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 the total age of all teachers year by year, we can track the average age directly, using the rule that a group's average age changes by the number of years passed once everyone in it ages, and changes further whenever people join or leave with a different average age. Once we get the final average, we check it against each option.

From 2000 to 2002 (2 years), everyone ages 2 years, so the average of the original 100 rises from 50 to 52. Removing 20 teachers with average age 60 changes the average of the remaining 80 to \[ \frac{100 \times 52 - 20 \times 60}{80} = \frac{5200 - 1200}{80} = \frac{4000}{80} = 50 \] From 2002 to 2005 (3 years), this group of 80 ages further, so their average rises from 50 to 53. Adding 40 new teachers with average age 38 changes the average of all 120 teachers to \[ \frac{80 \times 53 + 40 \times 38}{120} = \frac{4240 + 1520}{120} = \frac{5760}{120} = 48 \] From 2005 to 2008 (3 more years), this full group of 120 ages further, so the average rises from 48 to \( 48 + 3 = 51 \).

1. 54 years: This does not match the average worked out through this year-by-year tracking of the group's average age.
2. 49 years: This is also not what results from tracking the average age through each retirement, addition, and passage of time.
3. 51 years: This matches exactly the average age reached after accounting for the 2002 retirements, the 2005 additions, and the ageing of the group across all three periods.
4. 50 years: This was the average midway through the calculation, in 2002, right after the retirements, not the final 2008 average.

Therefore, the correct answer is 51 years.