Problem 20 · 2025 AMC 8
Hard
Algebra & Patterns
arithmetic-seriesfraction-to-decimal
Sarika, Dev, and Rajiv are sharing a large block of cheese. They take turns cutting off half of what remains and eating it: first Sarika eats half of the cheese, then Dev eats half of the remaining half, then Rajiv eats half of what remains, then back to Sarika, and so on. They stop when the cheese is too small to see. About what fraction of the original block of cheese does Sarika eat in total?
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Answer: A — 4/7.
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Hint 1 of 2
Sarika eats on turns 1, 4, 7, … — every third turn. Each bite is half of what's left, so list the sizes of her bites and look for a pattern from one of hers to the next.
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Hint 2 of 2
Her bites are 12, then 116, then 1128, … — each is 18 of the one before (three halvings between her turns). That's a geometric series.
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Approach: sum the geometric series of Sarika's bites
- Each turn eats half of what remains, so after turn n there's 1/2n left, and turn n ate 1/2n of the original. Sarika takes turns 1, 4, 7, …, so her bites are 12, 116, 1128, … — each 18 of the previous.
- So it's a geometric series, first term a = 12, ratio r = 18.
- Sum = a1 − r = 1/27/8 = 4/7.
- Why this transfers: a never-ending halving (or any |ratio| < 1) sums to a finite total a1 − r — pull out one person's terms, find the constant ratio between consecutive ones, and apply the formula.
Another way — ratio of bite sizes (no infinite sum):
- In every round of three turns, Sarika, Dev, and Rajiv eat in the fixed ratio 12 : 14 : 18 = 4 : 2 : 1 of whatever was there at the round's start.
- Since that 4 : 2 : 1 split repeats on every leftover chunk, it holds for the whole block too. Sarika's share is 44 + 2 + 1 = 4/7 — and the cheese is essentially all eaten.
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