Problem 6 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The mean age of 25 teachers in a school is 40 years. A teacher retires at the age of 60 years and a new teacher is appointed in his place. If now the mean age of the teachers in this school is 39 years, then the age (in years) of the newly appointed teacher is : [Online April 8, 2017] (a) 25 (b) 30 (c) 35 (d) 40

Step-by-Step Solution

Answer

The age of the newly appointed teacher is 35 years.

Step 1: Calculate the total age of all teachers initially

The mean age of 25 teachers is 40 years. This means the total age of all teachers initially is calculated as:\[\text{Total Age} = \text{Mean Age} \times \text{Number of Teachers} = 40 \times 25 = 1000 \text{ years}\]

Step 2: Calculate the total age after retirement and recruitment

When the teacher retires at age 60 and a new teacher is hired, the new mean age becomes 39 years. Hence, the total age for the 25 teachers now will become:\[39 \times 25 = 975 \text{ years}\]

Step 3: Determine the age of the newly appointed teacher

Initially, the total age with the retiring teacher was 1000 years. After one teacher retires, the sum becomes 975 years. Since the retiring teacher was 60 years old, replace him in calculations:\[\text{Total Age with New Teacher} = 1000 - 60 + \text{Age of New Teacher} \]Given this should equal 975, solve for the age of the newly appointed teacher:\[1000 - 60 + \text{Age of New Teacher} = 975\]\[940 + \text{Age of New Teacher} = 975\]\[\text{Age of New Teacher} = 975 - 940 = 35 \text{ years}\]

Step 4: Confirm the calculations

Verify by substituting back: Initial total was 1000 with retiring teacher, subtract the retired age (60) making it 940, add new teacher's age (35), totaling 975, confirming calculations are correct.

Key Concepts

Teacher Age Problem

In the teacher age problem, understanding the concept of mean age is crucial. The mean or average is a measure that sums up a dataset and divides it by the number of items in the dataset. In this exercise, we started by knowing the mean age of 25 teachers was 40 years. This gave us the total combined age of all teachers, calculated as:

• Total Age = Mean Age × Number of Teachers
• Total Age = 40 years × 25 = 1000 years

This calculation shows that the total age of all teachers combined was 1000 years initially. Understanding this initial setup is key when analyzing how changes affect the mean age.

Average Age Change

The average age change is an important concept to grasp, especially when substitutions occur, like retirements or new hirings. Initially, our total age with all teachers was 1000 years. When a teacher retired, the mean age dropped to 39. This means the new total age after replacing the retiring teacher and adjusting to the new average is as follows:

• New Total Age = New Mean Age × Number of Teachers
• New Total Age = 39 years × 25 = 975 years

Therefore, the total combined age dropped due to this shift. By understanding these calculations, we recognize how the average age of a group adjusts based on its members.

Retirement and Recruitment Calculation

Retirement and recruitment play a significant role in calculating changes in mean age. Initially acknowledged at 1000 years, the total age became 975 years after a teacher retired and a new one was hired. To find out the age of the new teacher, we adjust for the retired teacher's age:

• Subtract the retiring teacher's age from the total age: 1000 - 60 = 940 years
• Add the change due to the new teacher, making it 975 years

Solving this gives us:\[940 + \text{Age of New Teacher} = 975\]\[\text{Age of New Teacher} = 975 - 940 = 35 \text{ years}\]The age of the newly appointed teacher is 35 years, confirming the calculations are logical and match the mean age change.