Question

How do you convert 0.83 (3 repeating) to a fraction?

Answer 1

`" so "x= 0.8bar3 = 5/6`

Explanation:

To format this question type you write: 0.8bar3

But you use the hash key just before 0.8bar3
and also at the end. So you end up with

`" "0.8bar3`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let `x=0.8bar3`

Then `10x = 8.bar3`

So `(10x-x)=" " 8.3333bar3`
`color(white)("bbnnn.nnnnnnnbb")underline(0.8333bar3-` Subtracting
`color(white)("bbbbb.bbbbbbbbb")7.5`

`" "9x = 7.5`

Multiply both sides by 10

`" "90x=75`

Divide both sides by 90

`" "x=75/90 = 5/6`

`" so "x= 0.8bar3 = 5/6`

Answer 2

Here's a method using a calculator to help...

Explanation:

Here's another way you can convert decimals to fractions if you have a calculator to hand.

We use the calculator to find the terminating continued fraction expansion for the given number, then unwrap it to a regular fraction.

For our example, type `0.83333333` into your calculator.

Note that the portion before the decimal point is `0`, so write that down:

`color(blue)(0) +`

Take the reciprocal of the given number to get a result something like: `1.2000000048`. We can ignore the trailing digits `48` as they are just a rounding error. So with our new result `1.2` note that the number before the decimal point is `1`. Write that down as the next coefficient in the continued fraction:

`color(blue)(0) + 1/color(blue)(1)`

then subtract it to get `0.2`. Take the reciprocal, getting the result `5.0`. This has the number `5` before the decimal point and no remainder. So add that to our continued fraction as the next reciprocal to get:

`color(blue)(0) + 1/(color(blue)(1)+1/color(blue)(5)) = 0+1/(6/5) = 5/6`

`color(white)()`
Another example

Just to make the method a little clearer, let us consider a more complex example:

Given:

`3.82857142857`

Note the `color(blue)(3)`, subtract it and take the reciprocal to get:

`1.20689655173`

Note the `color(blue)(1)`, subtract it and take the reciprocal to get:

`4.83333333320" "color(lightgrey)"Note the rounding error"`

Note the `color(blue)(4)`, subtract it and take the reciprocal to get:

`1.20000000019`

Note the `color(blue)(1)`, subtract it and take the reciprocal to get:

`4.99999999525`

Let's call that `color(blue)(5)` and stop.

Taking the numbers we have found, we have:

`3.82857142857 = color(blue)(3) + 1/(color(blue)(1)+1/(color(blue)(4)+1/(color(blue)(1)+1/color(blue)(5))))`

`color(white)(3.82857142857) = color(blue)(3) + 1/(color(blue)(1)+1/(color(blue)(4)+5/6))`

`color(white)(3.82857142857) = color(blue)(3) + 1/(color(blue)(1)+6/29)`

`color(white)(3.82857142857) = color(blue)(3) + 29/35`

`color(white)(3.82857142857) = 134/35`