Question

Express each of the following decimals in the form $\frac{p}{q}$ :

(i) 0.$\overline{4}$ (ii) $0.\overline{37}$

(iii) $0.\overline{54}$ (iv) $0.\overline{621}$

(v) $125.\overline{3}$ (vi) $4.\overline{7}$

(vii) $0.4\overline{7}$

Answer 1

(i) 0.$\overline{4}$

Let $x=0.\overline{4}$ = 0.4444.... ....(i)
10 x = 4.4444... ....(ii)

Subtracting (i) from (ii)

9x = 4 $\Rightarrow$ x = $\frac{4}{9}$

$\therefore 0.\overline{4}=\frac{4}{9}$

(ii) $0.\overline{37}$

Let x = $0.\overline{37}$ = 0.373737... ....(i)

100 x = 37.373737... ....(ii)

Subtracting (i) from (ii)

99 x = 37 $\Rightarrow$ x = $\frac{37}{99}$

$\therefore$ $0.\overline{37}$ = $\frac{37}{99}$

(iii) $0.\overline{54}$

Let x = $0.\overline{54}$ $\Rightarrow$ x = $\frac{54}{99}$

$\Rightarrow$ x = $\frac{6}{11}$

$\therefore$ $0.\overline{54}$ = $\frac{6}{11}$

(iv) $0.\overline{621}$

Let x = $0.\overline{621}$ = 0621621621 .... ...(i)

1000x = 621.621621621... ...(ii)

Subtracting (i) from (ii)

999x = 621 $\Rightarrow$ x = $\frac{621}{999}$

$\Rightarrow$ x = $\frac{23}{37}$

(Dividing the numerator and denominator by 27)

(v) $125.\overline{3}$

Let x = $125.\overline{3}$ = 125.333.... ...(i)

10x = 1253.333... ...(ii)

Subtracting (i) from (ii)

9x = 1253 - 125 = 1128

x = $\frac{1128}{9}$ = $\frac{376}{3}$

$\therefore$ $125.\overline{3}$ = $\frac{376}{3}$

(vi) $4.\overline{7}$

Let x = $4.\overline{7}$ = 4.777... ...(i)

10x = 47.777... ...(ii)

Subtracting (i) from (ii)

9x = 47 - 4 = 43

$\Rightarrow$ x = $\frac{43}{9}$

$\therefore$ $4.\overline{7}$ = $\frac{43}{9}$

(vii) $0.4\overline{7}$

Let x = $0.4\overline{7}$

10 x = $0.4\overline{7}$ = 4.777... ...(i)

and 100x = 47.777... ...(ii)

Subtracting (i) from (ii)

90x = 43 $\Rightarrow$ x = $\frac{43}{90}$

$\therefore$ $0.4\overline{7}$ = $\frac{43}{90}$