Competition · AMC preparation · step 4 of 4
AMC 8 · 2022 · #11
Grade 4 arithmeticlogicPick an answer.
AMC 8 2022 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find the bite rule
Each bite removes a middle and leaves two ends, turning 1 piece into 2 — 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.
4.OA.C.5Look For A PatternFind the number of bites
The final pile has 10 pieces, so 1 + n = 10, meaning 9 bites.
Filling in the missing number in 1 + □ = 10 is a Grade 1 unknown-addend question.
1.OA.D.8Look For A PatternFind the length eaten
Each of the 9 bites ate 3 inches, so the total eaten is 9 × 3 = 27 inches.
Equal-groups multiplication (9 groups of 3 inches) is the Grade 3 way to total identical bites.
3.OA.A.3Work BackwardsAdd back what was eaten
Work backwards — add the eaten pasta back to the 17 inches still on the table: 17 + 27 = 44 inches.
Putting back what was removed is exactly the work-backwards move, and the final sum is a basic Grade 2 addition within 100.
The original length of the pasta equals the 17 inches still on the table plus every inch of pasta Henry ate.
▸ Why?
None of the original pasta disappears for any other reason — each inch is either still sitting in one of the leftover pieces or was eaten — so the leftover length and the eaten length are the two parts that add back to the original length.
▸ Why?
The eaten pasta is 9 bites that each removed the same 3 inches, and 9 equal helpings of 3 inches total 27 inches.
▸ Why?
Every bite eats the same 3 inches, so the whole amount eaten is 9 equal groups of 3 inches added up.
▸ Why?
The pieces grew from 1 to 10, and each bite makes exactly one new piece, so the number of bites equals the number of new pieces, which is 9.
▸ Why?
Biting the middle out of one piece removes the middle part and leaves its two end parts, so that single piece turns into two — one new piece for that bite.
▸ Why?
Each bite pairs with exactly the one extra piece it creates, so counting the bites is the same as counting the extra pieces beyond the starting one, namely 10 minus 1.
This AMC 8 problem only needs the Grade 4 idea of "find the pattern, then work backwards" you already know!
- Find the bite rule
- Find the number of bites
- Find the length eaten
- Add back what was eaten
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