## 8. Height

In a certain class, the average height of all twenty-two students was $163.5 \mathrm{~cm}$. Soon, one more student was enrolled, increasing the average height by $0.3 \mathrm{~cm}$, and the very next day, one student dropped out, causing the average height to increase again, this time by $0.4 \mathrm{~cm}$. How tall, in centimeters, was the student who dropped out?

Result: $\quad 155$
## Solution.

Let $x_{1}, x_{2}, \ldots, x_{22}$ be the heights of each of the 22 students. We have

$$
\frac{x_{1}+x_{2}+\cdots+x_{22}}{22}=163.5 \text{ cm}
$$

so the sum of the heights of all students is $x_{1}+x_{2}+\cdots+x_{22}=163.5 \cdot 22=3597 \text{ cm}$.

Let $x_{23}$ be the height of the student who later enrolled in the class. We have

$$
\frac{x_{1}+x_{2}+\ldots+x_{22}+x_{23}}{23}=163.5+0.3=163.8 \text{ cm}
$$

Substituting $x_{1}+x_{2}+\cdots+x_{22}=3597 \text{ cm}$ into this expression, we get:

$$
\begin{gathered}
\frac{3597+x_{23}}{23}=163.8 \\
x_{23}=170.4
\end{gathered}
$$

The height of the student who enrolled is $170.4 \text{ cm}$.

Now, the sum of the heights of all 23 students is:

$$
x_{1}+x_{2}+\cdots+x_{22}+x_{23}=3597+170.4=3767.4 \text{ cm}
$$

Let $x_{1}$ be the height of the student who dropped out of the class. Any of the original 22 students could have dropped out. The student who enrolled later could not have dropped out, as this would have reduced the average height.

We have:

$$
\frac{\left(x_{1}+x_{2}+\ldots+x_{22}+x_{23}\right)-x_{1}}{22}=163.8+0.4=164.2 \text{ cm}
$$

Substituting $x_{1}+x_{2}+\cdots+x_{23}=3767.4 \text{ cm}$, we get:

$$
\begin{gathered}
\frac{3767.4-x_{1}}{22}=164.2 \\
x_{1}=155
\end{gathered}
$$

The height of the student who dropped out is $155 \text{ cm}$.