Question

How do you write decimal notation for 5/6?

Answer 1

How I do it in my head is that I start from the fraction `6/6` and subtract `1/6`.

`6/6` is just a ratio that is equal to `1`. Any number divided by itself is equal to `1`. So, we know that `5/6` is less than `1.bar(00)`.

Next, to find `1/6`, what you can do is find what numbers get close to `100` upon multiplying by `6`, then look at the remainder and divide by `6`. So, you can:

1. Divide `100` by `6`.
2. Divide by `100` again.

Since `6xx16 = 96`, we have:

`100/6 = stackrel("quotient")(overbrace(16)) + stackrel("remainder")(overbrace(4/6)`

`= 16 + 2/3`

`= 16.bar(66)`

Therefore:

`((100)/6)/100 = 1/6 = 0.1bar(66)`

Finally, our answer is:

`color(blue)(5/6) = 6/6 - 1/6`

`= 1 - 0.1bar(66)`

`= color(blue)(0.8bar(33))`

Answer 2

`5/6 = 0.833` or shown as `0.8dot3`

Explanation:

6 cannot be multiplied by any number to give an exact power of 10,
so it is not possible to write an equivalent fraction to get a decimal.

Divide 5 by 6:

`5÷6 = 0.8333333...` The answer is a recurring decimal.

It is usually rounded to 2 or 3 decimal places.

Answer 3

`0.83bar3`

Explanation:

`color(blue)("This method is 'Long Division' in disguise")`

Think of 5 as `5+ 0/10+0/100+0/1000+....`

Observe that `5<6" "` ( 5 is less than 6 )

But 5 can be written as `50/10` so we have

`50/10-:6" " ->" " (50-:6)/10" " =" " 8/10+" remainder of " 2/10`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So now we have `5-:6 = 8/10+(2-:6)/10`

But `2/10`can be written as `20/100` giving

`8/10+(20-:6)/100" "->" "8/10+3/100+ "remainder of "2/100`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So now we have:

`5-:6=8/10+3/100+(2-:6)/100`

But `2/100` can be written as `20/1000` giving

`8/10+3/100+(20-:6)/1000` giving `8/10+3/100+3/1000+(2-:6)/1000`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
This process repeats indefinitely so we can write this in decimal form as:

`0.8333.."repeating "3`

Mathematically this is written as

`0.83bar3` or `0.83dot3` depending on your preference.

Notice the bar over the last 3 in `0.83bar3` and the dot over the last 3 in `0.83dot3`