Problem:
A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is 15. Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from 12 to 14. If Ash plays with the teachers, the average age on that team will decrease from 55 to 52. How old is Ash?
Answer Choices:
A. 28
B. 29
C. 30
D. 32
E. 33
Solution:
Let s = number of students and t = number of teachers. Then
s+t=15
We can compute the average age of the students as follows:
s+112s+A=14⇒12s+A=14s+14⇒A=2s+14(1)
and the average age of the teachers:
t+155t+A=52⇒55t+A=52t+52⇒A=52−3t(2)
Since we have 2 values for A in (1) and (2), we can equate them to get
2s+14=52−3t⇒2s+3t=38(3)
From s+t=15, t=15−s. We can substitute this into (3) to get
2s+3(15−s)=38⇒2s+45−3s=38⇒−s=−7⇒s=7,t=15−s=8
Using (1), we get
A=2s+14=2(7)+14=28
which gives us an answer of (A) 28
The problems on this page are the property of the MAA's American Mathematics Competitions