2. Marko and France each have their own orchard with apple and pear trees. France has $20 \%$ fewer apple trees, but 7 more pear trees than Marko. Together, they have 218 trees. Next year, France plans to plant 22 more apple trees and 24 more pear trees, so that he will have 1.5 times as many apple trees as pear trees. How many apple and pear trees does each of them have this year?

(10 points)

17th Mathematics Competition for Students of Secondary Technical and Vocational Schools

Selection Competition, March 16, 2017

Problems for 2nd Year Students

Time for solving: 45 minutes.
2. Mark has $x$ apple trees and $y$ pear trees. Then France has $0.8 x$ apple trees and $y+7$ pear trees. Together, they have $1.8 x+2 y+7=218$ trees, which gives us the equation $1.8 x+2 y=211$. The following year, France will have $0.8 x+22$ apple trees and $y+31$ pear trees. Since the number of apple trees is 1.5 times the number of pear trees, we get the equation $0.8 x+22=1.5(y+31)$. Rearranging gives the equation $1.6 x-3 y=49$. The solutions to the system of two linear equations with two variables ($1.8 x+2 y=211$ and $1.6 x-3 y=49$) are $x=85$ and $y=29$. This means Mark has 85 apple trees and 29 pear trees, while France has 68 apple trees and 36 pear trees.
Writing or using the number of apple trees, e.g., France has $0.8 x$ ..... 1 point
Writing or using the number of pear trees, e.g., France has $y+7$ ..... 1 point
The relationship for the total number of trees $1.8 x+2 y=211$ ..... 2 points
Writing the equation $0.8 x+22=1.5(y+31)$ ..... 1 point
Rearranging the equation to $1.6 x-3 y=49$ ..... 1 point
Solving the system of equations ..... 1* point
The solution $x=85$ and $y=29$ ..... 1 point
Writing the answer (Mark) ..... 1 point
Writing the answer (France) ..... 1 point

## 17th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools

Preliminary Competition, March 16, 2017

## Solutions and Scoring Guide

(March 16, 2017, 00:04)

A competitor who arrives at the solution by any correct method (even if the scoring guide does not anticipate it) receives all possible points.

A correct method is one that:

- sensibly takes into account the wording of the problem,
- 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 provided. However, if the solution is guessed (and cannot be reached without calculation), and there is no written verification or proof, it is scored with 0 points.

A competitor who has only partially solved the problem, with correct procedures but no clear path to the final solution, cannot receive more than half of the possible points.

The mark ${ }^{\prime * \prime}$ next to the points means that the point or points can be awarded for a correct procedure, even if the calculation is incorrect.

## Second Year