Quick way to do bit AND mapping By writing out a number in binary I can arrive at the value of taking the first x number of bits out of it.

For example 00001101 (13) taking the first 6 bit would be 3
                          taking the first 5 bit would be 1
                          taking the first 4 bit would be 0 etc


This results, I can easily compute by spelling out the bits that makes up the number I am operating on, take the first couple of bits I am interested in and convert to binary.

I was wondering if there is any common short hand step which allows the same operation to be done, but in decimals (more like how we can do addition on decimals by carrying digits over) where I can use it to calculate what the result of taking the first x bits from a value y would be without having to mentally convert to a binary representation.
If your number $x$ is $n$ bit long, the number formed by the first $k$ is $x/2^{n-k}$, rounded down. The number formed by the last $k$ is the remainder in that division.