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This problem involves determining how many workers belong to each of three categories (A, B, and C) based on the given salary data and changes in salary adjustments.

Let:
- \(x\) = number of workers in category A
- \(y\) = number of workers in category B
- \(z\) = number of workers in category C

We know:
1. The total number of workers is 57:
   \[
   x + y + z = 57
   \]

2. The salary expenditure this month is €62,000. The salaries per worker for the different categories are given as:
   - Category A: €800 per worker
   - Category B: €1,000 per worker
   - Category C: €2,000 per worker

   The total expenditure can be written as:
   \[
   800x + 1000y + 2000z = 62000
   \]

3. Salary adjustments for next month:
   - Category A workers get a 4% raise, so their new salary will be €832.
   - Category B workers' salary remains at €1,000.
   - Category C workers' salary is reduced by 10%, making their new salary €1,800.

   The total expenditure next month is supposed to be 2% less than this month, which means the expenditure will be:
   \[
   0.98 \times 62000 = 60760
   \]
   Thus, the equation for next month's salaries is:
   \[
   832x + 1000y + 1800z = 60760
   \]

Now, we have a system of three equations:
1. \(x + y + z = 57\)
2. \(800x + 1000y + 2000z = 62000\)
3. \(832x + 1000y + 1800z = 60760\)

We can solve this system to find \(x\), \(y\), and \(z\). Let me calculate that.The company has:
- 30 workers in category A,
- 16 workers in category B,
- 11 workers in category C.

Would you like a detailed breakdown of how the solution was obtained, or do you have any further questions?

Here are five related questions you might find helpful:
1. How would the total expenditure change if the salary adjustments were different?
2. What if the total number of workers increased by 10%—how would that affect the budget?
3. What would be the new total salary if the percentage decrease for category C was only 5% instead of 10%?
4. How does the overall salary structure impact the company's cost-saving strategies?
5. What is the percentage difference in salary between category A and category C?

**Tip**: When solving systems of equations, it's helpful to label each variable clearly and use substitution or elimination methods systematically for easier computation.

En una empresa de 57 trabajadores el gasto en salarios en este mes ha sido de 62000 euros. En la empresa hay trabajadores de tres categorías, denominadas A, B y C. Este mes el salario de los trabajadores de la categoría A ha sido de 800 euros, el de los trabajadores de la categoría B de 1000 euros y el de los trabajadores de la categoría C de 2000 euros. Una auditoría externa ha indicado que la desigualdad salarial entre los trabajadores de la empresa es excesiva, por lo que se ha decidido que el próximo mes se incrementará un 4% el salario a los trabajadores de la categoría A, se mantendrá el salario a los trabajadores de la categoría B y se rebajará en un 10% el salario a los trabajadores de la categoría C. De esta manera, el gasto de la empresa en salarios en el próximo mes será un 2% inferior al gasto en salarios de este mes. ¿Cuántos trabajadores de cada categoría tiene la empresa?

Solve a system of equations to find how many workers belong to three categories (A, B, and C) based on salary adjustments in a company with 57 employees. Includes detailed breakdown of salary expenditure and changes. Suitable for high school algebra students.

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