AMC 8 · 2006 · #2
Easy mode Grade 3Problem
Billy is taking the AMC 8 contest. Here is how scoring works: each correct answer is worth 1 point. A wrong answer is worth 0 points. A blank answer is also worth 0 points.
Billy answers 13 questions correctly. He answers 7 questions incorrectly. He leaves the last 5 questions blank.
What is Billy's score?
Pick an answer.
AMC 8 2006 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: On the AMC 8, Billy gets $13$ questions right, gets $7$ wrong, and leaves the last $5$ blank. Find his total score.
Givens: Billy answers $13$ questions correctly; Billy answers $7$ questions incorrectly; Billy leaves $5$ questions blank; Answer choices: (A) $1$, (B) $6$, (C) $13$, (D) $19$, (E) $26$
Unknowns: Billy's total AMC 8 score
Understand
Restated: On the AMC 8, Billy gets $13$ questions right, gets $7$ wrong, and leaves the last $5$ blank. Find his total score.
Givens: Billy answers $13$ questions correctly; Billy answers $7$ questions incorrectly; Billy leaves $5$ questions blank; Answer choices: (A) $1$, (B) $6$, (C) $13$, (D) $19$, (E) $26$
Plan
Primary tool: #8 Analyze the Units
Secondary: #7 Identify Subproblems
The unit on every question is "points". Each correct answer is worth $1$ point and each non-correct answer (wrong or blank) is worth $0$ points. Tool #8 (Analyze the Units) frames the question as a points-per-answer rate, which collapses everything to a single multiplication. Tool #7 (Identify Subproblems) splits the $25$ questions into three buckets — correct, incorrect, blank — so the points from each bucket can be counted separately and then added. Two buckets contribute zero, so the score is just the count of correct answers.
Execute — Answer: C
3.OA.A.3 Step 1 - Split Billy's $25$ answers into three buckets and write down the points-per-answer rate for each bucket.
- Two of the buckets carry a rate of $0$ points per answer, so they cannot contribute to the score.
💡 Sorting answers by their points-rate is a Grade 3 "multiplication word problem" setup — same rate within a group.
3.OA.B.5 Step 2 Compute the points in each bucket using the points-per-answer rate, then add the three buckets to get the total score.
💡 Anything times $0$ is $0$, so the wrong and blank buckets vanish and the score equals the count of correct answers.
3.OA.A.3 Split Billy's $25$ answers into three buckets and write down the points-per-answ 3.OA.B.5 Compute the points in each bucket using the points-per-answer rate, then add the Review
Reasonableness: Sanity check the totals: $13 + 7 + 5 = 25$, which is exactly the number of questions on the AMC 8 — the buckets account for every answer, nothing double-counted. The score $13$ also sits between the smallest possible non-zero score ($1$) and the maximum ($25$), so the magnitude is reasonable. Distractors hint at common mistakes: (D) $19 = 13 + 7 - 1$ confuses penalty rules, (E) $26 = 13 \times 2$ doubles the correct count, (B) $6 = 13 - 7$ subtracts wrong from right. The correct rule (wrong = $0$, blank = $0$) lands on (C).
Alternative: Tool #16 (Change Focus): instead of summing across three buckets, focus only on the correct bucket because it is the only one with a nonzero rate. The score is just "how many correct" $= 13$, giving (C) immediately. Same answer, one less line of arithmetic.
CCSS standards used (min grade 3)
3.OA.A.3Use multiplication and division within $100$ to solve word problems in situations involving equal groups (Treating each answer-type bucket as an equal-groups multiplication: number of answers in the bucket times the points-per-answer rate.)3.OA.B.5Apply properties of operations as strategies to multiply and divide, including the zero property of multiplication (Using $n \times 0 = 0$ to eliminate the wrong and blank buckets so the score equals $13 \times 1$.)
⭐ On the AMC 8, only correct answers add points — wrong and blank both score zero. So the total score is just the count of correct answers: $13$.
⭐ On the AMC 8, only correct answers add points — wrong and blank both score zero. So the total score is just the count of correct answers: $13$.
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