5. In the company, there are 22 workers who all have the same salary. In addition, there are two consultants who have the same salary as each other. The average salary is 775 euros. Next month, they will hire another consultant who will have the same salary as the two already employed consultants. The average salary will then increase to 852 euros. Calculate and write down the salary of the worker and the salary of the consultant.

## You have 120 minutes to solve.

Solve each problem independently on the provided paper, with each problem on a separate page. Do not sign the sheets, write only your code. Write your answers in ink, clearly and neatly. Draw graphs with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, cross it out.

The solution to each problem will be graded from 0 to 6 points.

The National Competition Committee wishes you great success.

## 9th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 18, 2009

## Problem Solutions and Scoring Guide

## A competitor who arrives at the correct solution by any valid method (even if not anticipated by the scoring guide) receives all possible points.

A correct method is one that

- sensibly takes into account the problem statement,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the competitor generally receives all the points anticipated. However, if the solution is guessed (and cannot be arrived at without calculation), such a solution is scored with 0 points.

A competitor who has only partially solved the problem, with correct but incomplete steps, cannot receive more than half of the possible points.

## First Year
5. We use the formula for the average - arithmetic mean $\bar{x}=\frac{f_{1} \cdot x_{1}+f_{2} \cdot x_{2}}{N}$ and get $\frac{22 \cdot a+2 \cdot b}{24}=775$ and $\frac{2 \cdot a+3 \cdot b}{25}=852$. We eliminate the denominators in both equations and get a system of two linear equations with two unknowns: $22 a+2 b=18600$ and $22 a+3 b=21300$. We subtract the equations and get $b=2700$, substitute into one of the equations and get $a=600$.

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-14.jpg?height=63&width=1636&top_left_y=454&top_left_x=210)
Writing the equation $\frac{22 \cdot a+2 \cdot b}{24}=775$ and $\frac{2 \cdot a+3 \cdot b}{25}=852 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .1+1^{*}$ point Calculation of $b=2700$ 1 point

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-14.jpg?height=52&width=1639&top_left_y=608&top_left_x=208)

Writing the answer: The worker's salary is 600 euros, and the consultant's salary is $2700 \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots$ point