Question

Express each of the following as a fraction in simplest form:

(i) $0.\overline{8}$
(ii) $2.\overline{4}$
(iii) $0.\overline{24}$
(iv) $0.\overline{12}$
(v) $0.2\overline{4}$
(vi) $0.3\overline{65}$

Answer 1

(i) Let x = 0.8
∴ x = 0.888 ...(1)
10x = 8.888 ...(2)
On subtracting equation (1) from (2), we get
9x = 8 ⇒ x = $\frac{8}{9}$

∴ 0.8 = $\frac{8}{9}$

(ii) Let x = 2.4
∴ x = 2.444 ...(1)
10x = 24.444 ...(2)
On subtracting equation (1) from (2), we get
9x = 22 ⇒ x = $\frac{22}{9}$

∴ 2.4 = $\frac{22}{9}$

(iii) Let x = 0.24
∴​ x = 0.2424 ...(1)
100x = 24.2424 ...(2)
On subtracting equation (1) from (2), we get
99x = 24 ⇒ x = $\frac{8}{33}$

∴ 0.24 = $\frac{8}{33}$

(iv) Let x = 0.12
∴ x = 0.1212 ...(1)
100x = 12.1212 ...(2)
On subtracting equation (1) from (2), we get
99x = 12 ⇒ x = $\frac{4}{33}$

∴ 0.12 = $\frac{4}{33}$

(v) Let x = 2.24
∴ x = 2.2444 ...(1)
10x = 22.444 ...(2)
100x = 224.444 ...(3)
On subtracting equation (2) from (3), we get
90x = 202 ⇒ x =$\frac{202}{90}$= $\frac{101}{45}$

Hence, 2.24 = $\frac{101}{45}$

(vi) Let x = 0.365
∴ x = 0.3656565 ...(1)
10x = 3.656565 ...(2)
1000x = 365.656565 ...(3)
On subtracting (2) from (3), we get
990x = 362 ⇒ x = $\frac{362}{990}=\frac{181}{495}$

Hence, 0.365 = $\frac{181}{495}$