AMC 8 · 2010 · #15

Easy mode Grade 6

Problem

Picture a jar full of gumdrops in five different colors. We do not know yet how many gumdrops there are in total.

We do know the breakdown by color. $30\%$ are blue. $20\%$ are brown. $15\%$ are red. $10\%$ are yellow. The remaining gumdrops are green, and there are exactly $30$ green gumdrops.

Now take half of the blue gumdrops out of the jar and replace them with brown gumdrops.

After the swap, how many brown gumdrops are in the jar?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Color shares: blue $30\%$, brown $20\%$, red $15\%$, yellow $10\%$; Green gumdrops: $30$ (the remaining color, count given directly); Half of the original blue gumdrops are replaced by brown gumdrops; Answer choices: (A) $35$, (B) $36$, (C) $42$, (D) $48$, (E) $64$

Unknowns: The final number of brown gumdrops in the jar after the swap

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units, #3 Eliminate Possibilities

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.

Execute — Answer: C

#7 Identify Subproblems 5.NBT.B.7 Step 1

Subproblem 1: find what percent is green.
The four named colors add to $30 + 20 + 15 + 10 = 75\%$, and all five colors must total $100\%$, so green fills the remaining $25\%$.

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

💡 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.

#8 Analyze the Units 6.RP.A.3 Step 2

Subproblem 2: find the total number of gumdrops.
We know $25\%$ of the total equals $30$ gumdrops.
Reading $25\%$ as the rate $\tfrac{25}{100} = \tfrac{1}{4}$ means "$1$ out of every $4$ gumdrops is green", so the total is $4$ times the green count.

$$\text{total} = 30 \div \tfrac{1}{4} = 30 \times 4 = 120$$

💡 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.

#7 Identify Subproblems 6.RP.A.3 Step 3

Subproblem 3: turn the blue and brown percents into actual counts using the total of $120$.
Multiply each color's decimal share by $120$.

$$\text{blue} = 0.30 \times 120 = 36, \quad \text{brown} = 0.20 \times 120 = 24$$

💡 "Percent of a quantity" as multiplication is the core Grade 6 ratio/percent skill; Tool #7 keeps each color as its own clean subproblem.

#7 Identify Subproblems 5.NF.B.4 Step 4

Subproblem 4: do the swap.
Half of the $36$ blue gumdrops — that is $18$ — leave the jar and $18$ brown gumdrops take their place.
The brown count rises from $24$ by $18$.

$$\text{new brown} = 24 + \tfrac{1}{2}(36) = 24 + 18 = 42 \;\Rightarrow\; \textbf{(C)}$$

💡 Taking "half of $36$" is fraction-times-whole-number from Grade 5; Tool #7 finishes the chain by combining the saved subresults.

#3 Eliminate Possibilities 6.RP.A.3 Step 5

Match $42$ to the choices and rule out the others.
$42$ lands on choice (C); (A) $35$ and (B) $36$ are too small (you can't end up with fewer brown than you started with plus $18$ added), and (D) $48$, (E) $64$ overshoot the only valid count.

$$42 \in \{35, 36, 42, 48, 64\} \;\Rightarrow\; \textbf{(C)}$$

💡 The AMC habit of locking the computed value onto the answer list (Tool #3) is the cheap safety net against arithmetic slips.

[1] #7 5.NBT.B.7 Subproblem 1: find what percent is green. The four named colors add to $30 + 20

[2] #8 6.RP.A.3 Subproblem 2: find the total number of gumdrops. We know $25\%$ of the total equ

[3] #7 6.RP.A.3 Subproblem 3: turn the blue and brown percents into actual counts using the tota

[4] #7 5.NF.B.4 Subproblem 4: do the swap. Half of the $36$ blue gumdrops — that is $18$ — leave

[5] #3 6.RP.A.3 Match $42$ to the choices and rule out the others. $42$ lands on choice (C); (A)

Review

Reasonableness: 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.

Alternative: Tool #6 (Guess and Check) on the total: if green's $25\%$ equals $30$ gumdrops, try total $= 120$ — then $25\%$ of $120 = 30$, matching. Brown starts at $20\% \times 120 = 24$; adding half of blue ($18$) gives $42$. Tool #13 (Convert to Algebra) also works: let $T$ be the total, then $0.25T = 30 \Rightarrow T = 120$ and $\text{brown}_\text{new} = 0.20T + 0.5(0.30T) = 0.20T + 0.15T = 0.35T = 0.35 \times 120 = 42$. We chose Tool #7 because the subproblem chain is shorter to explain than the algebraic compression.

CCSS standards used (min grade 6)

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Adding and subtracting the color percentages ($30 + 20 + 15 + 10 = 75$ and $100 - 75 = 25$) to find that green is $25\%$.)
5.NF.B.4 Multiply a fraction by a whole number (Taking half of the $36$ blue gumdrops to get $\tfrac{1}{2} \times 36 = 18$ swapped pieces.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, including percent problems (Treating $25\%$ as a rate to recover the total ($30 \div 0.25 = 120$), converting blue and brown percentages into counts ($0.30 \times 120 = 36$ and $0.20 \times 120 = 24$), and matching the final count to the multiple-choice list.)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 6)

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