Competition · AMC preparation · step 4 of 4
AMC 8 · 2001 · #22
Grade 6 number-theoryarithmeticPick an answer.
AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Five choices, one is bad — perfect setup for Tool #6 (Guess and Check) on the top end. The score formula 5C + U is small enough that we can list every score reachable by working C down from 20, with U ranging over what is left. Tool #2 (Make an Organized List) keeps the cases organized: for each C, the score sits in a clean interval [5C, 5C + (20 - C)]. That lets us see exactly which scores between 90 and 100 exist without ever solving an equation.
Set up the score formula
Set up the score formula: with C correct and U unanswered, the score is 5C + U, where C + U ≤ 20 since wrong answers add 0.
Naming the score with a short formula lets us check each choice by hand without missing cases.
6.EE.A.2Guess And CheckFind the top score
The best case is every answer correct: C = 20, U = 0, giving the maximum score 100.
Start at the ceiling — anything above 100 is impossible, period.
6.EE.A.2Guess And CheckStep the correct count down
Drop to C = 19: one question is left, giving 95 (wrong) or 96 (unanswered) — a gap already opens below 100.
From C = 19, the top reachable score is only 96 — already there is a gap between 96 and 100.
Once the number of correct answers drops from a perfect 20 down to 19, the score can reach 96 at most, so the totals sitting just below a perfect 100 stay out of reach.
▸ Why?
Every one of the 20 questions is exactly one of correct, unanswered, or wrong, and no question is counted twice, so fixing 19 of them as correct leaves just a single question to produce any further points.
▸ Why?
Each correct answer is worth 5 points, so 19 correct answers are 19 equal groups of 5, giving 95 points from the correct answers alone.
▸ Why?
The one leftover question can add at most 1 point — 1 if left unanswered, 0 if marked wrong — so the total climbs no higher than 95 + 1 = 96.
▸ Why?
A wrong answer adds 0 points, and adding zero leaves the total unchanged, so making the leftover question wrong keeps the score at 95 instead of lifting it toward 96.
Step down once more
Drop to C = 18: scores are 90, 91, 92; lower C tops out below 90. All reachable ≥ 90 = {90, 91, 92, 95, 96, 100}.
The organized list shows the only scores ≥ 90 that exist. Everything between 96 and 100 — and the score 93, 94 — is missing.
6.EE.A.2Make A Systematic ListFind the unreachable choice
Match the choices: 90, 91, 92, 95 all appear, but 97 is absent — the impossible score, (E).
Of the five choices, only 97 falls in the gap between 96 and 100.
6.EE.B.5Guess And CheckScore = 5C + U with 20 questions. Counting down from C = 20, the reachable scores at the top are 100, then 95 and 96, then 90, 91, 92. The choice 97 sits in the gap between 96 and 100, so it cannot be scored — answer (E).
- Set up the score formula
- Find the top score
- Step the correct count down
- Step down once more
- Find the unreachable choice
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