Question

Express each of the following as a fraction in simplest form.
(i) $0.\overline{3}$
(ii) $1.\overline{3}$
(iii) $0.\overline{34}$
(iv) $3.\overline{14}$
(v) $0.\overline{324}$
(vi) $0.\overline{17}$
(vii) $0.\overline{54}$
(viii) $0.\overline{163}$

Answer 1

(i) $0.\overline{3}$
Let $x=0.\overline{3}$
∴ x = 0.3333... ...(i)
10x = 3.3333... ...(ii)
On subtracting (i) from (ii), we get:
9x = 3
= x = $\frac{1}{3}$

∴ $0.\overline{3}=\frac{1}{3}$

(ii) $1.\overline{3}$
Let $x=1.\overline{3}$
∴ x = 1.3333... ...(i)
10x = 13.3333... ...(ii)
On subtracting (i) from (ii), we get:
9x = 12
= x = $\frac{4}{3}$

∴ $1.\overline{3}=\frac{4}{3}$

(iii) $0.\overline{34}$
Let x = $0.\overline{34}$
∴ x = 0.3434... ...(i)
100x = 34.3434... ...(ii)
On subtracting (i) from (ii), we get:
99x = 34
= x = $\frac{34}{99}$

∴ $0.\overline{34}=\frac{34}{99}$

(iv) $3.\overline{14}$
Let x = $3.\overline{14}$
∴ x = 3.1414... ...(i)
100x = 314.1414... ...(ii)
On subtracting (i) from (ii), we get:
99x = 311
= x = $\frac{311}{99}$

∴ $3.\overline{14}$ = $\frac{311}{99}$

(v) $0.\overline{324}$
Let x = $0.\overline{324}$
∴ x = 0.324324... ...(i)
1000x = 324.324324... ...(ii)
On subtracting (i) from (ii), we get:
999x = 324
= x = $\frac{324}{999}$=$\frac{12}{37}$

∴ $0.\overline{324}$= $\frac{12}{37}$

(vi) $0.1\overline{7}$
Let x = $0.1\overline{7}$
∴ x = 0.1777...
10x = 1.777... ...(i)
100x = 17.777.... ...(ii)
On subtracting (i) from (ii). we get:
90x = 16
= x = $\frac{8}{45}$
∴ $0.1\overline{7}$ = $\frac{8}{45}$

(vii) $0.5\overline{4}$
Let x = $0.5\overline{4}$
∴ x = 0.5444...
10x = 5.4444... ...(i)
100x = 54.4444... ...(ii)
On subtracting (i) from (ii), we get:
90x = 49
= x = $\frac{49}{90}$
∴ $0.5\overline{4}$= $\frac{49}{90}$

(viii) $0.1\overline{63}$
Let x = $0.1\overline{63}$
∴ x = 0.16363...
10x = 1.6363... ...(i)
1000x = 163.6363... ...(ii)
On subtracting (i) from (ii), we get:
990x = 162
= x = $\frac{162}{990}=\frac{9}{55}$

∴ $0.1\overline{63}$ = $\frac{9}{55}$