Question

Write the following rational numbers in $\frac{p}{q}$ form

$\left(\mathrm{i}\right)0.6^{°}\left(\mathrm{ii}\right)0.\overline{37}\left(\mathrm{iii}\right)3.\overline{17}\left(\mathrm{iv}\right)15.\overline{89}\left(\mathrm{v}\right)2.\overline{514}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{Let}x=0.6\mathbf{°}...\left(1\right) \\ x=0.666... \\ \mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}10,\mathrm{we}\mathrm{get} \\ 10x=6.666......\left(2\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get} \\ 9x=6 \\ \therefore x=\frac{6}{9} \\ \mathrm{So},0.6\mathbf{°}=\frac{2}{3} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{Let}x=0.\overline{37}...\left(1\right) \\ \mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}100,\mathrm{we}\mathrm{get} \\ 100x=37.\overline{37}...\left(2\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get} \\ 99x=37 \\ \therefore x=\frac{37}{99} \\ \mathrm{So},0.\overline{37}=\frac{37}{99} \end{gathered}$$​

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{Let}x=3.\overline{17}...\left(1\right) \\ \mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}100,\mathrm{we}\mathrm{get} \\ 100x=317.\overline{17}...\left(2\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get} \\ 99x=314 \\ \therefore x=\frac{314}{99} \\ \mathrm{So},3.\overline{17}=\frac{314}{99} \end{gathered}$$​

$$\begin{gathered} \left(\mathrm{iv}\right)\mathrm{Let}x=15.\overline{89}...\left(1\right) \\ \mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}100,\mathrm{we}\mathrm{get} \\ 100x=1589.\overline{89}...\left(2\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get} \\ 99x=1574 \\ \therefore x=\frac{1574}{99} \\ \mathrm{So},3.\overline{17}=\frac{1574}{99} \end{gathered}$$​

$$\begin{gathered} \left(\mathrm{v}\right)\mathrm{Let}x=2.\overline{514}...\left(1\right) \\ \mathrm{Multiplying}\mathrm{both}\mathrm{sides}\mathrm{by}1000,\mathrm{we}\mathrm{get} \\ 1000x=2514.\overline{514}...\left(2\right) \\ \mathrm{Subtracting}\left(1\right)\mathrm{from}\left(2\right),\mathrm{we}\mathrm{get} \\ 999x=2512 \\ \therefore x=\frac{2512}{999} \\ \mathrm{So},2.\overline{514}=\frac{2512}{999} \end{gathered}$$​