average age of 36 students in a group is 14

Sum of the ages of 36 students = $ 36 \times 14 $

When teachers age is included to it, the average increases by one

=> average = 15

Sum of the ages of 36 students and the teacher = $ 37 \times 15$

Hence teachers age

$= 37 \times 15 - 36 \times 14$

$= 37 \times 15 - 14(37 - 1)$

$= 37 \times 15 - 37 \times 14 + 14$

= 37(15 - 14) + 14

=37 + 14

= 51