The first thing to consider is training.
You can immediately tell that 3 x 27 is approximately 90, probably because you notice 27 is not so far from 30 and you know what 3 x 30 is.
Being trained as a physicist and having to divide things fairly often, I can immediately tell you tat 3/27 is not so far from 0.1, because I know that 3/30 = 0.1, and thanks to my experience I could recognize it.
However I think there is more than that.
I would say it hangs on is that multiplication and division are fundamentally different when you consider how they interact with ensemble of numbers. When you multiply two integers, you get an integer. When you divide an integer by another, you can get a rational number. Division is powerful enough that it can produce numbers of a new kind.
Now if you work in the space of rational numbers, and accept the true representation of number to be of the form
a/b, then the two operations are as simple.
a/b x c/d = (a x c)/(b x d) [2 integer multiplications]
(a/b) / (c/d) = (a x d)/(b x c) [2 integer multiplications]
Of course you then need more work if you want to convert them from or to decimal numbers. A work that is much simpler to do for multiplication. But this is not the division fault, we just prefer our base 10 representation to rational numbers.
To conclude, I think that division is more complicated on integers, because integers are fundamentally not suited to represent the result of a division. And since human brains and computer alike prefer integers over other number formats, division is harder for us.
Another related question is why we consider 0.1111 more informative than 3/27 to begin with, they are after all representing the same number. Strictly speaking there is no information gained by performing the computation.
 A poor estimate still, a more experience person would have recognized 3/27 = 3/3^3 = 1/9 = 0.1111 immediately.
 Because in decimal representation, numbers look like
a/10^n, so that multiplication is
a/10^n x b/10^m = (a x b)/10^(n + m) which is immediately a decimal number. While the division is
(a/10^n) / (b/10^m) = 10^(n - m) x a/b and we end up having the same problem as with integers, that
a/b can not be represented as a decimal number in general.
 In fact with training you start to appreciate other format, and for many scientists the answer to 3/27 really is 1/9.
 A decimal number is really just an integer with a dot.