1. September 2007, a language school was founded, where seven educators taught. On 1. September 2010, a new colleague joined these seven teachers, who was exactly 25 years old. By 1. September 2012, one of the teachers left the school, leaving seven teachers again. The average age of the educators at the school was the same on all three mentioned dates.

How old was the teacher who no longer worked at the school on 1. September 2012? What was the average age of the teachers at the school on that day?

(L. Šimünek)

Hint. Work with the sum of the ages of all employed teachers. Consider how this sum changes in relation to the mentioned dates.
The sum of the ages of all seven teachers at the school on September 1, 2007, is denoted by $c$. The sum of the ages of these seven people increased by $7 \cdot 3=21$ on September 1, 2010, so the sum of the ages of all eight teachers working at the school on that day was

$$
c+21+25=c+46.
$$

The sum of the ages of these eight people increased by $8 \cdot 2=16$ on September 1, 2012. On that day, one of them no longer worked at the school; let's denote his age at that time by $x$. The sum of the ages of the seven remaining teachers on that day was

$$
c+46+16-x=c+62-x.
$$

Since the average age of the teachers at the school on all the mentioned dates is the same, the following equalities hold:

$$
\frac{c}{7}=\frac{c+46}{8}=\frac{c+62-x}{7}.
$$

From the equality between the first and third fractions, it directly follows that $x=62$. The teacher who no longer worked at the school on September 1, 2012, was therefore exactly 62 years old.

By manipulating the equality between the first two fractions, we can find the value of $c$:

$$
\begin{aligned}
\frac{c}{7} & =\frac{c+46}{8} \\
8 c & =7 c+7 \cdot 46 \\
c & =322
\end{aligned}
$$

The average age of the teachers on all three mentioned dates was $322: 7=46$ years.