Question

The average age of 100 teachers in a college in 2000 was 50 years. In 2002, 20 teachers superannuated from their jobs, whose average age was 60 years. In 2005, 40 new teachers joined the college whose average age was 38 years. What was the average age of all the teachers in 2008?

• A. 54 years
• B. 49 years
• C. 51 years
• D. 50 years

Hint:

In average age problems, always convert averages into total ages first. Then adjust totals whenever people leave, join, or grow older.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The average age of 100 teachers in the year 2000 was 50 years. Some teachers retired in 2002 and new teachers joined in 2005. We need to find the average age of all teachers in 2008.

Step 2: Key Formula or Approach:
Use: \[ \text{Total Age} = \text{Average Age} \times \text{Number of Persons} \] Track the total ages after retirement, new appointments, and yearly increase in age.

Step 3: Detailed Explanation:
Initial situation in 2000: \[ \text{Total age} = 100 \times 50 = 5000 \] In 2002:
After 2 years, every teacher becomes 2 years older: \[ 5000 + (100 \times 2)=5200 \] Now, 20 teachers retire whose average age is 60 years. \[ \text{Age removed} = 20 \times 60 = 1200 \] Remaining total age: \[ 5200-1200=4000 \] Remaining teachers: \[ 100-20=80 \] From 2002 to 2005:
3 more years pass, so: \[ 4000 + (80 \times 3)=4240 \] In 2005:
40 new teachers join with average age 38 years. \[ \text{Added age} = 40 \times 38 = 1520 \] New total age: \[ 4240+1520=5760 \] Total teachers: \[ 80+40=120 \] From 2005 to 2008:
3 more years pass: \[ 5760 + (120 \times 3)=6120 \] Average age in 2008: \[ \frac{6120}{120}=51 \]

Step 4: Final Answer:
The average age of all teachers in 2008 was: \[ \boxed{51 \text{ years}} \] Hence, the correct option is: \[ \boxed{\text{(C) 51 years}} \]

Answer 2

Instead of tracking one continuous running total across the years, we can compute the final total age in 2008 by treating the surviving original teachers and the newly joined teachers as two separate groups, ageing each forward independently, and then check which option matches the resulting average.

1. Retained teachers: In 2000, the 100 teachers had a total age of \( 100 \times 50 = 5000 \). By 2002, each is 2 years older, so the total is \( 5000 + 200 = 5200 \). The 20 who retire in 2002 carry away \( 20 \times 60 = 1200 \) years, leaving 80 teachers with a total age of \( 5200 - 1200 = 4000 \) in 2002. By 2008, 6 more years have passed for this group, adding \( 80 \times 6 = 480 \), for a total of \( 4000 + 480 = 4480 \).
2. New teachers: The 40 teachers who join in 2005 have a total age of \( 40 \times 38 = 1520 \) at that point. By 2008, 3 more years have passed, adding \( 40 \times 3 = 120 \), for a total of \( 1520+120=1640 \).
3. Combining the groups: In 2008 there are \( 80+40=120 \) teachers in total, with a combined total age of \( 4480+1640=6120 \). Dividing gives an average age of \( \frac{6120}{120} = 51 \).
4. Checking the options: Of the given choices, 54, 49, 51, and 50 years, only 51 matches this independently computed total; the other three would require combined totals of \( 6480 \), \( 5880 \), or \( 6000 \) respectively, none of which arise from the group-by-group calculation above.

Building the total age group by group, rather than tracking one continuous sum, confirms the same result of 51 years.

Therefore, the correct answer is 51 years.