5. Janez, Marjeta, and Tine collected old paper, for which they received a monetary prize. Initially, the prize was to be divided in the ratio $7: 6: 5$. Later, they changed the agreement and divided the prize in the ratio $6: 5: 4$.

a) Which of the two divisions is more favorable for Tine? Justify your answer.

b) Janez received 120 euros more than Marjeta in the second division. How many euros did each of them receive?

## You have 120 minutes to solve the problems.

Solve the problems 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 pen, clearly and neatly. Draw graphs of functions with a pencil. If you solve a problem in multiple ways, clearly indicate which solution the evaluator should grade. If you make a mistake, strike it out.

The National Competition Committee wishes you great success.

# $\triangle \mathrm{MFA}$ 

11th competition in mathematics for students of secondary technical and vocational schools National Competition, April 16, 2011

## Problems for 2nd year

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-03.jpg?height=168&width=414&top_left_y=356&top_left_x=1501)
5. We assume that the total prize is $x$ initially divided into 18 parts, and the second division is into 15 parts. Initially, Janez gets $\frac{7}{18} x$, Marjeta gets $\frac{6}{18} x$, and Tine gets $\frac{5}{18} x$. After the second division, Janez gets $\frac{6}{15} x$, Marjeta gets $\frac{5}{15} x$, and Tine gets $\frac{4}{15} x$. We arrange the numbers in order of size, for example, by expanding to a common denominator: the first division $\frac{35}{90} x, \frac{30}{90} x, \frac{25}{90} x$ and the second division $\frac{36}{90} x, \frac{30}{90} x, \frac{24}{90} x$. We find that Tine gets more in the first division. For the second part of the problem, we set up the equation $\frac{36}{90} x = \frac{30}{90} x + 120$. The solution to the equation is $x = 1800$ euros. Janez gets 720 euros, Marjeta gets 600 euros, and Tine gets 480 euros.

Recording the shares of the first division for Janez $\frac{7}{18} x$, Marjeta $\frac{6}{18} x$, and Tine $\frac{5}{18} x \ldots \ldots \ldots \ldots .1$ point Recording the shares of the second division for Janez $\frac{6}{15} x$, Marjeta $\frac{5}{15} x$, and Tine $\frac{4}{15} x \ldots \ldots \ldots \ldots 1$ point Arranging in order of size: first division: $\frac{35}{90} x, \frac{30}{90} x, \frac{25}{90} x$ and second division $\frac{36}{90} x, \frac{30}{90} x, \frac{24}{90} x \ldots \ldots 1$ point Writing the answer: Tine gets more in the first division ......................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-07.jpg?height=63&width=1639&top_left_y=2196&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-07.jpg?height=46&width=1628&top_left_y=2253&top_left_x=208)

Answer: Janez gets 720 euros, Marjeta gets 600 euros, and Tine gets 480 euros................ 1 point

## Second Year