First, we can write:
x = 0.bar5
Next, we can multiply each side by 10 giving:
10x = 5.bar5
Then we can subtract each side of the first equation from each side of the second equation giving:
10x - x = 5.bar5 - 0.bar5
We can now solve for x as follows:
10x - 1x = (5 + 0.bar5) - 0.bar5
(10 - 1)x = 5 + 0.bar5 - 0.bar5
9x = 5 + (0.bar5 - 0.bar5)
9x = 5 + 0
9x = 5
(9x)/color(red)(9) = 5/color(red)(9)
(color(red)(cancel(color(black)(9)))x)/cancel(color(red)(9)) = 5/9
x = 5/9
Another "quick trick" way to do the specific problems where 1 or 2 or 3 through 8 are repeating is to remember to just put the repeating number over 9. This only works when it is a single digit repeating.:
0 .color(red)(11111).... = 0.barcolor(red)(1) = color(red)(1)/9
0 .color(red)(22222).... = 0.barcolor(red)(2) = color(red)(2)/9
0 .color(red)(33333).... = 0.barcolor(red)(3) = color(red)(3)/9
0 .color(red)(44444).... = 0.barcolor(red)(4) = color(red)(4)/9
0 .color(red)(55555).... = 0.barcolor(red)(5) = color(red)(5)/9
0 .color(red)(66666).... = 0.barcolor(red)(6) = color(red)(6)/9
0 .color(red)(77777).... = 0.barcolor(red)(7) = color(red)(7)/9
0 .color(red)(88888).... = 0.barcolor(red)(8) = color(red)(8)/9
0 .color(red)(99999).... = 0.barcolor(red)(9) = color(red)(9)/9 = 1
