Competition · AMC preparation · step 4 of 4

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% = 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

Find the value of 4 quarters

Subproblem 1: four quarters at $0.25 each add up to exactly $1.00 of face value.

4 × 0.25=0.25 =1.00
2STEP 2

Turn the percent into a multiplier

Convert the rate: "percent" means per hundred, so 2000% = 2000100\frac{2000}{100} = 20 times the face value.

2000% = 2000/100 = 20
3STEP 3

Multiply by the face value

Subproblem 2: multiply the $1.00 face value by 20, giving the payment $20 → (A).

20 × 1.00=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 2000100\frac{2000}{100} = 20 — twenty times the original. Four quarters add up to $1, and twenty times $1 is $20, answer (A).

  • Find the value of 4 quarters
  • Turn the percent into a multiplier
  • Multiply by the face value

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