Assuming you can't use a calculator:
5.575757... = 5 + 0.575757...
let 0.575757... = x
1) Substitute x for 0.575757...
5.575757... = 5 + x
2) Multiply both sides by 100
100(5.575757...) = 100(5+x)
557.575757... = 500 + 100x
3) Subtract 500 from both sides
57.575757... = 100x
4) 57.575757... can be written as 57 + 0.575757...
57 + 0.575757... = 100x
5) Substitute x for 0.575757...
57 + x = 100x
6) Subtract x from both sides
57 = 99x
7) divide both sides by 99
x = 57/99
8) plug in 57/99 for x in our original equation
5.575757... = 5 + (57/99)
9) simplify
5.575757... = ((5*99)/99) + (57/99)
5.575757... = (495/99) +(57/99)
5.575757... = (495+57)/99
5.575757... = 552/99
5.575757... as a fraction is 552/99
