Example 1. Two students took four tests, their average scores were different, but both were below 90 and were integers. They then took a fifth test, after which their average scores both increased to 90. What were the scores of the two students on the fifth test?
Let the average score of the first four tests for one of the students be $x$ points, and the score of the fifth test be $y$ points. According to the problem, we have
$\frac{4 x+y}{5}=90$, which means $y=450-4 x$.
However, it is clear that the score of the fifth test is higher than 90 points and no more than 100 points, so we have
$90<450-4 x \leqslant 100$.
Therefore, $87.5 \leqslant x<90$.
Since $x$ is an integer, $x=88$ or 89.
Moreover, the scores of the two students are not equal, meaning the average scores of the first four tests for the two students are 88 points and 89 points, respectively.

Thus, the scores of the fifth test are $y_{1}=98$ points and $y_{2}=94$ points.