AMC 8 · 2022 · #11
Easy mode Grade 4Problem
Imagine one very long piece of pasta. Henry the donkey takes bites out of it.
Each time Henry bites, he takes 3 inches out of the middle of one piece. That one piece becomes two pieces, and the bite is gone.
Henry keeps taking bites. When he is done, there are 10 pieces of pasta on the table. If you add up the lengths of all 10 pieces, the total is 17 inches.
How long was the piece of pasta Henry started with?
Pick an answer.
AMC 8 2022 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: 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
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
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
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.
💡 Each bite removes a middle and leaves two ends, so the rule "one bite, one extra piece" is a clean number 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.
💡 Filling in the missing number in $1 + \square = 10$ is a Grade 1 unknown-addend question.
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.
💡 Equal-groups multiplication ($9$ groups of $3$ inches) is the Grade 3 way to total identical bites.
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.
💡 Putting back what was removed is exactly the work-backwards move, and the final sum is a basic Grade 2 addition within $100$.
4.OA.C.5 Find the rule for how the number of pieces changes per bite. Imagine $1$ long pi 1.OA.D.8 Use the rule to find how many bites Henry took. The final pile has $10$ pieces, 3.OA.A.3 Compute how much pasta was eaten in total. Each of the $9$ bites ate $3$ inches, 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.5Generate 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.8Determine 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.3Solve 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.5Fluently 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!
⭐ This AMC 8 problem only needs the Grade 4 idea of "find the pattern, then work backwards" you already know!
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