AMC 10 · 2023 · #18
Grade 7 algebralogicPick an answer.
Every statement is about one number: D = Y_avg - Z_avg. So the real question is which values D can take. Tool #13 (Convert to Algebra) turns the three sentences into three equations and exposes the hidden symmetry — Zelda improved by 18 too. Tool #4 (Introduce a Variable) names the four quiz counts, which is where the whole difficulty lives: the year average is a WEIGHTED mean, and the two students may weight their semesters differently. That collapses D into D = 3 + 18(w_Y - w_Z), where each w is the share of that student's quizzes falling in semester two. Tool #14 (Extreme Principle) then does the decisive work: push w_Y and w_Z to their boundaries to find the largest and smallest D can ever be. Tool #6 (Guess and Check) supplies an explicit set of quiz counts and scores witnessing each statement that IS achievable, and Tool #3 (Eliminate Possibilities) crosses those four off, leaving the one that lies outside the range.
Turn the sentences into equations
The three sentences become three equations.
Yolanda stays 3 points ahead of Zelda in both semesters, so whatever jump Yolanda made, Zelda made the identical jump.
6.EE.B.6Convert To AlgebraThe year average is weighted
The year average is a weighted mean.
A semester with more quizzes pulls the year average harder toward that semester's score — quizzes are votes, and one semester can out-vote the other.
A semester with more quizzes pulls the year average harder toward its own score.
▸ Why?
An average is a total shared over a count, so each quiz counts once in both the total and the count.
▸ Why?
A semester's contribution rises in step with its share of the quizzes, so the shares are the weights.
Base plus a share
Write each as a base plus a share.
Getting an 18-point raise halfway through the year only lifts your yearly number by half of 18 — the raise is prorated by how much of the year came after it.
7.EE.A.1Introduce A VariableOne formula for the gap
The gap tidies into one formula.
Yolanda leads by a guaranteed 3; the only thing that can add to or eat into that lead is who had more of her quizzes in the high-scoring semester.
7.EE.A.1Convert To AlgebraPush the shares to their extremes
Pushing the shares gives a range.
The best case is Yolanda cramming all her quizzes into her strong semester while Zelda took all of hers in her weak one — and even that only buys 18 extra points on top of the built-in 3.
6.RP.A.3Extreme PrincipleBuild a witness for each
Build an actual example for each reachable claim.
Showing something is possible needs only one concrete example, so pick friendly numbers and count the quizzes honestly.
6.SP.B.5Guess And CheckKeep the impossible one
The only claim beyond the range is a twenty-two point gap.
One number, 21, is the ceiling on Yolanda's yearly lead, and 22 sits just above it.
7.EE.B.4Eliminate PossibilitiesA year average is a weighted average, so the same 18-point improvement is worth more to whoever took more quizzes afterward — and since that lever is worth at most 18 points on top of Yolanda's steady 3-point edge, her yearly lead can never pass 21.
- Turn the three sentences into equations
- The year average is a weighted mean
- Rewrite each average as base plus a share of 18
- One formula for the yearly gap
- Push the shares to their extremes
- Build a witness for each achievable claim
- Cross off four, keep the impossible one