AMC 8 · 2008 · #6

Grade 6 geometry-2d

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

ratio-proportionarea-rectanglessystematic-enumerationspatial-visualization identify-subproblemssystematic-enumeration ↑ Prerequisites: area-rectanglesratio-proportion

📏 Short solution 💡 2 insights 📊 Diagram

Problem

A large tilted square (a diamond) is divided by parallel lines into smaller identical tilted unit squares. Some of those unit squares are shaded gray and the rest are white. Find the ratio of the total gray area to the total white area.

Pick an answer.

(A)

3:10

(B)

3:8

(C)

3:7

(D)

3:5

(E)

1:1

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

How to solve

Strategy Use Structure / Transform

Tool #16 (Use Structure) is the key move: stop looking at the figure as three odd gray blobs and re-see it as one tidy 4 × 4 grid of identical tilted unit squares. Once the whole diamond is a grid, every region is just a pile of those unit squares. Tool #13 (Count Possibilities) then handles the easy part — count gray units, count total units, subtract for white, and read off the ratio.

1STEP 1

Re-see the whole diamond as a tilted 4 × 4 grid of 16 identical unit tiles, each of area 1.

total tiles = 4 × 4 = 16

2STEP 2

Count the gray tiles: 4 in the central 2 × 2 block, plus 1 far left and 1 far right, giving 6 gray tiles.

gray tiles = 4 + 1 + 1 = 6

3STEP 3

Every tile is gray or white, so subtract: 10 white tiles are left from the 16 total.

white tiles = 16 - 6 = 10

4STEP 4

Areas match tile counts, so gray : white = 6 : 10; divide both by 2 to get 3 : 5.

gray : white = 6 : 10 = 3 : 5 → (D)

Answer

3:5

Quick check: gray + white = 6 + 10 = 16 tiles, which matches the 4 × 4 = 16 total tiles in the big diamond. The ratio 3:5 also makes the gray area smaller than the white area, which matches what the picture looks like — most of the diamond is white.

💡Key takeaway

When a shape is cut into identical tiles, areas turn into tile counts — count the gray tiles, count the rest, and the ratio falls out.

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