When multiplying 2 positive integers, why when we round the larger number up and the smaller number down, the product will be lower and vice versa. I experimented with 2 digits multiplication and found out that when we rounded the larger number up and the smaller number down, the product will be lower. When we rounded the larger number down and the smaller number up, the product will be higher.
For example (We first multiply 97 by 84):
$97\times84=8148$
$98\times83=8134$
$99\times82=8118$  
$96\times85=8160$
$95\times86=8170$
$94\times87=8178$ 
$90\times90=8100$
$90\times91=8190$  
My questions are:
1. How can we explain this? Do we need to use any inequality such as the Geometric Mean or something?
2. It seems that $90\times91=8190$ is the highest, if we we plot a graph can we explain why is it the maximum?
3. Is it a general property of integer, are there any other systems with other operations that have this property, can we explain it more abstractly?
Thank you!

If $x$ and $y$ are integers with $x < y$, then $x + 1 \le y$, so $xy \le xy - x + y - 1 = (x + 1)(y - 1)$.  That's your basic result right there.  If $x + 1 < y$, then the same idea shows that $xy < (x + 1)(y - 1)$, so you can only get equality when $x$ and $y$ are at most $1$ apart.
