \left ( \dfrac{-4}{3}\right )}$

$\sf{~~~~~:\implies RHS= \bigg [ \dfrac{-7 + 6}{14} \bigg] + \left ( \dfrac{-4}{3}\right )}$

$\sf{~~~~~:\implies RHS= \dfrac{-1}{14} - \dfrac{4}{3}}$

$\sf{~~~~~:\implies RHS= \dfrac{-3 - 56}{42} }$

$\sf{~~~~~:\implies RHS= \dfrac{ - 59 \: \: }{ \: 42} }$

$\bf{LHS = RHS \: \: _{ \pink{Verified}}}$

$\\\pink{\sf (ii) \dfrac{2}{3} \times \bigg [ \dfrac{-6}{7} +\dfrac{4}{5} \bigg] \: = \bigg [ \dfrac{2}{3} \times \dfrac{4}{5} \bigg] \times \left ( \dfrac{-6}{7}\right )}$

Solution

$\\\bf{LHS= \dfrac{2}{3} \times \bigg [ \dfrac{-6}{7} +\dfrac{4}{5} \bigg] \:}$

$\bf{RHS= \bigg [ \dfrac{2}{3} \times \dfrac{4}{5} \bigg] \times \left ( \dfrac{-6}{7}\right ) }$

On Solving LHS we will

$\sf{~~~~~:\implies LHS= \dfrac{2}{3} \times \bigg [ \dfrac{-6}{7} +\dfrac{4}{5} \bigg] \:}$

$\sf{~~~~~:\implies.LHS= \dfrac{2}{3} \times \bigg [ \dfrac{-30 + 28}{35} \bigg] \:}$

$\sf{~~~~~:\implies LHS= \dfrac{2}{3} \times \dfrac{-2}{35} \:}$

$\sf{~~~~~:\implies LHS= \dfrac{ - 4}{105} \:}$

On Solving RHS we get

$\sf{~~~~~:\implies RHS= \bigg [ \dfrac{2}{3} \times \dfrac{4}{5} \bigg] \times \left ( \dfrac{-6}{7}\right ) }$

$\sf{~~~~~:\implies RHS= \dfrac{8}{15} \times \dfrac{-6}{7} }$

$\sf{~~~~~:\implies RHS= \dfrac{ - 48}{105} }$

$\sf{~~~~~:\implies RHS= \dfrac{ - 16}{35} }$

$~~~~~:\implies \bf{LHS ≠ RHS }$

(2) Find :

$\\\sf \dfrac{5}{22} + \dfrac{3}{7} + \left ( \dfrac{-8}{21}\right ) + \left ( \dfrac{-6}{11} \right)$

Solution

$~~~~~:\implies \sf \dfrac{5}{22} + \dfrac{3}{7} + \left ( \dfrac{-8}{21}\right ) + \left ( \dfrac{-6}{11} \right)$

Taking LCM of 22,7,21,11 = 462

$\sf ~~~~~:\implies \dfrac{105 + 198 - 176 - 252}{462}$

$\sf~~~~~:\implies \dfrac{303 - 428}{462}$

$\sf ~~~~~:\implies \dfrac{ - 125 \: \: }{462}$

$\sf ~~~~~:\implies\dfrac{ - 125}{462} \: \: \: \pink{\mathfrak{answer}}$

3 Find:

$\\\sf ~~~~~:\implies\left ( \dfrac{-14}{9}\right ) \times \dfrac{3}{5} \times \left ( \dfrac{-4}{7}\right ) \times \dfrac{15}{16}+ \left ( \dfrac{-6}{11} \right)$

$\sf~~~~~:\implies \dfrac{840}{5040} + \left ( \dfrac{-6}{11} \right)$

$\sf ~~~~~:\implies \dfrac{1}{6} - \dfrac{6}{11}$

$\sf ~~~~~:\implies \dfrac{11 - 36}{66}$

$\sf ~~~~~:\implies \dfrac{ - 25}{66}$

(4) Find :

$\\\\\sf{ ~~~~~:\implies \left ( \dfrac{-14}{9}\right ) \times \dfrac{3}{5} \times \left( \dfrac{-4}{7}\right) \times \dfrac{15}{16}}$

$\sf ~~~~~:\implies \dfrac{840}{5040}$

$\sf~~~~~:\implies \dfrac{1}{6}$

I Hope you Like My Answer

Happy Studying

Answer 2

$\huge\bold{1)Question:-}$

(1) Verify that :

$\sf (i) - \dfrac{1}{2} + \bigg [ \dfrac{-4}{3} +\dfrac{3}{7} \bigg] \: and \bigg [ \dfrac{-1}{2} + \dfrac{3}{7} \bigg] + \left ( \dfrac{-4}{3}\right ) are \: the \: same$

$\sf (ii) \dfrac{2}{3} \times \bigg [ \dfrac{-6}{7} +\dfrac{4}{5} \bigg] \: = \bigg [ \dfrac{2}{3} \times \dfrac{4}{5} \bigg] \times \left ( \dfrac{-6}{7}\right )$

$\huge\bold{Solution:-}$

Refer to the attachment 1 and 2

Verification

$\huge\bold{2)Question:-}$

(2) Find :

$\sf \dfrac{5}{22} + \dfrac{3}{7} + \left ( \dfrac{-8}{21}\right ) + \left ( \dfrac{-6}{11} \right)$

$\huge\bold{Solution:-}$

Refer to attachment-3

Final answer:-

$\frac{53}{231}$

$\huge\bold{3)Question:-}$

(3) Find :

$( \frac{ - 14}{11} ) \times (\frac{3}{5} ) \times ( - \frac{4}{7} ) \times \frac{15}{16} + ( \frac{ - 6}{11} )$

$+ ( \frac{ - 6}{11} )$

$\huge\bold{Solution:-}$

Refer to attachment- 4

Final answer:-

$\frac{-1}{22}$

$\huge\bold{4)Question:-}$

(4)Find:

$\sf \left ( \dfrac{-14}{9}\right ) \times \dfrac{3}{5} \times \left( \dfrac{-4}{7}\right) \times \dfrac{15}{16}$

$\huge\bold{Solution:-}$

Refer to the attachment -5

Final answer:-

$\frac{1}{2}$

Hope this is helpful $\huge\bold{✓}$