Question

What is `0.33%` (repeating) as a fraction?

Answer 1

`0.333...%" "=" "0.00bar3" "=" "1/300`.

Explanation:

Careful with percentages—they're not the same as regular decimal numbers! First, we convert the percentage to a decimal number, using this idea:

A percent symbol is short for "divide this number by 100".

For example, `50%` means one-half, right? So `50%` is short for `50 divide 100`, which equals `50/100`, or `5/10`, or `1/2` (one-half).

So we now know `0.333...%=0.00333...`, which can be concisely written as `0.00bar3`. But what is this decimal number as a fraction? To find this, we use the following rule.

The numerator (top number) of our fraction will be the digits under the bar (in this case, `3`).

The denominator (bottom number) is found by writing this string of digits:

1. Count the number of digits under the bar; write this many 9's.
2. Count the number of digits between the decimal and the bar; write this many 0's after the 9's.

So, for the number `0.00bar3`, our numerator is 3. For the denominator, we count one digit under the bar, so we write one 9. Since there are two digits between the decimal and the bar, we write this many 0's after the 9. Our denominator is then `900`.

(This simplified rule works because the digits before the 3 are all 0's. When these are non-zero digits, the fraction takes a bit more work.)

Okay—all of this means our final number is:

`0.333...%" "=" "0.00bar3" "=" "3/900`

But wait—`3/900` can be reduced! Both `3` and `900` are multiples of 3, so we can reduce it by dividing both top and bottom by 3:

`3/900=(3divide 3)/(900divide3)=1/300`

And there we have it! `" "0.bar3%=1/300`.

Answer 2

`1/300`

Explanation:

Write `0.3(3" repeating")%` as `color(white)("ddd")0.3(3" repeating")xx1/100`

`color(blue)("Consider just the 0.3(3 repeating) part")`

Let `x=0.3333333......`

Then `10x=3.3333....`

So `10x-x=3.3333....`
`ul( color(white)("ddddddddddd") 0.3333....larr" Subtract ")`
`color(white)("ddddddddddd") 3.0`

`10x-x=3`

`x(10-1)=3`

`x=3/9 = 1/3`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting it all together")`

But we have to include the `xx1/100` giving:

`1/3xx1/100 = 1/300`