## Task 3 - 030823

In an auditorium, 300 chairs are arranged in several equally long rows. If 3 chairs are removed from each row to form a central aisle and these chairs are used to create 5 new rows (with a central aisle), the total number of seats remains the same.

How many chairs were originally in each row? Justify your claim!
Let $r$ be the number of rows and $x$ the number of chairs per row. Then initially, $300 = r \cdot x$, i.e., $r = \frac{300}{x}$.

After rearranging the chairs, there are $x-3$ chairs in each of the now $r+5$ rows, i.e., $300 = (r+5) \cdot (x-3)$.

Substituting the first equation into the second, we get

$$
300 = \left(\frac{300}{x} + 5\right) \cdot (x-3)
$$

Rearranging yields the quadratic equation $x^2 - 3x - 180 = 0$, and using the quadratic formula,

$$
x_{1;2} = \frac{3}{2} \pm \sqrt{\frac{9}{4} + 180} = \frac{3}{2} \pm \frac{27}{2}
$$

Only $x_1 = 15$ is positive and thus the solution. Consequently, $r = \frac{300}{x} = \frac{300}{15} = 20$.

Initially, there were 20 rows with 15 chairs each. After rearranging, there are 25 rows with 12 chairs each.