How do you convert 0.16 (6 repeating) to a fraction?

Question

lenkiklisg7 · Accepted Answer

First multiply by 10(10−1)=(100−10) to get an integer.  The first multiplier of 10 is to shift the decimal representation one place to the left, so the repeating section begins just after the decimal point. Then the (10-1) multiplier is used to shift the digits one more place to the left (the length of the repeating pattern) and subtract the original to cancel the repeating tail.  (100−10)⋅0.16=16.6−1.6=15  Then divide both ends by (100−10) and simplify  0.16=15100−10=1590=1⋅156⋅15=16

Raymond Foley · Accepted Answer

Note: if the 6 is repeating then a way of showing this is to put a bar over the last 6 you write: →0.166―  Let x=0.166―  Also 100x=16.666―  100x−10x→16.666―  So 90x=15  Divide both sides by 90  x=1590  But 1590→15÷1590÷15=16  ⇒x=0.166―→16

karton · Accepted Answer

The details of how to convert a recurring decimal into a fraction are shown in the other answers.
However, sometimes you just want a quick method.
Here is the short cut.
If all the digits after the decimal point recur:
Write down the digits (without repeating) as the numerator.
Write a 9 for each digit in the denominator. Simplify if possibles
$0.7777 = 0. \overset{―}{7} = \frac{7}{9}$
$0.613613613 . . . = 0. \overset{―}{613} = \frac{613}{999}$
If only some digits recur
Numerator: write down all the digits - non-recurring digits
Denominator: a 9 for each recurring and a 0 for each non-recurring digit
In PSK0.524666... only the 6 recurs while the 524 do not.
$0.524666 = 0.524 \overset{―}{6} = \frac{5246 - 524}{9000} = \frac{4722}{9000}$
$0.1353535 . . . = 0.1 \overset{―}{35} = \frac{135 - 1}{990} = \frac{134}{900}$

How do you convert 0.16 (6 repeating) to a fraction?

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