Question

What is .194 repeating with the 94 repeating?

Answer 1

`0.1bar(94) = 193/990`

Explanation:

Using a viniculum (over bar) to indicate the sequence of decimals that repeat, we can write:

`0.194949494... = 0.1bar(94)`

We can make this into a fraction by multiplying by `10(100-1)` then dividing by the same:

`10(100-1) 0.1bar(94) = 194.bar(94) - 1.bar(94) = 193`

So:

`0.1bar(94) = 193/(10(100-1)) = 193/990`

This is in simplest form since the greatest common factor of `193` and `990` is `1`

Notice that multiplying by `10(100-1)` has the effect of:

• First shifting the number one place to the left so the repeating pattern starts immediately after the decimal point.

• Shifting the number two further places to the left (the length of the repeating pattern), then subtracting the original to cancel out the repeating tail.