How we can do this is by making the number (0.7bar8) equivalent to a pro numeral, and for this example, we'll use x
Note: barx just means that x is a reccurring /repeating number, so 0.7bar8 = 0.788888888888888888888....
So now we have:
x = 0.7888.... = 0.7bar8
What we can do is multiply x by 100 to get 100x, and obviously we have to do that to the other side.
x xx 100 = 78.bar8 xx 100
100x = 78.bar8
10x = 7.bar8
The reason why we do this is because now have two numbers, color(brown)(100x = 78.bar8, and color(brown)(10x = 7.bar8, so now we can cancel out the two repeating decimals, and then subtract the second number from the first, to get a whole integer.
(100x = 78. cancel(88bar8)) - (10x = 7. cancel(88bar8)) = (90x = 71)
Now we can find x by using algebra.
90x = 71
Divide each side by 90 to find x
(color(red)(cancel(color(black)90))x)/cancel(color(red)(90)) = 71/90
x= 71/90
color(blue)(0.7bar8 = 71/90
Because we can not simplify any further, this is our final answer.
I got all of this information from Khan Academy's videos on this , you can check it out here:
Converting Repeating Decimals to Fractions 1
Converting Repeating Decimals to Fractions 2
Hope this Helps :)