Péter and Tamás are racing from point $P$ to point $Q$ and back. Péter starts at a speed of $9 \mathrm{~km} /$ hour, and Tamás starts at a speed of $12 \mathrm{~km} /$ hour. When Péter meets Tamás, who is already running back to $P$, Péter tries to increase his speed. By how much did he increase his speed if he and Tamás arrived back at $P$ at the same time?
Solution. The data and symbols for the two stages of the race, before and after the meeting, are contained in the following table.

|  | Péter before after the meeting | Tamás before after the meeting |
| :---: | :---: | :---: |
|  | $\mathrm{x}$ | $\mathrm{y}$ |
| time  speed | $\mathrm{t}_{1}$ $\mathrm{t}_{2}$  $9 \mathrm{~km} /$ hour $v$ | $\mathrm{t}_{1}$ $\mathrm{t}_{2}$  $12 \mathrm{~km} /$ hour $12 \mathrm{~km} /$ hour |

Based on the relationship between distance, time, and speed,

$$
\begin{gathered}
t_{1}=\frac{x}{9 \mathrm{~km} / \text { hour }}=\frac{y}{12 \mathrm{~km} / \text { hour }} \\
t_{2}=\frac{y}{v}=\frac{x}{12 \mathrm{~km} / \text { hour }}
\end{gathered}
$$

From the system of equations, it follows that $v=16 \mathrm{~km} /$ hour, so Péter increased his speed by $7 \mathrm{~km} /$ hour.

Based on the work of Tamás Brezovszky (Váci Ipari SZKI, 3rd grade)