The average ages of boys and girls in a class are $14.5$ years and $15$ years respectively. If the ratio of the number of boys and girls is $2:3$, then what is the average age (in years) of all students in the class?

Let the number of boys be $2k$ and the number of girls be $3k$.Average age of boys = $14.5$ yearsTotal age of boys = $14.5\times 2k=29k$ yearsAverage age of girls = $15$ yearsTotal age of girls = $15\times 3k=45k$ yearsThe total age of all students combined is:
$$\text{Total Age}=29k+45k=74k\text{ years}$$
The total number of students is:
$$\text{Total Students}=2k+3k=5k$$
The overall average age of the class is:
$$\text{Average Age}=\frac{\text{Total Age}}{\text{Total Students}}=\frac{74k}{5k}=\frac{74}{5}=14.8\text{ years}$$