Competition · AMC preparation · step 4 of 4
AMC 8 · 2024 · #9
Grade 4 arithmeticPick an answer.
AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Once you notice G must be even, the most natural move is Tool #6 (Guess and Check): try the smallest even green counts G = 2, 4, 6, 8, … and just build the collection. After a few tries, Tool #5 (Look for a Pattern) reveals the totals are 7, 14, 21, 28, … — multiples of 7. Since the problem is multiple choice, Tool #3 (Eliminate Possibilities) finishes the job by keeping only the choice that is a multiple of 7. This avoids algebra entirely; we ‘build’ the answer with concrete numbers.
Try the smallest even green
Red is half of green, so green must be even; the smallest even case G = 2 gives R = 1, B = 4.
‘Half as many’ and ‘twice as many’ are multiplicative comparison phrases, which are introduced in the Grade 4 multiplicative-comparison standard.
4.OA.A.1Guess And CheckAdd the three colors
Adding the three counts gives the smallest total: 1 + 2 + 4 = 7.
Adding three small whole numbers to get a single total is the Grade 1 three-addend addition skill.
1.OA.A.2Guess And CheckSpot the multiples of 7
Trying G = 4, 6, 8 gives totals 14, 21, 28 — they rise by 7 each time, so every total is a multiple of 7.
Generating a number pattern from a rule (each new even G adds 7 to the total) is the Grade 4 ‘generate a pattern from a rule’ standard.
The total number of marbles is always a multiple of 7.
▸ Why?
The total is the three color counts added together, and once green is an even count 2x those counts are x, 2x, and 4x, so the total is x + 2x + 4x = 7x — a whole number of 7s.
▸ Why?
The marbles are exactly the red, green, and blue groups combined with none left over, so the total equals red plus green plus blue.
▸ Why?
The counts x, 2x, and 4x are all built from the same x, so adding them takes x one, two, and four times — that is x times (1 + 2 + 4), which is x times 7.
▸ Why?
x times 7 is x equal groups of 7 marbles, and any count made of whole groups of 7 is a multiple of 7.
Keep the choice divisible by 7
Among the choices, only 28 = 7 × 4 is a multiple of 7, and (R,G,B) = (4,8,16) confirms it works.
Recognizing which whole numbers are multiples of 7 is exactly the Grade 4 ‘factors and multiples’ standard.
4.OA.B.4Eliminate PossibilitiesThis AMC 8 problem only needs Grade 4 multiplicative comparison and multiples of 7 you already know!
- Try the smallest even green
- Add the three colors
- Spot the multiples of 7
- Keep the choice divisible by 7
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