AMC 10 · 2021 · #12
Grade 6 algebraPick an answer.
Tool #13 (Algebra) — the four sentences are tailor-made to be turned into four equations in n, Σ, G, L. Once written, the system isn't messy: the two "average over n-1" lines differ only in whether G or L is removed, so subtracting them isolates G - L = 8(n - 1) and uses condition (4) to get n in one step. Tool #7 (Identify Subproblems) sequences the unknowns: first n, then L (from the n - 2 equation), then Σ (back into one of the n - 1 equations). Finally divide Σ by n for the answer. Each sub-step is just careful arithmetic — no need for complex algebra moves.
Write the conditions
Write all four as equations.
"Average = sum ÷ count" turns each averaging condition into a single linear equation.
6.EE.B.7Convert To AlgebraSubtract two of them
Subtracting leaves only the size.
Two "n-1" averages differ only by which extreme is removed; their gap in sum is exactly G - L.
Two averages taken over the same count differ only by which extreme was removed.
▸ Why?
An average is a total shared over a count, so each one can be turned back into its own total.
▸ Why?
The two totals share everything but the two extremes, so subtracting leaves exactly their gap.
Solve for the size
The given difference fixes the size.
Divide both sides by 8 to get the count instantly.
6.EE.B.7Convert To AlgebraFind the least number
Two equations give the least number.
Once n is known, two equations differing only in L pin down L in one subtraction.
6.EE.B.7Identify SubproblemsFind the total
Compute the total.
Once L and n - 1 are known, the full sum is one addition.
6.EE.B.7Convert To AlgebraTake the average
The average is 36.8.
Divide by 10 — shift the decimal one place.
5.NBT.B.7Convert To AlgebraThis AMC 12 problem only needs Grade 6 "sum = average × count" equations you already know! Subtract two n - 1 averages to get G - L = 8(n-1) = 72, so n = 10. Then Σ = 368 and the answer is 368/10 = 36.8.