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Changing Decimals to Fractions



When we wish to approach changing decimals to fractions, this is a process that can be done following some standard steps.



Decimal to Fraction Steps


Take as an example  0.8.

To change this decimal to a fraction, a first step is to write  0.8  over  1  as a fraction.

1)    \frac{0.8}{1}


2)    For a next step we can multiply both the top and bottom by the same power of  10.

Which power of  10,  will depend on how many numbers there are beyond the decimal point in the relevant decimal number.

0.8  has only  1  number after the decimal point, so we can just do a multiplication by  10  here.

\frac{0.8 \space \times \space 10}{1 \space \times \space 10}    =    \frac{8}{10}


3)    After multiplication, the final step is to simplify the fraction obtained.

\frac{8}{10}   =   \frac{4}{5}

Thus:     0.8   =   \frac{4}{5}





Examples




1.1

Convert  0.75  to a fraction.

Solution

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

\frac{0.75 \space \times \space 100}{1 \space \times \space 100}    =    \frac{75}{100}

\frac{75}{100}    =    \frac{3}{4}



1.2

Convert  0.525  to a fraction.

Solution

0.525  has  3  numbers after decimal point, multiply by  10  to the power of  3,  which is  1000.

\frac{0.525 \space \times \space 1000}{1 \space \times \space 1000}    =    \frac{525}{1000}

\frac{525}{1000}    =    \frac{21}{40}



1.3

Convert  4.2  to a fraction.

Solution

Here the presence of a  4  in front of the decimal, means the fraction that the decimal is converted to will be a mixed number.

Initially we can leave the  3  to the side, and focus just on what sits after the decimal point.

\frac{0.2 \space \times \space 10}{1 \space \times \space 10}    =    \frac{2}{10}

\frac{2}{10}    =    \frac{1}{5}

Now we can bring the  4  back in.

4.2   =   4\bf{\frac{1}{5}}








Changing Decimals to Fractions,
Recurring Decimals


Recurring decimals, sometimes called repeating decimals, are decimal numbers where the numbers after the decimal point keep recurring on and on.

Such decimal numbers can also be converted to a fraction form.


Examples




2.1

Convert the recurring decimal   0.5555….   to a fraction.

Solution

First set the decimal equal to a variable, say  x.         0.5555….  =  x

Multiplying both sides by  10  gives a second equation.

( 0.5555….  =  x ) × 10     =>     5.5555….  =  10x


Now we subtract the first equation from the second.

\begin{array}{r} \space\space\space\space{{5.5555... \space = \space 10x}}\space\\ {\text{--}}\space\space\space\space{{0.5555... \space = \space\space\space x}}\space\space\space\\ \hline {{5 \space = \space 9x}}\space\space\space \end{array}


The result is a nice and neat equation that can be solved to give a fraction value.

5  =  9x        ( ÷ 9 )

\frac{5}{9}  =  x

So.     0.5555...  =  \frac{5}{9}




2.2

Convert the recurring decimal   2.783783783….   to a fraction.

Solution

Like before, set the decimal equal to a variable.         2.783783783….  =  x

Multiplying both sides by  1’000,
gives a second expression with the same decimal expansion.

( 2.783783783….  =  x ) × 1’000
=>     2783.783783783….  =  1’000x


We can then subtract the first equation from the second.
To get rid of the decimal expansion.

\begin{array}{r} \space\space\space\space{{2783.783783783... \space = \space 1'000x}}\\ {\text{--}}\space\space\space\space\space\space\space\space{{2.783783783... \space = \space\space\space x}}\space\space\space\space\space\space\space\\ \hline {{2781 \space = \space 999x}}\space\space\space \end{array}


Now.     2781  =  999x        ( ÷ 999 )

\frac{2781}{999}  =  x

Which can be simplified to   \frac{103}{37}.

2.783783783….  =  \frac{103}{37}








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