Answers:
a = 4
b = 11
The fraction is 4/11
==================================================
Work Shown:
Let x = 0.363636...
The "36" pattern goes on forever as indicated by the three dots.
Multiply both sides by 100 to get 100x = 36.363636...
Effectively, multiplying by 100 moves the decimal point over 2 spots to the right. The "36" goes on forever here too. The key is to match up the infinite sequences of "36"s and cancel them out through subtraction
36.363636... - 0.363636... = 36
while
100x - x = 99x
So after subtracting the equations, we end up with 99x = 36
From here we solve for x by dividing both sides by 99 and reducing the fraction
99x = 36
x = 36/99
x = (9*4)/(9*11)
x = 4/11
As a check, use your calculator to find that
4/11 = 0.36363636...
so the answer is confirmed
Note: your calculator may display something like 0.3636363636364, that 4 should not be there. It's a result of rounding (due to the next digit being a 6). Keep in mind that rounding error is often a possibility when it comes to approximate calculations. The calculator only has so much memory and display space.