## Task B-1.4.

In a certain quiz, the Blue, Red, and Yellow teams are participating. The average score of the members of the Blue team is 30, the Red team is 50, and the Yellow team is 90, while the average score of all participants is 60. The Blue team has scored a total of 360 points less than the Red team, and the Yellow team has scored 6 times more points than the Blue team. How many members are there in each team?
## Solution.

Let the Blue team have $n$ members and achieve $P$ points, the Red team have $m$ members and achieve $C$ points, and the Yellow team have $k$ members and achieve $\check{Z}$ points.

According to the conditions of the problem, we have

$$
\begin{aligned}
& \frac{P}{n}=30 \text { i.e., } P=30 n \\
& \frac{C}{m}=50 \text { i.e., } C=50 \mathrm{~m} \\
& \frac{\check{Z}}{k}=90 \text { i.e., } \check{Z}=90 k
\end{aligned}
$$

Furthermore, we have

$$
C=P+360, \check{Z}=6 P
$$

From the obtained equations, we get:

$$
\begin{gathered}
50 m=30 n+360, \text { i.e., } 5 m=3 n+36 \\
90 k=6 \cdot 30 n, \text { i.e., } k=2 n
\end{gathered}
$$

The average score of all participants is 60, which we write as

$$
\frac{P+C+\check{Z}}{n+m+k}=60
$$

Then, $\frac{30 n+50 m+90 k}{n+m+k}=60$, so $\frac{3 n+5 m+9 k}{n+m+k}=6$. From this, we sequentially obtain:

$$
\begin{gathered}
3 n+5 m+9 k=6 n+6 m+6 k \\
3 n+m-3 k=0 \\
3 n+\frac{3 n+36}{5}-6 n=0
\end{gathered}
$$

Thus, the final solution is

$$
n=3, m=9, k=6
$$

The Blue team has 3 members, the Red team has 9, and the Yellow team has 6.