Competition · AMC preparation · step 4 of 4
AMC 8 · 2010 · #15
Grade 6 arithmeticPick an answer.
AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find the green percent
Subproblem 1: the four named colors sum to 75%, and all five must total 100%, so green fills the remaining 25%.
Splitting off "find green's share" first is the Tool #7 move; the arithmetic itself is whole-number subtraction inside Grade 5 decimal/percent fluency.
5.NBT.B.7Identify SubproblemsFind the total gumdrops
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.
Tool #8 here means tracking that the 30 is a count but the 25% is a rate per total; recovering the total from a part and its rate is Grade 6 percent reasoning.
The jar must hold 120 gumdrops in all.
▸ Why?
The 30 green gumdrops fill exactly one quarter of the jar, and the four equal quarters together make the whole, so the total is four thirties.
▸ Why?
Green is 25% of the jar, and 25% means 25 out of 100 — the same value as 1 out of 4, because dividing both the top 25 and the bottom 100 by 25 leaves the fraction unchanged — so green is exactly one of four equal parts.
▸ Why?
The four quarter-shares cover the whole jar, so adding the four equal shares of 30 rebuilds the total count.
▸ Why?
The jar is split into the four quarter-shares with nothing left out and nothing counted twice, so those parts add back to the whole.
▸ Why?
Four equal shares of 30 gumdrops is 4 groups of 30, which is 120.
Turn percents into counts
Subproblem 3: apply the shares to the total of 120 — blue is 0.30 × 120 = 36 and brown is 0.20 × 120 = 24.
"Percent of a quantity" as multiplication is the core Grade 6 ratio/percent skill; Tool #7 keeps each color as its own clean subproblem.
6.RP.A.3Identify SubproblemsDo the color swap
Subproblem 4: half of the 36 blue — that is 18 — leave and 18 browns take their place, so brown rises from 24 to 42.
Taking "half of 36" is fraction-times-whole-number from Grade 5; Tool #7 finishes the chain by combining the saved subresults.
5.NF.B.4Identify SubproblemsMatch against the choices
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).
The AMC habit of locking the computed value onto the answer list (Tool #3) is the cheap safety net against arithmetic slips.
6.RP.A.3Eliminate PossibilitiesOnce 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.
- Find the green percent
- Find the total gumdrops
- Turn percents into counts
- Do the color swap
- Match against the choices
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