AMC 8 · 2008 · #25

Grade 7 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Nested Frame / Concentric Difference

area-circlesfraction-arithmeticpercentagepattern-recognition area-differenceidentify-subproblems ↑ Prerequisites: area-circlespercentage

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

A target-style design is built from six circles. The smallest has radius 2 inches, and each next circle's radius is 2 inches larger. The circles alternate black and white, with the outermost ring white. What percent of the whole design is black? Pick the closest choice.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The black area is not one shape — it is the innermost disk plus two rings. Tool #7 (Identify Subproblems) splits the work into computing each black piece separately, then adding them. Tool #15 (Visualize) helps read the picture: starting from the outside, the colors go white, black, white, black, white, black. That tells us exactly which radii bound each black region. Once the black total and the design total are in hand, the percent is a single division.

1STEP 1

Square each of the six radii 2, 4, 6, 8, 10, 12 with A = π r² to get the disk areas.

A₁=4π, A₂=16π, A₃=36π, A₄=64π, A₅=100π, A₆=144π

2STEP 2

Reading outward the colors go white, black, white, black, white, black, so the black is the inner disk plus two rings.

Black = (A₅ - A₄) + (A₃ - A₂) + A₁

3STEP 3

Add the three black pieces: (100π - 64π) + (36π - 16π) + 4π = 60π.

(100π - 64π) + (36π - 16π) + 4π = 36π + 20π + 4π = 60π

4STEP 4

The whole design is the outer circle 144π, so the black share is $\frac{60\pi}{144\pi}$ = $\frac{5}{12}$ .

\frac{60\pi}{144\pi}

\frac{60}{144}

\frac{5}{12}

5STEP 5

As a percent, $\frac{5}{12}$ ≈ 41.7%, whose closest choice is 42.

\frac{5}{12}

= 0.416 ≈ 41.7% → (A) 42

Answer

Quick gut check: the three white rings have areas 144π - 100π = 44π, 64π - 36π = 28π, and 16π - 4π = 12π, summing to 84π. Together black plus white is 60π + 84π = 144π, matching the total. White is the slightly bigger color because the outermost ring (the widest one) is white, which fits 58.3% white vs 41.7% black. Answer (A) 42% is the only choice below 50%, and that is exactly where the picture lands.

💡Key takeaway

Don't try to measure the black shape all at once. Cut it into a small disk and two rings, find each piece with π r², and the percent falls out.

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