AMC 10 · 2025 · #3
Grade 8 algebraPick an answer.
Ash's age and both team sizes are unknown, so Tool #4 (Introduce a Variable) names them A, S, and T and pins down the link S+T=15. Each 'if Ash joins' sentence is a separate average condition, so Tool #7 (Identify Subproblems) turns the two scenarios into two equations, and Tool #13 (Convert to Algebra) rewrites each average statement as a clean equation using total age = average × count. Two equations plus the head-count constraint pin down every unknown.
Name the unknowns
Let A be Ash's age, S the students, T the teachers, with S+T=15. Keep the sizes separate — Ash shifts each average differently.
Give each unknown a letter so the word problem becomes equations you can solve.
7.EE.B.4Introduce A VariableTurn the student scenario into an equation
The S students total 12S. After Ash joins, S+1 people average 14, so 12S+A=14(S+1), which expands to A=2S+14.
Total age is always the average times the number of people, so adding one person just adds that person's age to the total.
A total age is always the average times the number of people, so adding one person adds that person's age.
▸ Why?
An average is a total shared over a count, so multiplying it back by the count returns the total.
▸ Why?
That total is the individual ages added together, so a new member simply adds one more term.
Turn the teacher scenario into an equation
The T teachers total 55T. After Ash joins, T+1 people average 52, so 55T+A=52(T+1), which expands to A=-3T+52.
The same average-times-count rule works for the teachers, and here the average drops because Ash is younger than they are.
6.SP.B.5Convert To AlgebraSolve the system for the team sizes
Both equal A, so 2S+14=-3T+52, i.e. 2S+3T=38. Substitute S=15-T: 30+T=38, so T=8 and S=7.
Since both scenarios describe the same Ash, their two expressions for A must match, which pins down the team sizes.
8.EE.C.8Introduce A VariableCompute Ash's age
Put S=7 into A=2S+14: A=2(7)+14=28. Check: A=-3(8)+52=28 too, so Ash is 28 — choice (A).
Once a team size is known, the age equation hands you Ash's age directly.
8.EE.C.7Introduce A VariableTotal age is the average times the number of people, so adding one person just adds their age to the total — write that fact as an equation for each team and the sizes fall out.
- Name the unknowns
- Turn the student scenario into an equation
- Turn the teacher scenario into an equation
- Solve the system for the team sizes
- Compute Ash's age