Question

How do you convert repeating decimals into fractions?

Answer 1

Like so...

Explanation:

For example,

Basic repeating decimals

• `0.bar(18)`

`0.bar(18)=18/99`
`color(white)(0.bar(18))=2/11`

• `0.bar(123)`

`0.bar(123)=123/999`
`color(white)(0.bar(123))=41/333`

• `0.bar(4680`

`0.bar(4680)=4680/9999`
`color(white)(0.bar(4680))=52/1111`

Not-so-basic repeating decimals

• `1.0bar(42)`

Let `x=1.0bar(42)`

`1000x-10x=1042.bar(42)-10.bar(42)`
`color(white)(xxxx//)990x=1032`
`color(white)(xxxxxxx.)x=1032/990`
`color(white)(xxxxx)1.0bar(42)=1032/990`
`color(white)(xxxxx1.0bar(42))=172/165`