## Task 2 - 260732

The following is known about the holiday guests in a holiday home:

The number of girls is equal to half the number of holiday guests who are not girls.

The number of boys is equal to one-third the number of holiday guests who are not boys.

The number of women is equal to one-quarter the number of holiday guests who are not women.

In addition to these girls, boys, and women, there are exactly 26 men as holiday guests in this holiday home.

Determine whether the numbers of girls, boys, and women can be uniquely determined from these statements!

If this is the case, provide these numbers!
Let $m$ be the number of girls, $j$ be the number of boys, and $f$ be the number of women. The total number of holiday guests in the holiday camp is $x$. According to the problem statement,

$$
m=\frac{1}{2}(x-m) \quad \text { so } \quad m=\frac{x}{3}
$$

Similarly, from $j=\frac{1}{3}(x-j)$, thus $3 j=x-j$, the relationship $j=\frac{x}{4}$ (2) follows, and from $f=\frac{1}{4}(x-f)$ the relationship $f=\frac{x}{5}(3)$.

According to the problem statement, it also holds that $x=m+j+f+26$. From this, due to (1), (2), (3),

$$
x=\frac{x}{3}+\frac{x}{4}+\frac{x}{5}+26 \Rightarrow x=120
$$

From (1), (2), (3) it follows again that $m=40, j=30, f=24$.

Thus, it is shown that the numbers of girls, boys, and women can be uniquely determined from the given information.