# How do you write 27/99 as a decimal?

Jun 21, 2018

$0.27272727 \ldots \to 0.27 \overline{27}$

#### Explanation:

A fraction written as a decimal is either terminating (has a fixed number of decimal places) or has a cycle of digits that repeat for ever. As we have 99 in the denominator I suspect that the decimal has an infinitely repeating set of digits.

$\textcolor{b l u e}{\text{Using a calculator I get } 0.272727 \ldots .}$

The repeating part can be indicated by putting a bar over the appropriate digits. So I would chose to write this as:

$0.27 \overline{27}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{What if you do not have a calculator?}}$

We are dividing 99 into a number that is less (do NOT use the word 'smaller')

However I can do a sort of cheat. 27 is the same as $270 \times \frac{1}{10}$

This idea can be repeated as many times as you wish as long as you apply the $\times \frac{1}{10} \times \frac{1}{10} \times$ however many $\frac{1}{10}$ you end up with. This will be clearer when I use it.

$\textcolor{w h i t e}{}$
$\textcolor{w h i t e}{}$

$27 \to \textcolor{w h i t e}{\text{ddd}} 270 \textcolor{m a \ge n t a}{\times \frac{1}{10}}$

$\textcolor{red}{2} \times 99 \to \underline{198 \leftarrow \text{ Subtract}}$
$\textcolor{w h i t e}{\text{ddddddddd}} 72$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
But $72 < 99 \text{ so write it as } 720 \times \frac{1}{10}$

$\textcolor{w h i t e}{\text{ddddddddd}} 720 \textcolor{m a \ge n t a}{\times \frac{1}{10}}$

$\textcolor{red}{7} \times 99 \to \textcolor{w h i t e}{\text{d")ul( 693 larr" Subtract}}$
$\textcolor{w h i t e}{\text{dddddddddd}} 27$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$\textcolor{p u r p \le}{\text{And so the cycle goes on and on for ever.}}$

Thus so far we have:

$\textcolor{red}{27} \textcolor{m a \ge n t a}{\times \frac{1}{10} \times \frac{1}{10}} = 0.27$

But the repeats give: $0.27272727 \ldots \ldots \to 0.27 \overline{27}$