What is the best estimate of 44.53 divided by 4.66?

There is no "best" estimate other than the exact answer since any other estimate can be improved upon.

44.53/4.66 = 44.5/4.7

Now 44.5/4.7 < 47.0/4.7 = 10

Also 44.5/4.7 > (47-4.7)/4.7 = 4.7*9/4.7 = 9

So the answer is between 9 and 10 and I would go for 9.5

The exact answer is 9.56 to 2 dp.