Question

How do you convert 0.325 (25 being repeated) to a fraction?

Answer 1

`0.3bar(25) = 161/495`

Explanation:

A common notation to denote a repeating decimal is to put a bar over the repeating sequence. In this case, we would write the number in question as `0.3bar(25)`.

The trick we will use is to multiply by a power of `10` and subtract the original value to eliminate the repeating digits.

Let `x = 0.3bar(25)`

`=>100x = 32.5bar(25)`

`=>100x - x = 32.5bar(25)-0.3bar(25)`

`=>99x = 32.2 = 161/5`

`:.x = 161/(5*99) = 161/495`

Answer 2

Multiply by `(1000-10)` and rearrange to find:

`0.3bar(25) = 161/495`

Explanation:

`(1000-10)*0.3bar(25) = 325.bar(25) - 3.bar(25) = 322`

So, dividing both sides by `(1000-10)` we find:

`0.3bar(25) = 322/(1000-10) = 322/990 = 161/495`

Why `(1000-10)` ?

`1000-10 = 10 (100-1)`

The initial factor of `10` is to shift our starting number one place to the left so that the repeating portion starts just after the decimal point.

The `(100-1)` factor is to shift the number a couple more places to the left and subtract the original value. Since the repeating pattern is of length `2` this serves to cancel out the repeating part, leaving us with an integer.