what is the easiest way to multiply/divide two large numbers mentally?

Answer 1

if you want a less accurate answer, the best way is to round off the to numbers in such a way that it will be easily divided or multiplied then multiply or divide but if you want an accurate answer, there is no direct way to do it.

Answer 2

There is no direct sure way, sometimes it is better to substract the numerator from denominator till numerator becomes larger than denom.
sometimes it is good to approximate the number on top to a convenient value and later work on the remainder (or excess) in a similar manner.

remembering expansions such as 1/(1-x)= 1+x+x^2 + etc. will work (x should be less than 1 this could be invoked at a later stage when the numbers to use become smaller)

lots of other tricks could be useful to understand mutual divisibility and things , I came across a book called Vedic Mathematics (indian book i think author is KApoor) there are lots
of tricks one could use here.

Answer 3

use short division