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 wants to join. If Ash plays with the students, the average age increases from $12$ to $14$. If Ash plays with the teachers, the average age decreases from $55$ to $52$. How old is Ash?
$$\textbf{(A) } 28 \qquad \textbf{(B) } 29 \qquad \textbf{(C) } 30 \qquad \textbf{(D) } 32 \qquad \textbf{(E) } 33$$
Difficulty: 26/100 — Requires setting up a system of equations; computation is straightforward. Not immediately intuitive.
Core Concepts: mean, system of equations
Techniques: Use sums and averages, set up system with three unknowns.
Error-prone Steps: Setting up equations incorrectly (for example, forgetting that Ash adds one person to the team).
Ideal Time:
Experienced: ≤ 1 min
Intermediate: ≤ 2 min
Beginner: ≤ 3 min