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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