← Sensim Lab Sensim Math

AMC 8 · 2008 · #25

Grade 7 geometry-2d
Archetype Compound Area by Decomposition / Difference
area-circlesfraction-arithmeticpercentagepattern-recognition area-differenceidentify-subproblems ↑ Prerequisites: area-circlespercentage
📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the percent of the design that is black?

Pick an answer.

(A)
42
(B)
44
(C)
45
(D)
46
(E)
48
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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: Six circles with radii $2, 4, 6, 8, 10, 12$ inches (smallest to largest); Colors alternate; from the picture, the outermost ring is white and the innermost disk is black; Answer choices: (A) $42$, (B) $44$, (C) $45$, (D) $46$, (E) $48$

Unknowns: The percent of the total design area that is black

Understand

Restated: 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.

Givens: Six circles with radii $2, 4, 6, 8, 10, 12$ inches (smallest to largest); Colors alternate; from the picture, the outermost ring is white and the innermost disk is black; Answer choices: (A) $42$, (B) $44$, (C) $45$, (D) $46$, (E) $48$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #15 Visualize

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.

Execute — Answer: A

#7 Identify Subproblems 7.G.B.4 Step 1
  • List the six radii and their disk areas.
  • The radii grow by $2$ inches each, so they are $2, 4, 6, 8, 10, 12$.
  • Use $A = \pi r^2$ for each.
$$A_1=4\pi,\; A_2=16\pi,\; A_3=36\pi,\; A_4=64\pi,\; A_5=100\pi,\; A_6=144\pi$$

💡 Grade 7 area-of-a-circle formula applied six times. Squaring even numbers is easy: $r^2$ goes $4, 16, 36, 64, 100, 144$.

#15 Visualize 7.G.B.4 Step 2
  • Read the picture to find which regions are black.
  • Going from outside in, the colors alternate white, black, white, black, white, black.
  • So the black pieces are: (i) the ring between circle $5$ (radius $10$) and circle $4$ (radius $8$), (ii) the ring between circle $3$ (radius $6$) and circle $2$ (radius $4$), and (iii) the innermost disk (radius $2$).
$$\text{Black} = (A_5 - A_4) + (A_3 - A_2) + A_1$$

💡 Tool #15: trace the rings from the outside and label them W, B, W, B, W, B. Each ring's area is the outer disk minus the inner disk.

#7 Identify Subproblems 7.NS.A.3 Step 3
  • Compute each black piece and add them.
  • Use the disk areas from Step 1.
$$(100\pi - 64\pi) + (36\pi - 16\pi) + 4\pi = 36\pi + 20\pi + 4\pi = 60\pi$$

💡 Subtract, then add — Grade 7 arithmetic with the common factor $\pi$ kept as is.

#7 Identify Subproblems 7.NS.A.3 Step 4
  • The whole design is bounded by the outermost circle, so its total area is $A_6 = 144\pi$.
  • Divide the black area by the total to get the fraction that is black.
$$\dfrac{60\pi}{144\pi} = \dfrac{60}{144} = \dfrac{5}{12}$$

💡 $\pi$ cancels top and bottom. Divide $60$ and $144$ by their common factor $12$ to get $\tfrac{5}{12}$.

#7 Identify Subproblems 6.RP.A.3 Step 5

Convert $\tfrac{5}{12}$ to a percent and pick the closest choice.

$$\dfrac{5}{12} = 0.41\overline{6} \approx 41.7\% \;\Rightarrow\; \textbf{(A) } 42$$

💡 Grade 6 percent: multiply the fraction by $100\%$. $41.7\%$ is closest to $42$ on the list.

[1] #7 7.G.B.4 List the six radii and their disk areas. The radii grow by $2$ inches each, so t
[2] #15 7.G.B.4 Read the picture to find which regions are black. Going from outside in, the col
[3] #7 7.NS.A.3 Compute each black piece and add them. Use the disk areas from Step 1.
[4] #7 7.NS.A.3 The whole design is bounded by the outermost circle, so its total area is $A_6 =
[5] #7 6.RP.A.3 Convert $\tfrac{5}{12}$ to a percent and pick the closest choice.

Review

Reasonableness: Quick gut check: the three white rings have areas $144\pi - 100\pi = 44\pi$, $64\pi - 36\pi = 28\pi$, and $16\pi - 4\pi = 12\pi$, summing to $84\pi$. Together black plus white is $60\pi + 84\pi = 144\pi$, 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.

Alternative: Tool #5 (Find a Pattern): drop the units of $\pi$ and just track $r^2$. Black uses $r^2 = 100 - 64 + 36 - 16 + 4$, which alternates plus-minus from the outside. That equals $60$. Total is $r^2 = 144$. Ratio $\tfrac{60}{144} = \tfrac{5}{12} \approx 41.7\%$ — same answer (A) with no $\pi$ written down.

CCSS standards used (min grade 7)

  • 7.G.B.4 Know the formulas for the area and circumference of a circle and use them to solve problems (Computing each disk area as $\pi r^2$ for $r = 2, 4, 6, 8, 10, 12$ and turning each black ring into (outer disk) $-$ (inner disk).)
  • 7.NS.A.3 Solve real-world and mathematical problems involving the four operations with rational numbers (Adding and subtracting the $\pi$-multiples to get black area $= 60\pi$ and simplifying $\tfrac{60}{144}$ to $\tfrac{5}{12}$.)
  • 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, including percent (Converting the fraction $\tfrac{5}{12}$ to the percent $41.7\%$ and picking the closest answer choice.)

⭐ Don't try to measure the black shape all at once. Cut it into a small disk and two rings, find each piece with $\pi r^2$, and the percent falls out.

⭐ Don't try to measure the black shape all at once. Cut it into a small disk and two rings, find each piece with $\pi r^2$, and the percent falls out.

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