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 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?

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.

How to solve

Strategy 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% = $\frac{2000}{100}$ = 20, turning "2000% of" into plain multiplication by 20. No variables or equations are needed — just two short arithmetic steps in order.

1STEP 1

Subproblem 1: four quarters at $0.25 each add up to exactly <span class="mk-a"><span class="hl-ask">$ 1.00 of face value.

4 ×

0.25 =

1.00

2STEP 2

Convert the rate: "percent" means per hundred, so 2000% = $\frac{2000}{100}$ = 20 times the face value.

2000% =

\frac{2000}{100}

= 20

3STEP 3

Subproblem 2: multiply the $1.00 face value by 20, giving the payment <span class="mk-a"><span class="hl-ask">$ 20 → (A).

20 ×

1.00 =

20 → (A)

Answer

20 dollars

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.

💡Key takeaway

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

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