Competition · AMC preparation · step 4 of 4

AMC 8 · 2011 · #7

Grade 6 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

area-rectanglesarea-trianglesfraction-arithmetic identify-subproblemsarea-difference ↑ Prerequisites: fraction-arithmeticarea-rectangles

📏 Medium solution 💡 3 insights 📊 Diagram

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Problem

Four congruent large squares are each cut into congruent triangles or rectangles, and some pieces in each square are drawn with bold (thick) outlines. Taking the four squares together as the whole, what percent of the combined area is enclosed by the bold outlines?

Pick an answer.

(A)

$12\frac{1}{2}$

(B)

(C)

(D)

$33\frac{1}{3}$

(E)

$37\frac{1}{2}$

AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The whole problem is visual, so Tool #1 (Draw a Diagram) is the natural lead: redraw each square, mark its subdivisions, and shade the bolded piece so the fraction it covers becomes obvious. Tool #7 (Identify Subproblems) lets us handle one square at a time instead of fighting all four bolded regions at once. Tool #9 (Solve an Easier Related Problem) is the bookkeeping trick: set the side of one large square to 1 so each big-square area is 1 and the combined area is 4 — every bolded piece is then a simple fraction, no variables needed.

1STEP 1

Set one square's area to 1

Give one large square area 1, so the four squares together have combined area 4 — the easiest numbers that keep the proportions.

area of one large square = 1, combined area = 4

2STEP 2

Measure the top-left square

Top-left square: it splits into 4 equal vertical strips and 1 is bolded, so the bolded part is $\frac{1}{4}$ of a square.

bolded_TL = 1/4 × 1 = 1/4

3STEP 3

Measure the top-right square

Top-right square: the top-right quadrant ( $\frac{1}{4}$ ) is halved by a diagonal, so the bolded triangle is $\frac{1}{8}$ .

bolded_TR = 1/2 × 1/4 = 1/8

4STEP 4

Measure the bottom-left square

Bottom-left square: split the bolded region into the $\frac{1}{4}$ quadrant plus a $\frac{1}{8}$ triangle on its top edge, giving $\frac{3}{8}$ .

bolded_BL = 1/4 + 1/8 = 2/8 + 1/8 = 3/8

5STEP 5

Measure the bottom-right square

Bottom-right square: exactly one of the 4 equal quadrants is bolded, so the bolded part is $\frac{1}{4}$ .

bolded_BR = 1/4 × 1 = 1/4

6STEP 6

Add the four bolded pieces

Put every piece over denominator 8 and add: $\frac{2}{8}$ + $\frac{<span class="hl-ask">1</span>}{8}$ + $\frac{3}{8}$ + $\frac{2}{8}$ = 1 whole square.

total bolded = 1/4 + 1/8 + 3/8 + 1/4 = 2/8 + 1/8 + 3/8 + 2/8 = 8/8 = 1

7STEP 7

Turn the ratio into a percent

Compare the bolded 1 to the combined 4: that is $\frac{1}{4}$ of the whole, which as a percent is 25% → (C).

percent bolded = 1/4 × 100% = 25% → (C)

Answer

The four bolded pieces are

\frac{1}{4}

\frac{1}{8}

\frac{3}{8}

\frac{1}{4}

of one big square — none larger than half — so their sum should sit a little above 1 big square. It comes out to exactly 1, which divided by the 4 big squares gives

\frac{1}{4}

=25%. That value sits between choices (B) 20% and (D) 33

\frac{1}{3}

%, exactly where a "slightly more than a quarter of each" picture should land.

💡Key takeaway

Set each big square's area to 1, read each bolded piece as a fraction, then add — the four pieces sum to 1 out of 4, which is 25%.

Set one square's area to 1
Measure the top-left square
Measure the top-right square
Measure the bottom-left square
Measure the bottom-right square
Add the four bolded pieces
Turn the ratio into a percent

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