AMC 8 · 2001 · #22
Grade 6 number-theoryarithmeticProblem
On a twenty-question test, each correct answer is worth 5 points, each unanswered question is worth 1 point and each incorrect answer is worth 0 points. Which of the following scores is NOT possible?
Pick an answer.
AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A test has $20$ questions. Each correct answer scores $5$ points, each unanswered question scores $1$ point, and each wrong answer scores $0$ points. Which of the listed scores cannot be reached?
Givens: $20$ questions total; Correct $= 5$ points, Unanswered $= 1$ point, Wrong $= 0$ points; Answer choices: (A) $90$, (B) $91$, (C) $92$, (D) $95$, (E) $97$
Unknowns: Which choice is NOT an achievable total score
Understand
Restated: A test has $20$ questions. Each correct answer scores $5$ points, each unanswered question scores $1$ point, and each wrong answer scores $0$ points. Which of the listed scores cannot be reached?
Givens: $20$ questions total; Correct $= 5$ points, Unanswered $= 1$ point, Wrong $= 0$ points; Answer choices: (A) $90$, (B) $91$, (C) $92$, (D) $95$, (E) $97$
Plan
Primary tool: #6 Guess and Check
Secondary: #2 Make an Organized List
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.
Execute — Answer: E
6.EE.A.2 Step 1 - Set up the score formula.
- With $C$ correct and $U$ unanswered, the score is $5C + U$.
- Since wrong answers contribute $0$, only $C$ and $U$ matter, and they satisfy $C + U \le 20$ (the remaining $20 - C - U$ are wrong).
💡 Naming the score with a short formula lets us check each choice by hand without missing cases.
6.EE.A.2 Step 2 - Find the maximum score.
- The best case is every answer correct: $C = 20$, $U = 0$, so the score is $5 \times 20 = 100$.
- Nothing can beat that.
💡 Start at the ceiling — anything above $100$ is impossible, period.
6.EE.A.2 Step 3 - Step $C$ down by one and list the scores reachable.
- With $C = 19$, there are $20 - 19 = 1$ question left, which can be unanswered ($U = 1$) or wrong ($U = 0$).
- The scores are $5(19) + 1 = 96$ and $5(19) + 0 = 95$.
💡 From $C = 19$, the top reachable score is only $96$ — already there is a gap between $96$ and $100$.
6.EE.A.2 Step 4 - Take one more step down to confirm the gap.
- With $C = 18$, two questions remain, so $U$ can be $0$, $1$, or $2$.
- The scores are $90$, $91$, $92$.
- Now combine all reachable scores at or above $90$: from $C = 20$ we get $\{100\}$, from $C = 19$ we get $\{95, 96\}$, from $C = 18$ we get $\{90, 91, 92\}$, and lower $C$ caps out at $5 \cdot 17 + 3 = 88 < 90$.
💡 The organized list shows the only scores $\ge 90$ that exist. Everything between $96$ and $100$ — and the score $93$, $94$ — is missing.
6.EE.B.5 Step 5 - Match the choices against the reachable list.
- $90$, $91$, $92$, and $95$ all appear; $97$ does not.
- So $97$ is the impossible score.
💡 Of the five choices, only $97$ falls in the gap between $96$ and $100$.
6.EE.A.2 Set up the score formula. With $C$ correct and $U$ unanswered, the score is $5C 6.EE.A.2 Find the maximum score. The best case is every answer correct: $C = 20$, $U = 0$ 6.EE.A.2 Step $C$ down by one and list the scores reachable. With $C = 19$, there are $20 6.EE.A.2 Take one more step down to confirm the gap. With $C = 18$, two questions remain, 6.EE.B.5 Match the choices against the reachable list. $90$, $91$, $92$, and $95$ all app Review
Reasonableness: Confirm each surviving choice with an explicit case. $90 = 5(18) + 0$ ($18$ correct, $2$ wrong). $91 = 5(18) + 1$ ($18$ correct, $1$ unanswered, $1$ wrong). $92 = 5(18) + 2$ ($18$ correct, $2$ unanswered). $95 = 5(19) + 0$ ($19$ correct, $1$ wrong). For $97$, we would need $5C + U = 97$ with $C + U \le 20$. $C \le 19$ gives a top of $5(19) + 1 = 96 < 97$, and $C = 20$ forces $U = 0$ and score $100 \ne 97$. No case works, so $97$ really is unreachable — answer (E).
Alternative: Tool #9 (Solve an Easier Problem): rewrite the score as $\text{Score} = 100 - 5W - 4U$ by starting at the perfect $100$ and subtracting $5$ for each wrong answer and $4$ for each unanswered question (since unanswered loses $4$ compared to correct). Then the gap from $100$ down to the score must be a non-negative combination of $5$s and $4$s. From $100$ we can reach gaps of $0, 4, 5, 8, 9, 10, 12, \dots$ — but a gap of $3$ (which is what $97$ would need) is impossible with $4$s and $5$s. So $97$ is the only choice that cannot occur.
CCSS standards used (min grade 6)
6.EE.A.2Write, read, and evaluate expressions in which letters stand for numbers (Writing the score as $5C + U$ with the constraint $C + U \le 20$ so each case can be checked algebraically.)6.EE.B.5Understand solving an equation or inequality as a process of answering which values make the statement true (Testing each answer choice as a target value and asking whether any non-negative integers $C, U$ with $C + U \le 20$ satisfy $5C + U = \text{choice}$.)
⭐ Score $= 5C + U$ 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).
⭐ Score $= 5C + U$ 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).
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