How do people perform mental arithmetic for complicated expressions? 
This is the famous picture "Mental Arithmetic. In the Public School of S. Rachinsky." by the Russian artist Nikolay Bogdanov-Belsky.
The problem on the blackboard is:
$$
\dfrac{10^{2} + 11^{2} + 12^{2} + 13^{2} + 14^{2}}{365}
$$
The answer is easy using paper and pencil: $2$.
However, as the name of the picture implies, the expression ought be simplified only mentally.
My questions: 


*

*Are there general mental calculation techniques useful for performing basic arithmetic and exponents?

*Or is there some trick which works in this case? 

*If so, what is the class of problems this trick can be applied to?

Here is how it was done in my head:
$10^2 = 100$. Okay. $11^2 = 121$. Okay, thats $221$. $12^2 =144$. Aha, thats $365$ total. So far we have a total of $365/365 =1$. Now, $13^2 =169$ and $14^2 =196$, which sum to $365$. So the answer is $2$.
