Pat and Mat took a swim in the pond and then had a race to their cottage. At a certain moment, it was true that if Mat had covered half the distance he had already run, he would still be short of the cottage by three times that half distance. At the same moment, it was true that if Pat had covered twice the distance he had already run, he would still be short of the cottage by one third of that doubled distance.

Who was closer to the cottage at that moment?

(L. Hozová)
If Mat were to be a third of the distance to the cottage short of the distance he has already run, then he would be a quarter of the way. This case would occur if he had covered half the distance he has run so far. Thus, Mat was at the halfway point between the pond and the cottage at the given moment.

If Pat were to be a third of the distance to the cottage short of the distance he has already run, then he would be three-quarters of the way. This case would occur if he had covered twice the distance he has run so far. Thus, Pat was at three-eighths of the way between the pond and the cottage at the given moment.

At the given moment, Mat was closer to the cottage.

![](https://cdn.mathpix.com/cropped/2024_04_17_096aded327f4b802fe92g-1.jpg?height=97&width=1005&top_left_y=1279&top_left_x=523)

Notes. If $m$ and $p$ represent Mat's and Pat's distances from the pond at the given moment, and $c$ represents the total distance between the pond and the cottage, then the information from the problem can be written as follows:

$$
\frac{1}{2} m + 3\left(\frac{1}{2} m\right) = c, \quad 2 p + \frac{1}{3}(2 p) = c
$$

From this, we easily get $2 m = c$, so $m = \frac{1}{2} c$, and $\frac{8}{3} p = c$, so $p = \frac{3}{8} c$.