Question 8
The sum of the following seven numbers is exactly $19:$ $$
\begin{aligned} a_1&=2.56,\\ a_2&=2.61,\\ a_3&=2.65,\\ a_4&=2.71,\\ a_5&=2.79,\\ a_6&=2.82,\\ a_7&=2.86. \end{aligned}
$$ It is desired to replace each $a_i$ by an integer approximation $A_i,$ $1\leq i\leq7,$ so that the sum of the $A_i$’s is also $19$ and so that $M,$ the maximum of the “errors” $|A_i-a_i|,$ is as small as possible. For this minimum $M,$ what is $100M?$
\begin{aligned} a_1&=2.56,\\ a_2&=2.61,\\ a_3&=2.65,\\ a_4&=2.71,\\ a_5&=2.79,\\ a_6&=2.82,\\ a_7&=2.86. \end{aligned}
$$ It is desired to replace each $a_i$ by an integer approximation $A_i,$ $1\leq i\leq7,$ so that the sum of the $A_i$’s is also $19$ and so that $M,$ the maximum of the “errors” $|A_i-a_i|,$ is as small as possible. For this minimum $M,$ what is $100M?$