The average weight of $10$ students is $42\text{kg}$. If $3$ students of average weight of $43\text{kg}$ are added and two other students of average weight of $48\text{kg}$ are also added, then find the average weight of all the students.

To find the new average weight, calculate the total weight of all students and divide it by the total number of students.Initial total weight of $10$ students:
$$10\times 42=420\text{kg}$$
Total weight of the next $3$ students:
$$3\times 43=129\text{kg}$$
Total weight of the last $2$ students:
$$2\times 48=96\text{kg}$$
Total weight of all students:
$$\text{Total Weight}=420+129+96=645\text{kg}$$
Total number of students:
$$\text{Total Students}=10+3+2=15$$
New average weight:
$$\text{Average Weight}=\frac{645}{15}=43\text{kg}$$