How do you change repeating decimals into fractions?

1 Answer
Apr 8, 2015

Multiply so that the decimal parts subtract to leave #0#. Then subtract and divide. Finally simplify.

Example:

Change #0.743252525252525 . . . # to a fraction.

Solution: Let #x = 0.743252525252525 . . . #

We want two multiples of #x# whose decimal parts are #.25252525...#

#1000x = 10^3 x= 743.25252525...#

And #100000x = 10^5 x = 74325.252525...#.

Subtracting the first from the second, we get:

#99000x= 73582#

So #x= 73582/99000 = 36791 / 49500#

Easier example

Change #n = 0.246246246 . . . # to a fraction.

The repeating part is 246, so we want two multiples with that for their decimal parts.

We can use the 'multiple' #1n = 0.246246246. . . #
And #1000n = 246.246246246. . . #

Subtracting the first from the second gives us:

#999n=246#

So #n = 246/999 = 82/333#.