This page gives a brief overview of conversion of numbers having repeating digits after decimal point into equivalent fractions.

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Convert `1/3` into a decimal. The long-division method to convert the fraction into a decimal is illustrated in the figure.

- `0.3`
- `0.333cdots`
- `0.333cdots`

The answer is "`0.333cdots`". It is noted that the decimal number does not end and number `3` repeats.

Convert `23/9` into a decimal

- `2.555cdots`
- `2.555cdots`
- `0.2555cdots`

The answer is "`2.555cdots`".

Convert `371/990` into a decimal.

- `0.37474 cdots`
- `0.37474 cdots`
- `3.7474cdots`

The answer is "`0.37474cdots`". It is noted that the decimal number does not end and number `7` and `4` repeats.

`371/990 = 0.37474cdots`

In the above representation, it is not clear which part of the digits are repeating

• is `4` repeated? like `0.37474444444cdots`

• is `74` repeated? like `0.37474747474cdots`

• is `37474` repeated? like `0.3747437474cdots`

To avoid the confusion the following representation is adapted. The number is given as

`0.3bar(74)`

The line over `74` represents that `74` is repeated.

Convert `1/3` into a fraction.

- `0.bar(3)`
- `0.333cdots`
- both the above
- both the above

The answer is "both the above".

**Representation of Repeating Decimals** : The repetitive pattern in decimal digits is represented with an over-line.

convert `0.bar(4)` into a fraction

- `4/10 = 2/5`
- `4/9`
- `4/9`

The answer is "`4/9`".

This is explained in the next page.

To convert `0.bar(4)` into a fraction, the following steps are used

`x=0.bar(4)`

`10x = 4.bar(4)`

Subtracting the two equations

`9x = 4`

`x=4/9`

Convert `2.bar(4)` into a fraction.

- `2 2/5`
- `2 4/9`
- `2 4/9`

The answer is "`2 4/9`".

`x=2.bar(4)`

`10x = 24.bar(4)`

subtracting the two above

`9x = 22`

`x=22/9`

`x=2 4/9`

convert `2.23bar(43)` into a fraction

- `2212/990`
- `223/100 + 43/9900`
- both the above
- both the above

The answer is "both the above". This is explained in the next page.

To convert `2.23bar(43)` into a fraction:

`x=2.23bar(43)`

`100x = 223.43bar(43)`

Subtracting the above,

`99x = 221.20`

`990x = 2212`

`x = 2212/990`.

Another method is as follows.

`100x = 223 + 0.bar(43)`

`10000x =223xx100 + 43.bar(43)`

Subtracting the two equations

`9900x = 223xx (100-1) + 43 `

`x=223/100 + 43/9900`

**Conversion of Repeating Decimals to equivalent Fractions** : A decimal has two parts, a non-repeating part at the beginning and repetitive part. With some simple arithmetics, equivalent fraction is derived.

eg: `0.bar(3)` is represented as `10x-x = 3.bar(3) - 0.bar(3)`. The value of `x` is derived as fraction `1/3`.

*slide-show version coming soon*