Question

How to convert $2.4\stackrel{\bullet }{3}$ in $\frac{p}{q}$ form?

Answer 1

$2.4\overline{3}$

$2+.4\overline{3}$

$2+.40+.03+.003+.0003+.........$

$2+.40+\left(\frac{3}{100}+\frac{3}{1000}+\frac{3}{10000}+...........\infty \right)$

$2+.40+\frac{3}{100}\left(1+\frac{1}{10}+\frac{1}{100}+.......\infty \right)$

$2+.40+\frac{3}{100}\left(\frac{1}{1-\frac{1}{10}}\right)$

$2+\frac{4}{10}+\frac{3}{100}\times \frac{10}{9}$

$2+\frac{4}{10}+\frac{1}{30}$

$\frac{60+12+1}{30}=\frac{73}{30}$

Hence $\frac{p}{q}=\frac{73}{30}.$