Question

What is `0.583333333...` (repeating 3) as a fraction?

Answer 1

As others have said `0.583bar3 =7/12`. Here is another way to see this.

Explanation:

Let `x=0.5833bar3`

Multiply by powers of 10 to get repeated 3's only, on the right for two different multiples of `x`.

`1000x=583.3bar3`
`100x =58.3bar3`

Subtract to get
`900x=525`

So `x=525/900 = (25xx21)/(25xx36) = 21/36=7/12`

Answer 2

`7/12`

Explanation:

Here's yet another way:

Knowing that `0.bar(3) = 1/3`, try subtracting that first:

`0.58bar(3) - 0.bar(3) = 0.58bar(3)-0.33bar(3) = 0.25 = 1/4`

So `0.58bar(3) = 1/3+1/4 = 4/12+3/12 = 7/12`

Answer 3

`525/900 =7/12`

Explanation:

In addition to all the methods above which explain what to do and why it works, there is also just a simple rule which you can learn and apply quickly to get the fraction. It is based on one of the methods already described.

There are two cases:

• If ALL the digits recur: write a fraction made up as follows:

`"the recurring digits"/"a 9 for each recurring digit"`

Then simplify if possible.

Eg: `0.432432432.... = 432/999 =16/37`

Eg: `0.62626262.... = 62/99`

• If only SOME of the digits recur: write a fraction made up as:

`"all the digits - non-recurring digits"/"9 for each recurring and 0 for each non-recurring digit"`

Then simplify if possible.

We have:

`0color(blue)(.58)color(red)(3)33333... = (583-58)/(color(red)(9)color(blue)(00)) = 525/900`

`525/900 =7/12`

Another example

`7.4561616161... = 7 (4561-45)/9900 = 7 4516/9900`

`=7 1129/2475`

Answer 4

`0.58bar(3) = 7/12`

Explanation:

If you have access to a pocket calculator, then another way of finding fractions from decimal expansions uses continued fractions.

• Note down any whole number part of the given number.

• Subtract the whole number part.

• Take the reciprocal.

Repeat these steps until there is no remainder, rounding to eliminate rounding errors.

Use the numbers you have noted down to write a terminating continued fraction, then simplify it.

In our example, given `0.58bar(3)`, type as many digits of the expansion as will fit into your calculator, say:

`0.583333333`

This has no whole number part so just take the reciprocal to get:

`1.714285715`

Note down the whole number part `color(red)(1)` and subtract it to get:

`0.714285715`

Take the reciprocal to get:

`1.399999998`

There's an obvious rounding error here, so round to:

`1.4`

Note down the whole number part `color(red)(1)` and subtract it to get:

`0.4`

Take the reciprocal to get:

`2.5`

Note down the whole number part `color(red)(2)` and subtract it to get:

`0.5`

Take the reciprocal to get:

`2`

Since this is a whole number, note it, `color(red)(2)`, and stop.

Taking the numbers we found, we can deduce that:

`0.58bar(3) = 1/(color(red)(1)+1/(color(red)(1)+1/(color(red)(2)+1/color(red)(2)))) = 1/(1+1/(1+2/5)) = 1/(1+5/7) = 7/12`