Answer:
I will answer in English:
We first have a rational number, the quotient between 2 positive integer numbers is 0.57....
And we want to find which number we should add in order to get 1.
Then we have:
A/B = 0.5757...
So if we have:
x = 0.5757....
we have two periodic decimals, then we multiply by 100 (the number of zeros is equal to the number of periodic decimals)
100*x = 57.575757
Now we subtract the original number:
100*x - x = 99*x = 57.575757... - 0.5757 = 57.
Then we have:
99*0.5757... = 57
0.575757... = 57/99
Now that we know the original fraction, if we want to find a fraction such that:
0.57... + a/b = 1
57/99 + a/b = 1
we should use the same denominator in both fractions:
b = 99
57/99 + a/99 = 1
57 + a = 99
a = 99 - 57 = 42
Then the fraction is 42/99
