Explanation:

We know that,
Average = Sum of all quantities / Total no. of quantities
∴ Sum of ages of students = Average × number of students
The average age of 30 students is 13
∴ Sum of ages of 30 students = Average × number of students
⇒ Sum of ages of 30 students = 30 × 13 = 390 years
Average age of group of 9 students is 12.
⇒ Sum of ages of 9 students = 12 × 9 = 108 years
Average age of group of 10 students is 14.
⇒ Sum of ages of 10 students = 14 × 10 = 140 years
Let the number of remaining students be x
∴ x + 9 + 10 = 30
⇒ x + 19 = 30
⇒ x = 11 students
∴ Sum of ages of 11 students = (Sum of ages of 30 students) – (Sum of ages of 9 students + Sum of ages of 10 students)
⇒ Sum of age of 11 students = 390 – (108 + 140) = 390 – 248 = 142 years
∴ Average age of 11 students  =  Sum of ages of 11 students /11
⇒ Average age of 11 students = 142 /11 = 12.91
∴ Average age of remaining students in 12.91 years