AMC 8 · 2010 · #10

Grade 7 geometry-2d

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

area-circlesratio-proportionfraction-arithmetic identify-subproblems ↑ Prerequisites: area-circlesfraction-arithmetic

📏 Medium solution 💡 3 insights

Problem

A pizza is a circle with diameter 12 inches. Six pepperoni circles, all the same size, fit edge-to-edge exactly across that diameter. Twenty-four such pepperoni circles are placed on the pizza without any two overlapping. What fraction of the pizza's area is covered by pepperoni?

Pick an answer.

(A)

$\frac 12$

(B)

$\frac 23$

(C)

$\frac 34$

(D)

$\frac 56$

(E)

$\frac 78$

AMC 8 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

Tool #1 (Draw a Diagram) turns the words into a labeled picture: a big circle (the pizza) with 6 small circles lined up across its diameter. That picture immediately exposes the key length — each pepperoni's diameter — by simple division. Tool #11 (Look for Symmetry / Structure) then handles the finish: both the pizza area and the total pepperoni area are multiples of π, so the π cancels and the answer is just a clean number ratio. No algebra needed.

1STEP 1

Six pepperoni span the 12-in diameter, so each pepperoni's diameter is 12 ÷ 6 = 2 in, giving radius 1 in.

pepperoni diameter =

\frac{12}{6}

= 2 in → r_pep = 1 in

2STEP 2

The pizza's radius is 6 in, so its area is π · 6² = 36π square inches.

A_pizza = π · 6² = 36π in²

3STEP 3

Each pepperoni has radius 1 in and area π, so 24 non-overlapping pepperoni cover 24π square inches.

A_pep,total = 24 · π · 1² = 24π in²

4STEP 4

The covered fraction is $\frac{24π}{36π}$ ; the π cancels top and bottom, leaving $\frac{24}{36}$ .

\frac{A_pep,total}{A_pizza}

\frac{24π}{36π}

\frac{24}{36}

5STEP 5

Both 24 and 36 are divisible by 12, so $\frac{24}{36}$ reduces to $\frac{2}{3}$ — choice (B).

\frac{24}{36}

\frac{(24 ÷ 12)}{(36 ÷ 12)}

\frac{2}{3}

→ (B)

Answer

\frac{2}{3}

The pepperoni are circles packed into a circle — circle packing always leaves gaps, so the covered fraction must be less than 1. Also, the 24 small circles each have radius

\frac{1}{6}

of the pizza's radius, so each pepperoni's area is (

\frac{1}{6}

)² =

\frac{1}{36}

of the pizza. Twenty-four of them give

\frac{24}{36}

\frac{2}{3}

— same answer.

\frac{2}{3}

sits sensibly between the other choices and rules out the larger ones

\frac{3}{4}

\frac{5}{6}

\frac{7}{8}

💡Key takeaway

This AMC 8 problem just needs the Grade 7 circle-area formula A = π r² — and the π cancels out, leaving a clean fraction!

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