AMC 8 · 2025 · #3

Grade 3 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

multi-digit-arithmeticrateratio-proportion identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmetic

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Problem

In Buffalo Shuffle-o, the whole deck is dealt evenly among the players. With Annika and 3 friends (4 players total), each player gets 15 cards. If 2 more friends join next time so there are 6 players, how many cards will each player get?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The big question ("how many cards each in Game 2?") hides two smaller questions: (1) how big is the deck? and (2) how does that deck split among the new number of players? Tool #7 (Identify Subproblems) names those two pieces so we can solve them one at a time. Tool #8 (Analyze the Units) keeps the bookkeeping honest — players × cards/player = cards, and then cards/players = cards/player — which guarantees we multiply when we should and divide when we should.

1STEP 1

Subproblem 1 — the deck stays fixed: Game 1's 4 players × 15 cards each means the deck holds 60 cards.

4 players × 15 cards/player = 60 cards

2STEP 2

Game 2 keeps the same 4 players and adds 2 friends, so now 6 players share the deck.

4 + 2 = 6 players

3STEP 3

Subproblem 2 — share that fixed 60-card deck among 6 players; 60 ÷ 6 gives 10 cards each → (C).

(60 cards)/(6 players) = 10 cards/player → (C)

Answer

More players sharing the same deck must mean fewer cards each, so the Game 2 answer should be less than 15. It is: 10 ＜ 15. Also, 10 cards per player × 6 players = 60 cards, which matches the deck size from Game 1 — the deck didn't shrink or grow, just got resliced. Choices (D) 11 and (E) 12 would need a bigger deck; (A) 8 and (B) 9 would leave cards left over. Only (C) 10 divides 60 evenly into 6 shares.

💡Key takeaway

This AMC 8 problem only needs Grade 3 multiplication and equal-sharing division you already know!

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