Task 11.4. (12 points)

Employees of the company are divided into hard workers and slackers. In 2016, the average salary of hard workers was twice the average salary of slackers. After improving their qualifications, hard workers in 2017 began to earn $50 \%$ more, while the salary of slackers remained unchanged. At the end of 2016, some slackers were laid off. The average salary of all employees in 2017 became $20 \%$ higher than it was in 2016. Find what percentage of the total number of employees were hard workers in 2017, if in 2016 they were $10 \%$.
# Solution.

Let the number of hard workers in the company in 2016 be $x$, then the number of slackers was $9x$, and the total number of people in the company was $10x$. Let the average salary of a slacker be $s$, then the average hard worker received $2s$, and the average salary across the entire company was $\frac{2sx + 9xs}{x + 9x} = \frac{11s}{10}$.

Let the number of slackers remaining in 2017 be $y$, then the average salary across the entire company (considering the 50% salary increase for hard workers) became: $\frac{3sx + ys}{x + y}$.

Thus, we get the equation:

$\frac{3sx + ys}{x + y} = \frac{11s}{10} \cdot \frac{12}{10}$

$100(3sx + ys) = 132(x + y)$

$50(3sx + ys) = 66(x + y)$

$150x + 50y = 66x + 66y$

$84x = 16y$

$y = \frac{21}{4}x$

In the end: $\frac{x \cdot 100\%}{x + y} = \frac{x \cdot 100\%}{x + \frac{21}{4}x} = \frac{4 \cdot 100\%}{4 + 21} = \frac{4 \cdot 100\%}{25} = 16\%$.

Answer. $16\%$.

All-Russian School Olympiad "Mission Possible. Your Calling - Financier!"

| Content of the criterion | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | 12 |
| Correct answer. All main logical steps of the solution are provided. The solution contains arithmetic errors or typos that did not affect the overall course of the solution. | .+ | 10 |
| Correct answer. All main logical steps of the solution are provided. The solution lacks a strict justification for some conclusions. | $\pm$ | 9 |
| The idea of the solution is found, but it is not completed or contains errors. |  |  |
| At the same time, a significant part of the solution is completed, in particular, the system of equations is correctly set up. | $+/ 2$ | 6 |
| The solution is generally incorrect or incomplete, but contains a certain advancement in the right direction. The corresponding system of equations is set up incorrectly. | $\mp$ | 2 |
| The idea of the solution is found, but it is not completed or contains errors. A significant part of the solution is completed. | $+/ 2$ | 6 |
| The solution is generally incorrect or incomplete, but contains a certain advancement in the right direction. | $\mp$ | 2 |
| The solution does not meet any of the criteria described above. | $-/ 0$ | 0 |
| Maximum score |  | 12 |