Question

Express each of the following decimals in the form $\frac{p}{q}$, where p, q are integers and q ≠ 0.
(i) $0.\overline{2}$
(ii) $0.\overline{53}$
(iii) $2.\overline{93}$
(iv) $18.\overline{48}$
(v) $0.\overline{235}$
(vi) $0.00\overline{32}$
(vii) $1.3\overline{23}$
(viii) $0.3\overline{178}$
(ix) $32.12\overline{35}$
(x) $0.40\overline{7}$

Answer 1

(i) $0.\overline{2}$
Let x = 0.222... .....(i)
Only one digit is repeated so, we multiply x by 10.
10x = 2.222... .....(ii)
Subtracting (i) from (ii) we get
$$\begin{gathered} 9x=2 \\ \Rightarrow x=\frac{2}{9} \end{gathered}$$

(ii) $0.\overline{53}$
Let x = 0.5353... .....(i)
Two digits are repeated so, we multiply x by 100.
100x = 53.5353... .....(ii)
Subtracting (i) from (ii) we get
$$\begin{gathered} 99x=53 \\ \Rightarrow x=\frac{53}{99} \end{gathered}$$

(iii) $2.\overline{93}$
Let x = 2.9393... .....(i)
Two digits are repeated so, we multiply x by 100.
100x = 293.9393... .....(ii)
Subtracting (i) from (ii) we get
$$\begin{gathered} 99x=291 \\ \Rightarrow x=\frac{291}{99}=\frac{97}{33} \end{gathered}$$

(iv) $18.\overline{48}$
Let x = 18.4848... .....(i)
Two digits are repeated so, we multiply x by 100.
100x = 1848.4848... .....(ii)
Subtracting (i) from (ii) we get
$$\begin{gathered} 99x=1830 \\ \Rightarrow x=\frac{1830}{99}=\frac{610}{33} \end{gathered}$$

(v) $0.\overline{235}$
Let x = 0.235235... .....(i)
Three digits are repeated so, we multiply x by 1000.
1000x = 235.235235... .....(ii)
Subtracting (i) from (ii) we get
$$\begin{gathered} 999x=235 \\ \Rightarrow x=\frac{235}{999} \end{gathered}$$

(vi) $0.00\overline{32}$
Let x = 0.003232... .....(i)
we multiply x by 100.
100x = 0.3232... .....(ii)
Again multiplying by 100 as there are 2 repeating numbers after decimals we get
10000x = 32.3232... .....(iii)
Subtracting (ii) from (iii) we get
$$\begin{gathered} 9900x=32 \\ \Rightarrow x=\frac{32}{9900}=\frac{8}{2475} \end{gathered}$$

(vii) $1.3\overline{23}$
Let x = 1.32323... .....(i)
we multiply x by 10.
10x = 13.2323... .....(ii)
Again multiplying by 100 as there are 2 repeating numbers after decimals we get
1000x = 1323.2323... .....(iii)
Subtracting (ii) from (iii) we get
$$\begin{gathered} 990x=1310 \\ \Rightarrow x=\frac{131}{99} \end{gathered}$$

(viii) $0.3\overline{178}$
Let x = 0.3178178... .....(i)
we multiply x by 10.
10x = 3.178178... .....(ii)
Again multiplying by 1000 as there are 3 repeating numbers after decimals we get
10000x = 3178.178178... .....(iii)
Subtracting (ii) from (iii) we get
$$\begin{gathered} 9990x=3175 \\ \Rightarrow x=\frac{3175}{9990}=\frac{635}{1998} \end{gathered}$$

(ix) $32.12\overline{35}$
Let x = 32.123535... .....(i)
we multiply x by 100.
100x = 3212.3535... .....(ii)
Again multiplying by 100 as there are 2 repeating numbers after decimals we get
10000x = 321235.35... .....(iii)
Subtracting (ii) from (iii) we get
$$\begin{gathered} 9900x=318023 \\ \Rightarrow x=\frac{318023}{9900} \end{gathered}$$

(x) $0.40\overline{7}$
Let x = 0.40777... .....(i)
we multiply x by 100.
100x = 40.7777... .....(ii)
Again multiplying by 10 as there is 1 repeating number after decimals we get
1000x = 407.777... .....(iii)
Subtracting (ii) from (iii) we get
$$\begin{gathered} 900x=367 \\ \Rightarrow x=\frac{367}{900} \end{gathered}$$