The average age of $29$ students of a class and their teacher is $14\text{ years}$. When the teacher's age is excluded, the average age of the students decreases by $1\text{ year}$. What is the age (in years) of the teacher?

Method 1: Total Sum MethodTotal number of people initially (students + teacher) = $29+1=30$ Initial average age = $14\text{ years}$ Sum of initial ages = $30\times 14=420\text{ years}$ When the teacher is excluded:Remaining number of people (students only) = $29$ New average age = $14-1=13\text{ years}$ Sum of students' ages = $29\times 13=377\text{ years}$ Age of the teacher:
$$\text{Age of Teacher}=\text{Sum of initial ages}-\text{Sum of students’ ages}$$
$$\text{Age of Teacher}=420-377=43\text{ years}$$
Method 2: Deviation Method
$$\text{Excluded Value}=\text{Old Average}+(\text{Change in Average}\times \text{New Number of Items})$$
$$\text{Age of Teacher}=14+(1\times 29)=14+29=43\text{ years}$$