decimal to fractions

Question

When being asked how to solve the Arithmetic Means of $8, 7, 7, 5, 3, 2,$ and $2$, I understand that adding these numbers then dividing by $7$ (the amount of numbers) gives me the decimal $4.85714...$ But when being asked to change this number into a fraction, I do not understand where the $\frac{6}{7}$ comes from in the answer of $4\frac{6}{7}$. Can someone please help me understand?

Answer 1

Adding $8, 7, 7, 5, 3, 2,$ and $2$ gives you $34$. Divide that by $7$. (manually, not with a calculator)

you will get a quotient of $4$ and a remainder of $6$

that gives you the answer $4\frac{6}{7}$ = 4.85714...

Answer 2

Another way to see it:

Let $a = 0.\overline{857142}$ (the overline part repeats).

Then $1,000,000a = 857,142.\overline{857142}$, so:

$999,999a = 857,142$, and then:

$a = \dfrac{857,142}{999,999} = \dfrac{95,238}{111,111} = \dfrac{8,658}{10,101}$

$= \dfrac{666}{777} = \dfrac{6}{7}$