Question

How do you convert 2.136 (36 repeating) as a fraction?

Answer 1

`2.1bar(36) = 47/22`

Explanation:

There is a notation for repeating decimals that places a bar above the repeating pattern.

Using that notation, the original decimal representation is written:

`2.1bar(36)`

To make this into an integer, multiply by `10(100-1) = 1000-10`. The factor `10` here is to shift the repeating section to just after the decimal point. the factor `(100-1)` shifts the number by an additional `2` places (the length of the repeating pattern) and subtracts the original number to cancel out the repeating part:

`(1000-10)*2.1bar(36) = 2136.bar(36) - 21.bar(36) = 2115`

Then divide both sides by `(1000-10)` and simplify to find:

`2.1bar(36) = 2115/(1000-10) = 2115/990 = (color(red)(cancel(color(black)(3*3*5)))*47)/(color(red)(cancel(color(black)(3*3*5)))*22) = 47/22`