Question

How to express 0.2545454... as a fraction in it's simplest form?

Answer 1

`:.=(1,272,727)/(5,000,000)`

Explanation:

`0.2545454`

`:.cancel2545454^1272727/(cancel10000000)^5000000`

`:.=(1,272,727)/(5,000,000)`

Answer 2

`0.2bar(54) = 14/55`

Explanation:

A common notation to indicate the repeating part of a decimal representation is a viniculum - a bar placed above the group of digits that repeat.

So we can write `0.2545454...` as `0.2bar(54)`

To find an equivalent fraction, first multiply by an integer that will give an integer product.

A suitable multiplier is `10(100-1)`. The first factor `10` shifts the number one place to the left so that the repeating part starts just after the decimal point. Then the `100` shifts a further two places to the left - the length of the repeating pattern. Then the `-1` subtracts the unshifted number to cause the tail to be cancelled out...

`10(100-1)0.2bar(54) = (100-1)2.bar(54) = 254.bar(54) - 2.bar(54) = 252`

Then divide both ends by `10(100-1)` to find:

`0.2bar(54) = 252/(10(100-1)) = 252/990 = (14 * color(red)(cancel(color(black)(18))))/(55 * color(red)(cancel(color(black)(18)))) = 14/55`