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

📘 View easy version →

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

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

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 =

\frac{1}{4}

× 1 =

\frac{1}{4}

3STEP 3

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 =

\frac{1}{2}

\frac{1}{4}

\frac{1}{8}

4STEP 4

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 =

\frac{1}{4}

\frac{1}{8}

\frac{2}{8}

\frac{1}{8}

\frac{3}{8}

5STEP 5

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

bolded_BR =

\frac{1}{4}

× 1 =

\frac{1}{4}

6STEP 6

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 =

\frac{1}{4}

\frac{1}{8}

\frac{3}{8}

\frac{1}{4}

\frac{2}{8}

\frac{1}{8}

\frac{3}{8}

\frac{2}{8}

\frac{8}{8}

= 1

7STEP 7

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 =

\frac{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%.

More like this