We were on a class trip. If 5 more of us had pooled our money and each of us had paid $6 \mathrm{Ft}$ more, then $792 \mathrm{Ft}$ more would have been collected. If, however, 4 fewer of us and each of us had paid $4 \mathrm{Ft}$ more, then we would have $388 \mathrm{Ft}$ less. How many of us went on the trip and how much was the payment per person?
Let the number of excursionists be $x$, and their per capita payment be $y$ Ft. Thus, the total collected money was $x y$ Ft, and in the first variant, it would have been $(x+5)(y+6)$, and in the second variant, it would have been $(x-4)(y+4)$ Ft, hence, based on the given differences,

$$
\begin{aligned}
& (x+5)(y+6)=x y+792 \\
& (x-4)(y+4)=x y-388
\end{aligned}
$$

After rearrangement,

$$
\begin{aligned}
& 6 x+5 y=762 \\
& 4 x-4 y=-372
\end{aligned}
$$

and from this, $x=27, y=120$. So, 27 students each paid 120 Ft, totaling 3240 Ft. In the first variant, it would have been $32 \cdot 126=4032$ Ft, and in the second variant, it would have been $23 \cdot 124=2852$ Ft, with the surplus indeed being 792 Ft, and the shortfall 388 Ft.

István Fülöp (Kővágóörs, VIII. grade, elementary school)

Remarks. 1. The problem can also be solved using a single-variable equation, if we express the actual payment of one student based on both data points with the number of participants $x$. The surplus came from two sources: the increase in the per capita payment by $6$ Ft, which, with the increased number of participants, amounts to $6(x+5)$ Ft; the remaining part, $792-6(x+5)$, comes from the original planned payment of the 5 new participants. Thus, the original payment is $[792-6(x+5)] / 5=158.40-1.2(x+5)$. In the second case, the shortfall was caused by the 4 participants who dropped out, but this was reduced by the additional $4(x-4)$ Ft paid by the remaining participants; thus, the total shortfall due to the 4 participants who dropped out would have been $388+4(x-4)$ Ft, and the payment of one participant is $[388+4(x-4)] / 4=93+x$. From the equality of the two expressions, $x=27$, and with this, the value of the expressions is $120$ Ft.

Antónia Földes (Budapest, I. grade, J. Apáczai Csere secondary school)

2. With similar reasoning, János Markó (Pannonhalma, II. grade, Bencés secondary school) set up a single-variable equation for the actual payment of one student. We can observe that the equation mentioned in Remark 1 is essentially the elimination of $y$ from the system of equations in the solution, not by mechanical transformation, but by reasoning related to the text of the problem.