Alright, we want the fraction to end up with whole numbers instead of decimals, so let's start by multiplying each by 100.
(38.56/19.84)(100/100) = 3856/1984
From here, we can keep dividing. Both are even, so...
(3856/1984)/(2/2) = 1928/992
Keep going until you get your answer.
Remember that a ratio stays as a fraction, even if it's irregular. You wouldn't change 5/2 to 2 1/2 or even 6/1 to 6.
