Express each of the following decimals in the form

Solution:

(i) Let $x=0 . \overline{4}$

$\Rightarrow x=0.44444 \ldots$

$10 x=4.444 \ldots$

$\Rightarrow 10 x=4+x$

$\Rightarrow 9 x=4$

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

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

$\Rightarrow x=0.373737 \ldots$

$\Rightarrow 100 x=37.3737 \ldots$

$\Rightarrow 100 x=37+0.3737 \ldots$

$\Rightarrow 100 x=37+x$

$\Rightarrow 99 x=37$

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

(iii) Let $x=0 . \overline{54}$

$\Rightarrow x=0.545454 \ldots$

$\Rightarrow 100 x=54.5454 \ldots$

$\Rightarrow 100 x=54+0.5454 \ldots$

$\Rightarrow 100 x=54+x$

$\Rightarrow 99 x=54$

$\Rightarrow x=\frac{54}{99}=\frac{6}{11}$

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

$\Rightarrow x=0.621621621 \ldots$

$\Rightarrow 1000 x=621.621621 \ldots$

$\Rightarrow 1000 x=621+0.621621 \ldots$

$\Rightarrow 1000 x=621+x$

$\Rightarrow 999 x=621$

$\Rightarrow x=\frac{621}{999}=\frac{207}{333}$

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

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

$\Rightarrow x=125+0 . \overline{3}$

$\Rightarrow x=125+\frac{1}{3}\left(\because 0 . \overline{3}=\frac{1}{3}\right)$

$\Rightarrow x=\frac{375+1}{3}$

$\Rightarrow x=\frac{376}{3}$

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

$x=4+0 . \overline{7}$

Let $y=0 . \overline{7}=0.777 \ldots$

$\Rightarrow 10 y=7+0.777 \ldots$

$\Rightarrow 10 y=7+y$

$\Rightarrow y=\frac{7}{9}$

Therefore,

$x=4+\frac{7}{9}=\frac{43}{9}$

(vii) Let $x=0.4 \overline{7}$

$\Rightarrow 10 x=4+0 . \overline{7}$

Since, $0 . \overline{7}=\frac{7}{9}$

Therefore,

$\Rightarrow 10 x=4+\frac{7}{9}=\frac{43}{9}$

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