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Let's solve the problem step by step. Here is a summary of the problem:

1. **Company Details**:
   - There are three categories of workers: A, B, and C.
   - The current salaries are:
     - Category A: 800 euros
     - Category B: 1000 euros
     - Category C: 2000 euros
   - The total number of workers is 57.
   - The total salary expense this month is 52,800 euros.

2. **Changes to Salaries**:
   - Next month:
     - Salaries for Category A workers will increase by 4%.
     - Salaries for Category B workers will remain the same.
     - Salaries for Category C workers will decrease by 10%.
   - The new total salary expense will be 2% less than this month's expense.

3. **Objective**: Find the number of workers in each category (A, B, and C).

### Step-by-step Solution:

Let's denote:
- \(x\) = number of workers in Category A,
- \(y\) = number of workers in Category B,
- \(z\) = number of workers in Category C.

We have the following equations based on the given information:

1. **Equation for the total number of workers**:
   \[
   x + y + z = 57
   \]

2. **Equation for the total salary expense this month**:
   \[
   800x + 1000y + 2000z = 52800
   \]

3. **New salaries after the changes**:
   - Category A: \(800 \times 1.04 = 832\) euros
   - Category B: \(1000\) euros (no change)
   - Category C: \(2000 \times 0.9 = 1800\) euros

   The total salary expense next month is 2% less than this month's expense. So, the new expense will be:
   \[
   52800 \times 0.98 = 51744 \text{ euros}
   \]

   **Equation for the total salary expense next month**:
   \[
   832x + 1000y + 1800z = 51744
   \]

### Solving the Equations

Let's solve the system of equations:

1. \(x + y + z = 57\)
2. \(800x + 1000y + 2000z = 52800\)
3. \(832x + 1000y + 1800z = 51744\)

We'll solve this system using substitution or elimination. Let me perform the calculations.

En una empresa de 57 trabajadores, el gasto en salarios este mes ha sido de 52,800 euros. Los trabajadores están clasificados en tres categorías: A, B y C. El salario actual es de 800 euros para la categoría A, 1000 euros para la categoría B y 2000 euros para la categoría C. Se ha decidido aumentar en un 4% el salario de los trabajadores de la categoría A, mantener el salario de los de la categoría B y reducir en un 10% el salario de la categoría C. El gasto total en salarios el próximo mes será un 2% inferior. ¿Cuántos trabajadores hay en cada categoría?

This problem involves 57 workers in a company with salaries distributed across three categories. Using algebraic equations, we solve for the number of workers in each category based on salary adjustments and a total payroll reduction. Suitable for students in Grades 9-12.

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