7. (10 points) The correspondence stage of the Innopolis University Olympiad was held among 9th, 10th, and 11th grade students, and among them, $a \%$, $b \%$, and $c \%$ respectively advanced to the in-person stage. Moreover, among the 9th and 10th grade students together, $k \%$ advanced to the in-person stage, and among the 10th and 11th grade students together, $-l \%$ advanced. Determine the percentage of students who advanced to the in-person stage from all three grades combined ( $a<k<b<l<c$ ).
Solution: Let $x$ be the number of 9th grade students, $y$ be the number of 10th grade students, and $z$ be the number of 11th grade students. Then

$$
k=\frac{a x+b y}{x+y} \Longrightarrow \frac{y}{x}=\frac{k-a}{b-k} \quad \text { and } \quad l=\frac{b y+c z}{y+z} \Longrightarrow \frac{z}{y}=\frac{l-b}{c-l}
$$

Express $x$ and $z$ in terms of $y$

$$
x=y \cdot \frac{b-k}{k-a} \quad \text { and } \quad z=y \cdot \frac{l-b}{c-l}
$$

The percentage of students who advanced to the in-person stage for all three grades combined is

$$
\begin{gathered}
n=\frac{a x+b y+c z}{x+y+z}=\frac{a y \cdot \frac{b-k}{k-a}+b y+c y \cdot \frac{l-b}{c-l}}{y \cdot \frac{b-k}{k-a}+y+y \cdot \frac{l-b}{c-}}= \\
=\frac{a(b-k)(c-l)+b(k-a)(c-l)+c(l-b)(k-a)}{(b-k)(c-l)+(k-a)(c-l)+(l-b)(k-a)}
\end{gathered}
$$

## Answers:

| Option | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Answer | 18 | 28 | 36 | 10 | 24 | 14.5 | 34 | 23.74 | 15.4 | 33 |