B3. Tone, Luka, and Tine collected old paper and 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 $6: 5: 4$. Both ratios are listed in the same order as the names.

(a) Which division is more favorable for Tine? Justify your answer.

(b) Tone received 216 euros more than Tine in the second division. How many euros did each of them receive?

## 22nd Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 23, 2022

## Problems for 2nd Year

Time for solving: 120 minutes. In section A, we will award three points for each correct answer, and deduct one point for each incorrect answer. Write your answers for section A in the left table, leave the right table blank.
![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-06.jpg?height=168&width=804&top_left_y=590&top_left_x=638)
B3. According to the original agreement, Tine received $\frac{5}{7+6+5}=\frac{5}{18}=\frac{25}{90}$ of the total prize, while after the change, he received $\frac{4}{6+5+4}=\frac{4}{15}=\frac{24}{90}$ of the prize. Therefore, the first division is more favorable for Tine.

We know that the prize was divided in the ratio $6: 5: 4$. Thus, Tonet's share is $6x$, Luka's is $5x$, and Tine's is $4x$. We write the relationship between Tonet's and Tine's shares as $6x - 216 = 4x$ and get $x = 108$. Therefore, Tone received $6x = 6 \cdot 108 = 648$ euros, Luka received $5x = 5 \cdot 108 = 540$ euros, and Tine received $4x = 4 \cdot 108 = 432$ euros.
Writing Tine's share according to the original agreement $\frac{5}{18}$ ..... 1 point
Writing Tine's share after the agreement change $\frac{4}{15}$ ..... 1 point
Conclusion that the original agreement would be more favorable for Tine ..... 1 point
Writing the shares of Tone $6x$, Luka $5x$, Tine $4x$ ..... 1 point
Writing the relationship between Tonet's and Tine's shares $6x - 216 = 4x$ ..... 1 point
Solving the equation $x = 108$ ..... 1 point
Calculating the shares:
Tone $6x = 6 \cdot 108 = 648$ euros, Luka $5x = 5 \cdot 108 = 540$ euros, Tine $4x = 4 \cdot 108 = 432$ euros. 1
point
Answer: Tone received 648 euros, Luka 540 euros, and Tine 432 euros. ..... 1 point

## 22nd Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 23, 2022  Solutions to Problems for 2nd Year

| $\mathrm{A} 1$ | $\mathrm{~A} 2$ | $\mathrm{~A} 3$ |
| :---: | :---: | :---: |
| $\mathrm{A}$ | $\mathrm{D}$ | $\mathrm{A}$ |