Problem 11 · 2022 AMC 8
Easy
Arithmetic & Operations
off-by-one
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?
Show answer
Answer: D — 44 inches.
Show hints
Hint 1 of 2
Each bite takes one piece and leaves two behind — so every bite raises the piece-count by exactly 1. Work backward from 10 pieces to the number of bites.
Still stuck? Show hint 2 →
Hint 2 of 2
Going from 1 piece to 10 takes 9 bites. Each bite removes 3 inches, and the original = what's left + what was eaten.
Show solution
Approach: every bite adds one piece, so pieces − 1 = number of bites
- Insight: don't think about lengths yet — count the bites. A bite from the middle splits one piece into two, so each bite raises the piece count by exactly 1. Starting at 1 piece and ending at 10 means 10 − 1 = 9 bites.
- Each bite removed 3 inches: 9 × 3 = 27 inches eaten.
- Original = what remains + what was eaten = 17 + 27 = 44 inches.
- You'll see this again: this is the “fenceposts vs. gaps” idea — cutting a piece into N parts always takes N − 1 cuts. Count the separations, not the parts.
Mark:
· log in to save