AMC 8 · 2010 · #15

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

percentagefraction-arithmeticratio-proportion identify-subproblems ↑ Prerequisites: percentagefraction-arithmetic

📏 Medium solution 💡 4 insights

Problem

A jar holds 5 colors of gumdrops with these shares: 30% blue, 20% brown, 15% red, 10% yellow, and the rest green. The green pile is 30 gumdrops. After half of the blue gumdrops are swapped out for brown ones, how many brown gumdrops end up in the jar?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The question hides four small steps inside one sentence, so Tool #7 (Identify Subproblems) breaks it into clean pieces: (a) find green's percent, (b) use green's percent and its 30-count to find the total, (c) turn percents into counts for blue and brown, (d) do the swap. Tool #8 (Analyze the Units) keeps "percent of the whole" and "actual count" from getting mixed up — 25% is not 25 gumdrops. Tool #3 (Eliminate) is the AMC habit of checking the final count against the choices.

1STEP 1

Subproblem 1: the four named colors sum to 75%, and all five must total 100%, so green fills the remaining 25%.

100% - (30% + 20% + 15% + 10%) = 100% - 75% = 25%

2STEP 2

Subproblem 2: 25% of the total is the 30 green, and 25% means 1 out of every 4, so the total is 4 × 30 = 120.

total = 30 ÷

\frac{1}{4}

= 30 × 4 = 120

3STEP 3

Subproblem 3: apply the shares to the total of 120 — blue is 0.30 × 120 = 36 and brown is 0.20 × 120 = 24.

blue = 0.30 × 120 = 36, brown = 0.20 × 120 = 24

4STEP 4

Subproblem 4: half of the 36 blue — that is 18 — leave and 18 browns take their place, so brown rises from 24 to 42.

new brown = 24 +

\frac{1}{2}

(36) = 24 + 18 = 42 → (C)

5STEP 5

Match 42 to the list — it lands on choice (C); the others are ruled out as too small (35, 36) or too big (48, 64).

42 ∈ {35, 36, 42, 48, 64} → (C)

Answer

Quick sanity pass on the totals. After the swap the jar should still hold 120 gumdrops: blue drops from 36 to 18, brown rises from 24 to 42, red stays 18, yellow stays 12, green stays 30. Sum: 18 + 42 + 18 + 12 + 30 = 120. The jar count is preserved and brown = 42 is the only choice that fits.

💡Key takeaway

Once you know the green pile is 25% and equals 30 gumdrops, the rest is just Grade 6 percent reasoning — find the total, take percents of it, and add the swap.

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