Question

Express the following as a fraction.

`0.bar(213)`

Answer 1

There are two possible types of repeating decimals with notation like this:

1. Pure repeating decimal: where the digits after the decimal point entirely repeat in blocks of the repeating pattern.

For `0.bar(213)`, this means 0.213213213...

This is what I initially calculated, giving `x = 213/999 = 71/333`

2. Mixed repeating decimal: where some digits are non-repeating, followed by a repeating pattern.

If the notation `0.bar(213)` is being interpreted as `0.2bar(13)` meaning the 2 is non-repeating and only 13 repeats, that is 0.213131313... then the fraction form will differ.

Let’s convert `0.2bar(13)` i.e., 0.2131313... to a fraction:

Let x = 0.2131313...

Since the repeating part has 2 digits 13, multiply by 10³ = 1000 to shift three decimal places 1000x = 213.131313...

Also multiply by 10¹ = 10 to shift the non-repeating part 10x = 2.131313...

Subtract 1000x – 10x = 213.131313... – 2.131313... = 211

Thus 990x = 211

Solving for x: x = `211/990`

`0.bar(213)` as a pure repeating decimal 213 repeating equals `71/333`.

`0.2bar(13)`, 2 non-repeating digit, 13 repeating equals `211/990`.