If we were given $\frac{7}{9}$79 and asked to express it as a decimal, we could enter it into our calculator and get $0.777777777$0.777777777. Our calculator eventually runs out of space, but we know that we have a recurring decimal $0.\overline{7}$0.7 and that the sevens in $0.777777777$0.777777777... go on forever.
Now, what if we were given the recurring decimal $0.0\overline{12}=0.012121212$0.012=0.012121212... and asked to convert it into a fraction? We could type $0.012121212$0.012121212 into our calculator and press the button to convert it into a fraction, but most calculators won't do it, and even if they do, it will give you the fraction for the terminating decimal $0.012121212$0.012121212, NOT for the recurring decimal $0.0\overline{12}$0.012.
Fortunately, we can use what we've learnt about basic algebra to convert any recurring decimal to a fraction. To do this, we use a little trick to get rid of the endless recurring part of the decimal.
Say we have $59.\overline{4}=59.444444444$59.4=59.444444444... and $32.\overline{4}=32.444444444$32.4=32.444444444... .
If we subtract $32.\overline{4}$32.4 from $59.\overline{4}$59.4 then we have:
$59.\overline{4}-32.\overline{4}$59.4−32.4 | $=$= | $59.444444444$59.444444444... |
$=$= | $-32.444444444$−32.444444444.... | |
$=$= | $27$27 |
The recurring part cancels out! This makes sense, since $0.\overline{4}-0.\overline{4}$0.4−0.4 equals nothing.
This fact will come in handy when we are converting recurring decimals to fractions.
Let's learn out conversion method using the fraction $0.0\overline{12}$0.012.
Let $x$x equal the recurring decimal you are trying to convert.
$x=0.0\overline{12}$x=0.012
Look for the repeating digits in the recurring decimal. In $0.012121212$0.012121212... , they are '$12$12'.
Shift one lot of the repeating digits to the left of the decimal place. Remember that multiplying by $10$10, $100$100, $1000$1000, etc. will shift the digits left across the decimal place by $1$1 place, $2$2 places, $3$3 places, etc.
Therefore, $1000\times0.012121212$1000×0.012121212...$=$=$12.1212121$12.1212121... and so we have a new equation for $x$x.
$1000x=12.1212121$1000x=12.1212121...
Check your original equation $x=0.012121212$x=0.012121212... . Do the repeating digits start on the right? In this case, they don't, so we'll have to make a new equation where they do.
$10x=0.12121212$10x=0.12121212...
We have our two equations $1000x=12.1212121$1000x=12.1212121... and $10x=0.12121212$10x=0.12121212... .
We subtract the corresponding sides of the equations like so.
$1000x-10x$1000x−10x | $=$= | $12.\overline{12}-0.\overline{12}$12.12−0.12 |
$990x$990x | $=$= | $12$12 |
$x$x | $=$= | $\frac{12}{990}$12990 |
$x$x | $=$= | $\frac{2}{165}$2165 |
And so our recurring decimal $0.0\overline{12}$0.012 is in fact the fraction $\frac{2}{165}$2165.
If you follow the method above, you can convert any recurring decimal into a fraction!
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