AMC 8 · 2006 · #2

Grade 3 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

mental-arithmeticmulti-digit-arithmeticif-then-reasoning identify-subproblems ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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.

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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

#7 Identify Subproblems 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.

$$\text{correct}: 13 \text{ answers} \times 1 \text{ pt} \quad|\quad \text{wrong}: 7 \times 0 \quad|\quad \text{blank}: 5 \times 0$$

💡 Sorting answers by their points-rate is a Grade 3 "multiplication word problem" setup — same rate within a group.

#8 Analyze the Units 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.

$$13 \times 1 + 7 \times 0 + 5 \times 0 = 13 + 0 + 0 = 13 \;\Rightarrow\; \textbf{(C)}$$

💡 Anything times $0$ is $0$, so the wrong and blank buckets vanish and the score equals the count of correct answers.

[1] #7 3.OA.A.3 Split Billy's $25$ answers into three buckets and write down the points-per-answ

[2] #8 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.3 Use 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.5 Apply 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: $13$.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 3)

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