AMC 8 · 2017 · #9

Grade 4 rate-rationumber-theory

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Integrality / Divisibility Screen

fraction-arithmeticmultipleslcmratio-proportion systematic-enumerationratio-proportion ↑ Prerequisites: fraction-arithmeticmultiples

📏 Medium solution 💡 3 insights

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Problem

Marcy's marbles come in four colors: blue, red, green, yellow. Exactly

\frac{1}{3}

of them are blue, exactly

\frac{1}{4}

are red, and exactly 6 are green. We want the smallest possible number of yellow marbles she could have.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

Instead of writing an equation with an unknown total, we use Tool #9: replace the abstract "total" with the smallest concrete totals that work, and check each one. Because $\frac{1}{3}$ and $\frac{1}{4}$ of the total must both be whole numbers, the total has to be a multiple of 3 and of 4 — so the candidates are 12, 24, 36, …. Tool #6 (Guess and Check) walks through these smallest totals in order; the first total that leaves a non-negative number of yellow marbles is the answer. Tool #3 (Eliminate) keeps us honest: once we find Y = 4, we can confirm that the smaller answer choices (1, 2, 3) are impossible.

1STEP 1

Blue needs $\frac{1}{3}$ and red needs $\frac{1}{4}$ to be whole, so the total is a multiple of both 3 and 4 — candidates are 12, 24, 36, ...

Possible totals ∈ {12, 24, 36, …}

2STEP 2

Test total 12: blue 4 + red 3 + green 6 = 13, already 1 over 12 before any yellow — so 12 is impossible.

4 + 3 + 6 = 13 ＞ 12 → no room for yellow

3STEP 3

Test total 24: blue 8 + red 6 + green 6 = 20, leaving 24 - 20 = 4 yellow — a valid non-negative whole number.

8 + 6 + 6 = 20, 24 - 20 = 4 yellow

4STEP 4

The only smaller total, 12, already failed and bigger totals like 36 give 9 yellow, so 4 is the smallest.

T=12 → invalid; T=24 → Y=4; T=36 → Y=9; …

5STEP 5

The smallest number of yellow marbles is 4, matching choice (D).

Y_min = 4 → (D)

Answer

Sanity-check with T = 24: 8 blue + 6 red + 6 green + 4 yellow = 24. Fractions check out:

\frac{8}{24}

\frac{1}{3}

blue and

\frac{6}{24}

\frac{1}{4}

red, exactly as the problem says. Yellow is the smallest of all four counts (4 ＜ 6 ＜ 6 ≤ 8), which fits the question's hint that yellow is the leftover, not the dominant, color. Answer 4 is also one of the listed choices, so we are not off by a factor.

💡Key takeaway

This AMC 8 problem only needs Grade 4 thinking about multiples of 3 and 4 that you already know!

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