Competition · AMC preparation · step 4 of 4
AMC 8 · 2024 · #8
Grade 4 arithmeticPick an answer.
AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
There are only two choices per day for three days, so at most 2³ = 8 possible paths — small enough to write down every single one (Tool #2). Drawing the day-by-day possibilities as a branching tree (Tool #1) makes duplicate amounts (different paths, same total) impossible to miss. At the end we cross out duplicates and match against the answer choices (Tool #3). No algebra is needed.
List Tuesday's amounts
Apply +3 and double to Monday's $2 — 2 + 3 = 5 and 2 × 2 = 4 — so Tuesday's amounts are {4, 5}.
Adding within 20 and doubling small numbers are fluent Grade 2 mental-math operations.
2.OA.B.2Make A Systematic ListBranch to Wednesday
Branch each Tuesday amount: 4→7,8 and 5→8,10; the raw list {7,8,8,10} repeats 8, so distinct Wednesday amounts are {7, 8, 10}.
A branching tree diagram makes the two paths joining at 8 obvious. Single-digit doublings (4 × 2, 5 × 2) are part of Grade 3 multiplication fluency.
3.OA.C.7Draw A DiagramBranch to Thursday
Apply both actions to each Wednesday amount: 7→10,14; 8→11,16; 10→13,20, giving the Thursday list {10, 14, 11, 16, 13, 20} — six values.
Carrying out the two operations within 100 step by step on each prior amount is the multi-step, four-operation reasoning of Grade 3.
3.OA.D.8Make A Systematic ListCount the distinct amounts
All six candidates are distinct, so the count of possible amounts is 6 — choice (D); 3, 4, 5, 7 don't match and are eliminated.
Generating the terms of a sequence by repeatedly applying a given rule (+3 or × 2) and counting distinct outputs is exactly the Grade 4 'generate a pattern following a rule' standard.
The number of different dollar amounts possible on Thursday is found by listing the amount produced by every three-day sequence of actions and then counting each distinct value once.
▸ Why?
Every amount Taye can reach on Thursday comes from choosing one of the two actions on each of the three days, so listing the result of each such choice-sequence captures every amount that can occur.
▸ Why?
Each day exactly one of the two actions is taken, so the full set of outcomes is built by branching each current amount into its two next amounts, one more day at a time.
▸ Why?
Applying an action gives one definite new amount: adding 3 is a single sum, and doubling is taking two equal groups of that amount.
▸ Why?
Splitting every amount into its two next cases leaves no path out and lets no path be counted twice, so these cases together are exactly the whole set of reachable amounts.
▸ Why?
Two different action-sequences can land on the very same dollar amount, so to count different amounts each value must be counted only once no matter how many sequences reach it.
▸ Why?
Counting the different amounts means pairing each distinct value with one count number, so a value reached by several sequences still adds only one to the total.
This AMC 8 problem only needs the Grade 4 skill of generating number patterns by following a given rule that you already know!
- List Tuesday's amounts
- Branch to Wednesday
- Branch to Thursday
- Count the distinct amounts
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