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Change Recurring Decimals to Fractions

Before looking at how to change recurring decimals to fractions, it’s best looking at some more basic cases.

For the times when we want to change some shorter decimal numbers to fractions, we can go about this process in some standard stages.

Change a Short Decimal to a Fraction Steps

We can look at a standard decimal such as   0.6.

In order to change this decimal number to fraction form, the first stage would be to write  0.6  over  1.


The next step is to multiply both the top and bottom by the same power of  10.
Which power of  10,  depends on how many digits are after the point in the decimal number.

The decimal  0.6  has only 1 digit after the decimal point, so we can just do a multiplication by the number  1010  to power  1.

\bf{\frac{0.6 \space \times \space 10}{1 \space \times \space 10}}   =   \bf{\frac{6}{10}}

After multiplication, the final step is to simplify the fraction obtained if possible.     \bf{\frac{6}{10}}  =  \bf{\frac{3}{5}}

So:   0.6  =  \bf{\frac{3}{5}}



Convert  0.75  to fraction form.


0.75  has 2 numbers after the decimal point, so we multiply by 10 to the power of 2, which is 100.

\bf{\frac{0.75 \space \times \space 100}{1 \space \times \space 100}}   =   \bf{\frac{75}{100}}       ,     \bf{\frac{75}{100}}  =  \bf{\frac{3}{4}}


Convert  0.425  to fraction form.


\bf{\frac{0.425 \space \times \space 1000}{1 \space \times \space 1000}}   =   \bf{\frac{425}{1000}}       ,     \bf{\frac{425}{1000}}  =  \bf{\frac{17}{40}}


Convert  4.75  to fraction form.


We can initially leave the  4  out to the side.

\bf{\frac{0.75 \space \times \space 100}{1 \space \times \space 100}}   =   \bf{\frac{75}{100}}       ,     \bf{\frac{75}{100}}  =  \bf{\frac{3}{4}}

Now bringing the  4  back in.

4.75  =  4\bf{\frac{3}{4}}

Change Recurring Decimals to Fractions

In Math we can have repeating decimals, sometimes referred to as recurring decimals, which are decimal numbers where the digits after the decimal point keep on repeating in a pattern.

Such decimal numbers can also be converted to a fraction, so we can change recurring decimals to fractions when we wish.



Convert   0.7777…   to fraction form.


The first step with change decimals to fractions examples such as this, is to set the decimal number equal to a variable,  x  for example.         0.7777….  =  x

We can then multiply both sides by  10  to get a second equation, that importantly also follows the same repeating pattern.

( 0.7777….  =  x ) × 10     =>     7.7777….  =  10x

Now at this point, we can subtract the first equation away from the second, as this will get rid of the decimal expansion.

\begin{array}{r} \space\space\space\space{{7.7777... \space = \space 10x}}\space\\ {\text{--}}\space\space\space\space{{0.7777... \space = \space\space\space x}}\space\space\space\\ \hline {{7 \space = \space 9x}}\space\space\space \end{array}

The result is now a basic equation that can be easily solved to give us our fraction.
7  =  9x       =>       {\frac{7}{9}}  =  x

So  0.7777  =  {\frac{7}{9}}   as a fraction.


Convert   2.78787878…   to fraction form.


Like before, we first set the repeating decimal number to be equal to a variable.         2.78787878…  =  x

Now we can see that the pattern after the decimal point repeats every  2  places, so this time to obtain the 2nd equation featuring the same repeating pattern after the decimal.
The multiplication will be by  100,  as this moves the decimal point  2  places right.

( 2.78787878….  =  x ) × 100     =>     278.78787878….  =  100x

Now again subtracting the 1st equation from the 2nd removes the repeating decimal expansion.

\begin{array}{r} \space\space\space\space\space{{278.787878... \space = \space 100x}}\space\\ -\space\space\space\space\space\space\space{{2.787878... \space = \space\space\space x}}\space\space\space\space\space\\ \hline {{276 \space = \space 99x}}\space\space\space \end{array}

276  =  99x       =>       {\frac{276}{99}}  =  x

Which can be simplified to   {\frac{92}{33}}  =  x.

So  2.78787878  =  {\frac{92}{33}}   as a fraction.


Convert   3.4090909…   to fraction form.


The extra digit  4  just after the decimal before the repeating pattern starts does give us a bit more to think about.
But it really just means that we’ll need an extra multiplication in order to carry out the decimal conversion.

3.4090909…  =  x

Firstly a multiplication by  10  brings the  4  out from the decimal expansion and leaves a recurring pattern.

( 3.4090909….  =  x ) × 10     =>     34.090909….  =  10x

The original expression and our new one from the multiplication with  10  don’t however have the same decimal expansion.
We can multiply again by  1’000  though, and this will give us a suitable extra expression.

( 3.4090909….  =  x ) × 1’000     =>     3409.090909….  =  1’000x

\begin{array}{r} \space\space\space\space\space{{3409.090909... \space = \space 1'000x}}\space\\ {\text{--}}\space\space\space\space\space\space\space\space{{34.090909... \space = \space\space\space 10x}}\space\space\space\space\\ \hline {{3375 \space = \space 990x}}\space\space\space\space \end{array}

3375  =  990x       =>       {\frac{3375}{990}}  =  x

Which can be simplified to   {\frac{75}{22}}  =  x.

So  3.4090909  =  {\frac{75}{22}}   as a fraction.

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