Question

How do you convert 0.47 (47 being repeated) to a fraction?

Answer 1

`0.bar(47) = 47/99`

Explanation:

First note that we can indicate the repeating part of a decimal representation by placing a bar over the repeating pattern:

`0.47474747... = 0.bar(47)`

To find the equivalent fraction, first multiply by `(100-1)` to get an integer:

`(100-1) 0.bar(47) = 47.bar(47)-0.bar(47) = 47`

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

`0.bar(47) = 47/(100-1) = 47/99`

This is in simplest form, since `47` and `99` have no common factor larger than `1`

Why `(100-1)` ?

Multiplying by `100` has the effect of shifting the number `2` places to the left, which is the length of the repeating pattern. Then the `-1` has the effect of subtracting the original, resulting in the cancellation of the recurring tail.

Alternative method

If you know that `1 = 0.bar(9)`, then you can say:

`0.bar(47) = 47/99 * 0.bar(99) = 47/99 * 1 = 47/99`