Question

How do you convert $0.21$ ($21$ repeating) as a fraction?

Answer 1

See a solution process below :

First:

Let $x=0.\overline{21}$

Then let, $100x=21.\overline{21}$

We can now subtract each side of the first equation from each side of the second equation and solve for $x$:

$100x-x=21.\overline{21}-0.\overline{21}$

$100x-1x=(21+0.\overline{21})-0.\overline{21}$

$(100-1)x=21+0.\overline{21}-0.\overline{21}$

$99x=21+0$

$99x=21$

$\frac{99x}{99}=\frac{21}{99}$

$\frac{99x}{99}=\frac{3\times 7}{3\times 33}$

$x=\frac{7}{33}$

$0.\overline{21}=\frac{7}{33}$