The recurring decimal 0.363636… (the block 36 repeating) is written as a fraction in its lowest terms. What is the sum of the numerator and the denominator?

Let $x=0.363636$…. The repeating block has two digits, so multiply by 100: $100x=36.363636$…. Subtracting gives $99x=36$, so $x=36/99$, which cancels by 9 down to 4/11. The sum asked for is 4 + 11 = 15. Leaving the fraction unreduced gives 36 + 99 = 135; cancelling only by 3 gives 12/33 and the total 45; and treating the decimal as the terminating 0.36 = 9/25 gives 34.