Question#1: Identify which of the following are rational and irrational numbers.
\[\begin{align}
\left( \text{i} \right)\qquad &\sqrt{3}\qquad &&\text{Irrational number} \\
\left( \text{ii} \right) \qquad &\frac{1}{6}\qquad &&\text{Rational number} \\
\left( \text{iii} \right) \qquad &\pi \qquad &&\text{Irrational number} \\
\left( \text{iv} \right) \qquad &\frac{15}{2}\qquad &&\text{Rational number} \\
\left( \text{v} \right) \qquad &7.25\qquad &&\text{Rational number} \\
\left( \text{vi} \right) \qquad &\sqrt{29}\qquad &&\text{Irrational number} \\
\end{align}\]
Question#2: Convert the following fractions into decimal fractions.
\(i\)
\[\frac{17}{25}\]
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:
\[\begin{array}{c}
4.75\\
4\enclose{longdiv}{19000}\\
-16\\ \hline
30\\-28\\ \hline
20\\-20\\ \hline
0\\
\frac{19}{4}=4.75
\end{array}\]
\(iii\)
\[\frac{57}{8}\]
Solution:
\[\begin{array}{c}
7.125\\
8\enclose{longdiv}{57000}\\
-56\\ \hline
10\\-8\\ \hline
20\\-16\\ \hline
40\\-40\\ \hline
0\\
\frac{57}{8}=7.125
\end{array}\]
\(iv\)
\[\frac{205}{18}\]
Solution:
\[\begin{array}{c}
11.388…\\
18\enclose{longdiv}{205000}\\
-198\\ \hline
70\\-54\\ \hline
160\\-144\\ \hline
160\\-144\\ \hline
16\\
\frac{208}{18}=11.388…
\end{array}\]
\(v\)
\[\frac{5}{8}\]
Solution:
\[\begin{array}{c}
0.625\\
8\enclose{longdiv}{500}\\
-48\\ \hline
20\\-16\\ \hline
40\\-40\\ \hline
0\\
\frac{5}{8}=0.625
\end{array}\]
\(vi\)
\[\frac{25}{38}\]
Solution:
\[\begin{array}{c}
0.6578…\\
38\enclose{longdiv}{250000}\\
-228\\ \hline
220\\-190\\ \hline
300\\-266\\ \hline
340\\-304\\ \hline
36\\
\frac{25}{38}=0.6578…
\end{array}\]
Question#3: Which statements are true and which are false?
\[\begin{align}
\left( \text{i} \right)\qquad &\frac{2}{3}\text{ is an irrational number}\qquad &&\textbf{False} \\
\left( \text{ii} \right) \qquad &\pi \text{ is an irrational number.}\qquad &&\textbf{True} \\
\left( \text{iii} \right) \qquad &\frac{1}{9} \text{ is a terminating fraction.} \qquad &&\textbf{False} \\
\left( \text{iv} \right) \qquad &\frac{3}{4} \text{ is a terminating fraction.}\qquad &&\textbf{True} \\
\left( \text{v} \right) \qquad &\frac{4}{5} \text{ is a recurring fraction.}\qquad &&\textbf{False} \\
\end{align}\]
Question#4: Represent the following number on a number line.
Question#5: Give a rational number between \(\frac{3}{4}\) and \(\frac{5}{9}\).
Solution:
\[\begin{align}
& =\left( \frac{3}{4}+\frac{5}{9} \right)\div 2 \\
& =\left( \frac{27+20}{36} \right)\div 2 \\
& =\left( \frac{47}{36} \right)\times \frac{1}{2} \\
& =\frac{47}{72} \\
\end{align}\]
Question#6: Express the following recurring decimals as the rational number \(\frac{p}{q}\) where \(p,q\) are integer and \(q\ne0\).
\(i\)
\[0.\bar{5}\]
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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