Question

How do you convert 9.12 (2 repeating) to a fraction?

Answer 1

`9.1bar(2) = 821/90`

Explanation:

Multiply by `10(10-1) = (100-10)` then divide by it.

The first factor `10` is to shift the repeating portion to just after the decimal point. The factor `(10-1)` is designed to shift the number left by `1` digit (the length of the repeating pattern), then subtract the original to cancel out the repeating tail.

`(100-10)9.1bar(2) = 912.bar(2) - 91.bar(2) = 821`

So:

`9.1bar(2) = 821/(100-10) = 821/90`

This simplifies no further since `821` and `90` have no common factor.

Answer 2

`9 11/90`

Explanation:

In addition to George C's explanation, here is a simple rule to use.

If ALL the decimal digits recur, write a fraction as follows:

`("numerator")/("denominator") = ("the digit(s) which recur")/(" a 9 for each digit which recurs")`

`a-51 = 443`
`0.4444...... = 4/9`
`5.131313.... = 5 13/99`
`6.742742742... = 6 742/999`

If only SOME of the decimal digits recur:

`("numerator")/("denominator")`

= `("all the decimal digits minus those which do not recur")/(" a 9 for each digit which recurs, 0 for each that does not")`

`9.1color(red)(2)222222... = 9 (1color(red)(2)-color(blue)(1))/(color(red)(9)color(blue)(0)) = 9 11/90`

`4.1color(red)(35)3535.. = 4 (1color(red)(35)-color(blue)(1))/(color(red)(99)color(blue)(0)) = 4 134/990`

`0.267777... = (267-26)/900 = 241/900`