Problem 2. A total of 2400 students competed in five different categories in a competition. The total number of students from the first and second categories is equal to $\frac{3}{5}$ of the total number of students from the third, fourth, and fifth categories. The ratio of the number of students from the first and second categories is 2:3, and the ratio of the number of students from the third, fourth, and fifth categories is 5:4:6. How many students competed in each category.
Solution. First method. Let $a, b, c, d, e$ denote the number of students who competed in the first, second, third, fourth, and fifth category, respectively. Let $x=a+b$ and $y=b+c+d$. Then $x=\frac{3}{5} y$ and $x+y=2400$, from which we get $\frac{3}{5} y+y=2400$, i.e., $y=1500$ and therefore $x=2400-1500=900$. Further, from $a: b=2: 3$ and $a+b=900$ it follows that $a=\frac{900}{2+3} \cdot 2=360$ and $b=\frac{900}{2+3} \cdot 3=540$. From $c: d: e=5: 4: 6$ and $c+d+e=1500$, it follows that $c=\frac{1500}{5+4+6} \cdot 5=500, d=\frac{1500}{5+4+6} \cdot 4=400$ and $e=\frac{1500}{5+4+6} \cdot 6=600$.

Finally, 360 students competed in the first category, 540 in the second, 500 in the third, 400 in the fourth, and 600 in the fifth category.

Second method. Let $a, b, c, d, e$ denote the number of students who competed in the first, second, third, fourth, and fifth category, respectively. From the conditions of the problem, we have

$$
\left\{\begin{array}{l}
a+b+c+d+e=2400 \\
a+b=\frac{3}{5}(c+d+e) \\
a: b=2: 3 \\
c: d: e=5: 4: 6
\end{array}\right.
$$

From the last two equations, it follows that $a=2 k, b=3 k, c=5 m, d=4 m, e=6 m$ and if we substitute in the first two equations, we get $5 k+15 m=2400$ and $5 k=\frac{3}{5} \cdot 15 m$, i.e., $5 k+15 m=2400$ and $5 k=9 m$. If we substitute the second equation into the first, we get $24 m=2400$, i.e., $m=100$ and from there $5 k=900$, i.e., $k=180$. Now, it is easy to obtain that $a=360, b=540, c=500, d=400$ and $e=600$.

Finally, 360 students competed in the first category, 540 in the second, 500 in the third, 400 in the fourth, and 600 in the fifth category.