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: Find the total age, in 2000, of the 80 teachers who did NOT retire.
The 20 teachers who retired in 2002 had an average age of 60 in 2002, so 2 years earlier, in 2000, their average age was \( 60 - 2 = 58 \), giving a total age of \( 20 \times 58 = 1160 \). Since the total age of all 100 teachers in 2000 was \( 100 \times 50 = 5000 \), the 80 teachers who stayed had a total age in 2000 of \[ 5000 - 1160 = 3840 \]

Step 2: Age this group of 80 directly up to 2008.
From 2000 to 2008 is 8 years, so each of these 80 teachers is 8 years older by 2008, adding \( 80 \times 8 = 640 \) to their total age: \[ 3840 + 640 = 4480 \]

Step 3: Age the 40 new teachers directly up to 2008.
The 40 teachers who joined in 2005 had an average age of 38 that year, and from 2005 to 2008 is 3 years, so by 2008 their total age is \[ 40 \times (38 + 3) = 40 \times 41 = 1640 \]

Step 4: Add both totals and divide by the total number of teachers in 2008.
\[ \text{Total age in 2008} = 4480 + 1640 = 6120, \quad \text{Total teachers} = 80 + 40 = 120 \] \[ \text{Average age} = \frac{6120}{120} = \boxed{51} \]