class 9th math 2.1 solution

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\begin{align}  \left( \text{i} \right)\qquad &\sqrt{3}\qquad &&\text{Irrational number} \\  \left( \text{ii} \right) \qquad &\frac{1}{6}\qquad &&\text{Rational number} \\  \left( \text{iii} \right) \qquad &\pi \qquad &&\text{Irrational number} \\  \left( \text{iv} \right) \qquad &\frac{15}{2}\qquad &&\text{Rational number} \\  \left( \text{v} \right) \qquad &7.25\qquad &&\text{Rational number} \\  \left( \text{vi} \right) \qquad &\sqrt{29}\qquad &&\text{Irrational number} \\\end{align}

\(i\) \[\frac{17}{25}\] Solution: \[\begin{array}{c}0.68\\ 25\enclose{longdiv}{170}\\-150\\ \hline 200\\-200\\ \hline 0\\\frac{17}{25}=0.68\end{array}\]

\(ii\) \[\frac{19}{4}\] Solution: \[\begin{array}{c}4.75\\ 4\enclose{longdiv}{19000}\\-16\\ \hline30\\-28\\ \hline 20\\-20\\ \hline 0\\\frac{19}{4}=4.75\end{array}\]

\(iii\) \[\frac{57}{8}\] Solution: \[\begin{array}{c}7.125\\ 8\enclose{longdiv}{57000}\\-56\\ \hline10\\-8\\ \hline 20\\-16\\ \hline40\\-40\\ \hline 0\\\frac{57}{8}=7.125\end{array}\]

\(iv\) \[\frac{205}{18}\] Solution: \[\begin{array}{c}11.388…\\ 18\enclose{longdiv}{205000}\\-198\\ \hline70\\-54\\ \hline 160\\-144\\ \hline160\\-144\\ \hline 16\\\frac{208}{18}=11.388…\end{array}\]

\(v\) \[\frac{5}{8}\] Solution: \[\begin{array}{c}0.625\\ 8\enclose{longdiv}{500}\\-48\\ \hline20\\-16\\ \hline 40\\-40\\ \hline 0\\\frac{5}{8}=0.625\end{array}\]

\(vi\) \[\frac{25}{38}\] Solution: \[\begin{array}{c}0.6578…\\ 38\enclose{longdiv}{250000}\\-228\\ \hline220\\-190\\ \hline 300\\-266\\ \hline340\\-304\\ \hline 36\\\frac{25}{38}=0.6578…\end{array}\]

\begin{align}  \left( \text{i} \right)\qquad &\frac{2}{3}\text{ is an irrational number}\qquad &&\textbf{False} \\  \left( \text{ii} \right) \qquad &\pi \text{ is an irrational number.}\qquad &&\textbf{True} \\  \left( \text{iii} \right) \qquad &\frac{1}{9} \text{ is a terminating fraction.} \qquad &&\textbf{False} \\  \left( \text{iv} \right) \qquad &\frac{3}{4} \text{ is a terminating fraction.}\qquad &&\textbf{True} \\  \left( \text{v} \right) \qquad &\frac{4}{5} \text{ is a recurring fraction.}\qquad &&\textbf{False} \\\end{align}

Solution: \[\begin{align}& =\left( \frac{3}{4}+\frac{5}{9} \right)\div 2 \\ & =\left( \frac{27+20}{36} \right)\div 2 \\ & =\left( \frac{47}{36} \right)\times \frac{1}{2} \\ & =\frac{47}{72} \\\end{align}\]

\(i\) \[0.\bar{5}\] Solution: \[\begin{align}   \text{Let }\qquad & x=0.\bar{5} \\ & x=0.555… \\ & \text{Multiply with }10 \\ & 10x=5.555… \\ & 10x=5+0.555… \\ & 10x=5+x \\ & 10x-x=5 \\ & 9x=5 \\ & x=\frac{5}{9} \\\end{align}\]

\(ii\) \[0.\bar{13}\] Solution: \[\begin{align}   \text{Let }\qquad & x=0.\bar{13} \\ & x=0.131313… \\ & \text{Multiply with }100 \\ & 100x=13.131313… \\ & 100x=13+0.131313… \\ & 100x=13+x \\ & 100x-x=13 \\ & 99x=13 \\ & x=\frac{13}{99} \\\end{align}\]

\(iii\) \[0.\bar{67}\] Solution: \[\begin{align}   \text{Let }\qquad & x=0.\bar{67} \\ & x=0.676767… \\ & \text{Multiply with }100 \\ & 100x=67.676767… \\ & 100x=67+0.676767… \\ & 100x=67+x \\ & 100x-x=67 \\ & 99x=67 \\ & x=\frac{67}{99} \\\end{align}\]

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