The class teacher calculated the class average grades for each subject, and Kati helped her by recalculating the grades based on how many students received each consecutive grade. When comparing the results of the first subject, it turned out that Kati had used the data for consecutive fives in reverse order, taking the number of excellent grades as the number of poor grades. Feeling disheartened, Kati wanted to start the entire calculation over, but the class teacher said that this calculation would still be good for verification. Explain this statement.
Let the number of students who received grades 5, 4, 3, 2, 1 in a certain subject be $a, b, c, d, e$ respectively. Thus, the average grade calculated by the class teacher and Kati is:

$$
\frac{5 a+4 b+3 c+2 d+e}{a+b+c+d+e}, \quad \text { or } \quad \frac{5 e+4 d+3 c+2 b+a}{a+b+c+d+e}
$$

It is easy to see that the sum of these two averages is 6 for any distribution of grades $a, b, c, d, e$.

Therefore, any result calculated by Kati that, when added to the correct result calculated by the class teacher, equals 6, indicates that both have calculated correctly. If, however, this does not hold for a certain subject, those calculations need to be rechecked.