AMC 8 · 2001 · #10

Grade 6 arithmeticrate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

percentagefraction-decimal-conversionmulti-digit-arithmetic identify-subproblems ↑ Prerequisites: percentagefraction-decimal-conversion

📏 Short solution 💡 2 insights

Problem

A collector offers to buy state quarters for 2000% of their face value. At that rate how much will Bryden get for his four state quarters?

Pick an answer.

(A)

$20\text{ dollars}$

(B)

$50\text{ dollars}$

(C)

$200\text{ dollars}$

(D)

$500\text{ dollars}$

(E)

$2000\text{ dollars}$

AMC 8 2001 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: A U.S. state quarter is worth $\$0.25$. A collector offers to pay $2000\%$ of the face value. How much money does Bryden get for his $4$ state quarters?

Givens: Each state quarter has face value $\$0.25$; Bryden has $4$ quarters; The collector pays $2000\%$ of the total face value; Answer choices: (A) $\$20$, (B) $\$50$, (C) $\$200$, (D) $\$500$, (E) $\$2000$

Unknowns: The dollar amount the collector pays Bryden

Understand

Restated: A U.S. state quarter is worth $\$0.25$. A collector offers to pay $2000\%$ of the face value. How much money does Bryden get for his $4$ state quarters?

Plan

Primary tool: #7 Break Into Subproblems

The problem packs two simple ideas into one sentence: a money total and a percent rate. Tool #7 (Break Into Subproblems) splits the work cleanly: (a) find the total face value of $4$ quarters, then (b) take $2000\%$ of that total. The percent step shrinks once you notice $2000\% = \dfrac{2000}{100} = 20$, turning "$2000\%$ of" into plain multiplication by $20$. No variables or equations are needed — just two short arithmetic steps in order.

Execute — Answer: A

#7 Break Into Subproblems 4.MD.A.2 Step 1

Subproblem 1: find the total face value of $4$ quarters.
Each quarter is $\$0.25$, and four quarters make one whole dollar.

$4 \times \$0.25 = \$1.00$

💡 Four quarters make a dollar — the Grade 4 money fact that anchors the rest of the problem.

#7 Break Into Subproblems 6.RP.A.3 Step 2

Convert $2000\%$ into a plain multiplier.
"Percent" means "per hundred", so $2000\%$ is $\dfrac{2000}{100}$, which simplifies to $20$.
Paying $2000\%$ of the face value is the same as paying $20$ times the face value.

$$2000\% = \dfrac{2000}{100} = 20$$

💡 $100\%$ is one whole, so $2000\%$ is $20$ wholes — twenty times the original amount.

#7 Break Into Subproblems 6.RP.A.3 Step 3

Subproblem 2: apply the multiplier to the face value.
Multiply $\$1.00$ by $20$ to get the collector's payment.

$20 \times \$1.00 = \$20 \;\Rightarrow\; \textbf{(A)}$

💡 Twenty times one dollar is twenty dollars — the two subproblems combine into the final answer.

[1] #7 4.MD.A.2 Subproblem 1: find the total face value of $4$ quarters. Each quarter is $\$0.25

[2] #7 6.RP.A.3 Convert $2000\%$ into a plain multiplier. "Percent" means "per hundred", so $200

[3] #7 6.RP.A.3 Subproblem 2: apply the multiplier to the face value. Multiply $\$1.00$ by $20$

Review

Reasonableness: Quick size check: $2000\%$ is $20$ times the face value, and the face value is $\$1$, so the payment is $\$20$ — choice (A). The trap answers track common slips. (B) $\$50$ would come from misreading the rate. (C) $\$200$ matches treating the rate as $200$ times instead of $20$ times (dividing $2000$ by $10$ instead of $100$). (D) $\$500$ uses $500\%$ on $4$ quarters in a confused way. (E) $\$2000$ treats $2000\%$ as if it meant "multiply by $2000$" — forgetting the "per hundred" in "percent". Choosing (A) means the percent was converted correctly.

Alternative: Tool #5 (Find a Pattern) using benchmark percents. $100\%$ of $\$1$ is $\$1$. $200\%$ of $\$1$ is $\$2$. $1000\%$ of $\$1$ is $\$10$. Each time the percent multiplies by $10$, so does the dollar amount. So $2000\%$ of $\$1$ is $\$20$ — same answer (A), reached by scaling instead of formula.

CCSS standards used (min grade 6)

4.MD.A.2 Use the four operations to solve word problems involving money, including problems involving simple fractions or decimals (Computing the total face value $4 \times \$0.25 = \$1.00$ — a Grade 4 money fact (four quarters make a dollar).)
6.RP.A.3 Use ratio and rate reasoning to solve real-world problems, including finding a percent of a quantity (Converting $2000\%$ to the multiplier $20$ and computing $20 \times \$1.00 = \$20$ as "$2000\%$ of $\$1$".)

⭐ "Percent" means "per hundred", so $2000\%$ is just $\dfrac{2000}{100} = 20$ — twenty times the original. Four quarters add up to $\$1$, and twenty times $\$1$ is $\$20$, answer (A).

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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