# Assignment 1. (10 points)

Each of the 2017 middle school students studies English or German. English is studied by $70 \%$ to $85 \%$ of the total number of students, and both languages are studied by $5 \%$ to $8 \%$. What is the maximum number of students who can study German.
# Solution.

Let $A$ be the number of people who study English, $N$ be the number of people who study German, and $AN$ be the number of people who study both languages.

Then $N=2017-A+AN$.

It is known that $A \geq 2017 \cdot 0.7=1411.9$ and $AN \leq 0.08 \cdot 2017=161.36$.

Therefore, $N \leq 2017-1412+161=766$.

Note that $N=766$, if $A=1412, AN=161$.

Answer: 766.

| Criterion | Evaluation | Points |
| :---: | :---: | :---: |
| Complete solution. | + | 10 |
| A minor computational error was made. Main estimates are provided. Lack of necessary rigor in obtaining the answer. The answer is correct. | .+ | 8 |
| Main estimates are provided. Lack of necessary rigor in obtaining the answer. The answer is incorrect. | $+/ 2$ | 7 |
| Not all estimates are provided. The answer is not obtained or is incorrect. | $\mp$ | 2 |
| The answer is correct, but the solution is missing. | + | $-/ 0$ |
| The solution does not meet any of the criteria described above. | 0 |  |
| Maximum score |  | 10 |