# How can you convert a decimal to fraction?

Apr 5, 2016

See explanation...

#### Explanation:

If it is a terminating decimal, then multiply it by a power of $10$ to make it into an integer, use that power of $10$ as the denominator, then simplify it by dividing the numerator and denominator by any common factors.

For example:

$0.16 = \frac{16}{100} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} \cdot 4}{\textcolor{red}{\cancel{\textcolor{b l a c k}{4}}} \cdot 25} = \frac{4}{25}$

If it is a repeating decimal, then multiply by a power of $10$ to shift the repeating part to just after the decimal point and by a power of $10$ corresponding to the length of the repeating section, minus $1$, to get an integer, then divide and simplify.

For example:

$0.2345345345 \ldots = 0.2 \overline{345}$

Multiply by $10 \left(1000 - 1\right) = 10000 - 10$ to get an integer:

$\left(10000 - 10\right) 0.2 \overline{345} = 2345. \overline{345} - 2. \overline{345} = 2343$

So, dividing by $10000 - 10$ we find:

$0.2 \overline{345} = \frac{2343}{10000 - 10} = \frac{2343}{9990} = \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \cdot 781}{\textcolor{red}{\cancel{\textcolor{b l a c k}{3}}} \cdot 3330} = \frac{781}{3330}$