The students of two schools achieved the following results on a test:

The average score of students from the first school is 74 points, with boys scoring an average of 71 and girls 76 points. The average score of students from the second school is 84 points, with boys scoring an average of 81 and girls 90 points. If the overall average score of boys from both schools was 79 points, what was the average score of the girls?
Let the number of boys attending the first school be $f_{1}$, and the number of girls be $l_{1}$; the number of boys attending the second school be $f_{2}$, and the number of girls be $l_{2}$. Since the average score of students attending the first school was 74 points, the following relationship can be written:

$$
\frac{71 f_{1}+76 l_{1}}{f_{1}+l_{1}}=74
$$

Similarly:
(2) $\frac{81 f_{2}+90 l_{2}}{f_{2}+l_{2}}=84$,
(3) $\frac{71 f_{1}+81 f_{2}}{f_{1}+f_{2}}=79$.

Rearranging the (1) equation, we get

$$
l_{1}=\frac{3}{2} f_{1}
$$

Rearranging the (2) equation, we get $f_{2}=2 l_{2}$, and from the (3) equation, we get $f_{2}=4 f_{1}$, and thus $2 l_{2}=4 f_{1}$, that is,

$$
l_{2}=2 f_{1}
$$

Using equations (4) and (5), the average score of the girls attending the two schools is:

$$
\frac{76 l_{1}+90 l_{2}}{l_{1}+l_{2}}=\frac{76 \cdot \frac{3}{2} f_{1}+90 \cdot 2 f_{1}}{\frac{3}{2} f_{1}+2 f_{1}}=\frac{294}{3.5}=84
$$

Peter Kucsera (Bekescsaba, Belvarosi Secondary School and Gymnasium, 11th grade)