# How do you change 0.244444444 into a fraction?

Nov 6, 2015

This is the technique also for dealing more difficult ones. You have to decide what to multiply by guided by the repeat cycle.

#### Explanation:

Let $x = 0.244444 \ldots .$..............(1)

so $10 x = 2.44444$.....(2)

(2) - (1)

$10 x - x = 2.2$
$9 x = 2 \frac{2}{10}$

$x = \frac{2}{9} + \frac{2}{90}$

$x = \frac{20 + 2}{90}$

$x = \frac{22}{90}$

$x = \frac{11}{45}$

Nov 6, 2015

Convert into a terminating continued fraction, then simplify to find

$0.2 \dot{4} = \frac{11}{45}$

#### Explanation:

$0.2 \dot{4}$ is less than $1$ so the continued fraction starts $0 + \frac{1}{\ldots}$

Calculate $\frac{1}{0.2 \dot{4}} = 4. \dot{0} \dot{9}$

So our continued fraction looks like $0 + \frac{1}{4 + \frac{1}{\ldots}}$

Subtract $4$ then calculate $\frac{1}{0. \dot{0} \dot{9}} = 11$

So our fraction terminates here in the form: $0 + \frac{1}{4 + \frac{1}{11}}$

$0 + \frac{1}{4 + \frac{1}{11}} = \frac{1}{\frac{44}{11} + \frac{1}{11}} = \frac{1}{\frac{45}{11}} = \frac{11}{45}$