In a class of $88$ students, the number of girls is $20\%$ more than the number of boys. The average height of the boys is $164\text{ cm}$, and the average height of girls is $4\text{ cm}$ less than the average height of all the boys and girls in the class. What is the average height (in $\text{cm}$) of all the students in the class?

Let the number of boys be $5x$. Then the number of girls is $6x$ (since $20\%=\frac{1}{5}$ more).Total students = $5x+6x=11x=88⟹x=8$.Number of boys = $40$ Number of girls = $48$ The ratio of boys to girls is $40:48=5:6$.Let the average height of all students combined be $H\text{ cm}$.According to the problem, the average height of girls is $H-4$.The average height of boys is given as $164\text{ cm}$.Using the weighted average formula:
$$H=\frac{5\times 164+6\times (H-4)}{5+6}$$
$$11H=820+6H-24$$
$$11H-6H=796$$
$$5H=796⟹H=\frac{796}{5}=159.2\text{ cm}$$