# I am wondering how I can change a repeating decimal to a fraction?

##### 3 Answers

If all the decimal digits recur, then multiply the number by

#### Explanation:

Let's take an example.

This number has three repeating digits. So you can multiply by

which is

so you have

or

and thus

While the answer can be worked out by a full process as explained by another contributor, there is a useful short cut which is quick to use.

**If all the decimals recur:**

Write the fraction as:

eg:

eg:

**If only some of the decimals recur**

Write the fraction as:

eg:

eg:

eg:

Here's an alternative method if you have a calculator...

#### Explanation:

An alternative method if you have a calculator, but not all of the digits is to use continued fractions.

Find the coefficients of the (terminating) continued fraction by repeatedly separating off the whole number part and taking the reciprocal. Then write down the continued fraction and simplify:

For example, given:

#2.596638655462#

Write down the whole number part

#1.67605633803#

Write down the whole number part

#1.47916666666#

Write down the whole number part

#2.08695652177#

Write down the whole number part

#11.4999999959#

There's obviously a truncation error here, so round to:

#11.5#

Write down the whole number part

#2#

Our final whole number part is

So:

#2.596638655462 ~~ color(red)(2) + 1/(color(red)(1)+1/(color(red)(1)+1/(color(red)(2)+1/(color(red)(11)+1/(color(red)(2))))))#

#color(white)(2.596638655462) = 2+1/(1+1/(1+1/(2+2/23)))#

#color(white)(2.596638655462) = 2+1/(1+1/(1+23/48))#

#color(white)(2.596638655462) = 2+1/(1+48/71)#

#color(white)(2.596638655462) = 2+71/119#

#color(white)(2.596638655462) = 309/119#