AMC 8 · 2022 · #11

Grade 4 arithmeticlogic

Archetype Arithmetic Expression Decomposition

pattern-recognitionmulti-digit-arithmeticlogical-deduction pattern-recognitionidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticlogical-deduction

📏 Short solution 💡 2 insights

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Problem

Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating $3$ inches of pasta from the middle of one piece. In the end, he has $10$ pieces of pasta whose total length is $17$ inches. How long, in inches, was the piece of pasta he started with?

$\textbf{(A) } 34\qquad\textbf{(B) } 38\qquad\textbf{(C) } 41\qquad\textbf{(D) } 44\qquad\textbf{(E) } 47$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Henry starts with one long piece of pasta. Each time he bites $3$ inches out of the middle of some piece, that piece splits into two shorter pieces. After several bites, he is left with $10$ pieces whose lengths add up to $17$ inches. How long, in inches, was the original single piece?

Givens: Henry starts with exactly $1$ piece of pasta; Each bite eats $3$ inches from the middle of one existing piece; Each bite turns that one piece into $2$ pieces (the middle is gone); Final number of pieces $= 10$; Final total length of all pieces $= 17$ inches; Answer choices: (A) $34$, (B) $38$, (C) $41$, (D) $44$, (E) $47$ inches

Unknowns: The length, in inches, of the original piece of pasta before any bites

Understand

Plan

Primary tool: #11 Work Backwards

Secondary: #5 Look for a Pattern

The end state (how many pieces and how much pasta is left) is given, and we want the start state (original length). That is the textbook trigger for Tool #11 (Work Backwards): replay each bite in reverse to put back what was eaten. Before we can do that, we need Tool #5 (Look for a Pattern) to notice a clean rule about how the piece count changes with each bite — every bite turns one piece into two, so the number of pieces grows by exactly $1$ per bite. That single pattern tells us how many bites happened, which is what unlocks the work-backwards calculation.

Execute — Answer: D

#5 Look for a Pattern 4.OA.C.5 Step 1

Find the rule for how the number of pieces changes per bite.
Imagine $1$ long piece.
After bite $1$ the middle is eaten and the piece splits into $2$ pieces.
After bite $2$ one of those pieces is bitten and splits, giving $3$ pieces.
Each bite adds exactly $1$ piece, so after $n$ bites there are $1 + n$ pieces.

$$\text{pieces after } n \text{ bites} = 1 + n$$

💡 Each bite removes a middle and leaves two ends, so the rule "one bite, one extra piece" is a clean number pattern.

#5 Look for a Pattern 1.OA.D.8 Step 2

Use the rule to find how many bites Henry took.
The final pile has $10$ pieces, so $1 + n = 10$, which means $n = 9$ bites.

$1 + n = 10 \;\Rightarrow\; n = 9$ bites

💡 Filling in the missing number in $1 + \square = 10$ is a Grade 1 unknown-addend question.

#11 Work Backwards 3.OA.A.3 Step 3

Compute how much pasta was eaten in total.
Each of the $9$ bites ate $3$ inches, so the total eaten is $9 \times 3 = 27$ inches.

$$9 \times 3 = 27 \text{ inches eaten}$$

💡 Equal-groups multiplication ($9$ groups of $3$ inches) is the Grade 3 way to total identical bites.

#11 Work Backwards 2.NBT.B.5 Step 4

Work backwards: add the eaten pasta back to what is still on the table.
The original length is the $17$ inches still visible plus the $27$ inches that disappeared into Henry's mouth.

$$\text{original length} = 17 + 27 = 44 \text{ inches} \;\Rightarrow\; \textbf{(D)}$$

💡 Putting back what was removed is exactly the work-backwards move, and the final sum is a basic Grade 2 addition within $100$.

[1] #5 4.OA.C.5 Find the rule for how the number of pieces changes per bite. Imagine $1$ long pi

[2] #5 1.OA.D.8 Use the rule to find how many bites Henry took. The final pile has $10$ pieces,

[3] #11 3.OA.A.3 Compute how much pasta was eaten in total. Each of the $9$ bites ate $3$ inches,

[4] #11 2.NBT.B.5 Work backwards: add the eaten pasta back to what is still on the table. The orig

Review

Reasonableness: Sanity-check the size: $44$ inches is almost $4$ feet, which is plausible for "a very long piece of pasta" and is the only answer choice that satisfies $17 + 3n = (\text{original})$ with $n = 9$. The other choices fail this test: $34 - 17 = 17$, $38 - 17 = 21$, $41 - 17 = 24$, $47 - 17 = 30$ — none of these is a multiple of $3$, so they could not be "$3$ inches per bite, exactly $9$ bites." Only $44 - 17 = 27 = 9 \times 3$ works. Answer (D) is confirmed.

Alternative: Tool #3 (Eliminate Possibilities) directly on the choices: the original length must equal $17 + 3n$ for some whole number of bites $n$, so $(\text{choice}) - 17$ must be a multiple of $3$. Checking each choice gives $17, 21, 24, 27, 30$; only $27$ is divisible by $3$ with a sensible bite count ($n = 9$), so (D) is forced without computing anything else.

CCSS standards used (min grade 4)

4.OA.C.5 Generate a number or shape pattern following a given rule (Discovering the rule "each bite adds exactly one piece," giving the pattern $\text{pieces} = 1 + n$ after $n$ bites.)
1.OA.D.8 Determine the unknown whole number in an addition or subtraction equation (Solving $1 + n = 10$ to find that Henry took $n = 9$ bites.)
3.OA.A.3 Solve multiplication and division word problems within 100 (Multiplying $9 \times 3 = 27$ inches to total the pasta eaten across all $9$ bites.)
2.NBT.B.5 Fluently add and subtract within 100 (Adding the leftover $17$ inches to the eaten $27$ inches to recover the original length $17 + 27 = 44$.)

⭐ This AMC 8 problem only needs the Grade 4 idea of "find the pattern, then work backwards" you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Review

CCSS standards used (min grade 4)

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