Find minimum score from the average score

Question

How can I find the lowest score achieved by a class of 12 pupils in a 100 point exam when the average score is 95?

Answer 1

Hint: Assume that all the other students scored 100.

Answer 2

Assume that $12$ students received $12$ grades in increasing order (but not strictly increasing): $x_1\le x_2\le x_3\le x_4\le x_5\le x_6\le x_7\le x_8\le x_9\le x_{10}\le x_{11}\le x_{12}$.

Total score of $12$ students: $95 \times 12 =1140$.

So the maximum amout of students can have a perfect score of $100$ is $11$, it can't be $12$ because that will go over the total score of the class.

Then we will have $x_1=1140-(x_2+x_3+...+x_{11}+x_{12})$

Use the inequality below:

$x_2+x_3+...+x_{11}+x_{12}\le 100 \times 11 =1100 \Rightarrow x_1 \ge 1140-1100=40$.

also

The equality holds if and only if $11$ of the $12$ students get $100$ points.