Fill in the blanks :

(i) $\dfrac{2}{3} = ……………$

(ii) $\dfrac{11}{30} = ……………$

(iii) $\dfrac{13}{11} = ……………$

(iv) $\dfrac{23}{55} = ……………$

(i) $\dfrac{2}{3}$ = $0.\overline{6}$

(ii) (i) $\dfrac{11}{30}$ = $0.3\overline{6}$

(iii) $\dfrac{13}{11}$ = $1.\overline{18}$

(iv) $\dfrac{23}{55}$ = $0.4\overline{18}$

Explanation

(i) $\dfrac{2}{3}$

By actual division:

$\begin{array}{r} 0.666… \ 3 \overline{) 2.000} \ \underline{-0\phantom{.000}} \ 2.0\phantom{00} \ \underline{-1.8\phantom{00}} \ 20\phantom{0} \ \underline{-18\phantom{0}} \ 20 \ \underline{-18} \ 2… \end{array}$

∴ $\dfrac{2}{3} = 0.666… = 0.\overline{6}$

(ii) $\dfrac{11}{30}$

By actual division:

$\begin{array}{r} 0.366… \ 30 \overline{) 11.000} \ \underline{-0\phantom{.000}} \ 110\phantom{00} \ \underline{-90\phantom{00}} \ 200\phantom{0} \ \underline{-180\phantom{0}} \ 200 \ \underline{-180} \ 20… \end{array}$

∴ $\dfrac{11}{30} = 0.366… = 0.3\overline{6}$

(iii) $\dfrac{13}{11}$

By actual division:

$\begin{array}{r} 1.1818… \ 11 \overline{) 13.0000} \ \underline{-11\phantom{.0000}} \ 20\phantom{000} \ \underline{-11\phantom{000}} \ 90\phantom{00} \ \underline{-88\phantom{00}} \ 20\phantom{0} \ \underline{-11\phantom{0}} \ 90 \ \underline{-88} \ 2… \end{array}$

∴ $\dfrac{13}{11} = 1.1818… = 1.\overline{18}$

(iv) $\dfrac{23}{55}$

By actual division:

$\begin{array}{r} 0.41818… \ 55 \overline{) 23.00000} \ \underline{-0\phantom{.00000}} \ 230\phantom{0000} \ \underline{-220\phantom{0000}} \ 100\phantom{000} \ \underline{-55\phantom{000}} \ 450\phantom{00} \ \underline{-440\phantom{00}} \ 100\phantom{0} \ \underline{-55\phantom{0}} \ 450 \ \underline{-440} \ 10… \end{array}$

∴ $\dfrac{23}{55} = 0.41818… = 0.4\overline{18}$