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?
Best Answer
$$\begin{align}\\&\frac{10^2+11^2+12^2+13^2+14^2}{365}\\&=\frac{(12-2)^2+(12-1)^2+12^2+(12+1)^2+(12+2)^2}{365}\\&=\frac{5\times 12^2+10}{365}\\&=\frac{5(144+2)}{5\times 73}\\&=2\end{align}$$