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AMC 8 · 2008 · #6

Grade 6 geometry-2d
Archetype Compound Area by Decomposition / Difference
ratio-proportionarea-rectanglessystematic-enumerationspatial-visualization identify-subproblemssystematic-enumeration ↑ Prerequisites: area-rectanglesratio-proportion
📏 Short solution 💡 2 insights 📊 Diagram

Problem

In the figure, what is the ratio of the area of the gray squares to the area of the white squares?

Pick an answer.

(A)
3:10
(B)
3:8
(C)
3:7
(D)
3:5
(E)
1:1
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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: The large diamond is sliced by lines parallel to its sides into a grid of identical small tilted squares; Three gray regions are visible: a $2 \times 2$ gray block in the center, one gray unit square on the far left, and one gray unit square on the far right; Every other small tilted square is white; Answer choices: (A) $3:10$, (B) $3:8$, (C) $3:7$, (D) $3:5$, (E) $1:1$

Unknowns: The ratio (gray area) $:$ (white area)

Understand

Restated: 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.

Givens: The large diamond is sliced by lines parallel to its sides into a grid of identical small tilted squares; Three gray regions are visible: a $2 \times 2$ gray block in the center, one gray unit square on the far left, and one gray unit square on the far right; Every other small tilted square is white; Answer choices: (A) $3:10$, (B) $3:8$, (C) $3:7$, (D) $3:5$, (E) $1:1$

Plan

Primary tool: #16 Use Structure / Transform

Secondary: #13 Count Possibilities

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 $4 \times 4$ 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.

Execute — Answer: D

#16 Use Structure / Transform 3.MD.C.6 Step 1
  • Re-see the figure as a tilted $4 \times 4$ grid.
  • The lines drawn inside the big diamond run parallel to its sides and chop it into $16$ identical small tilted squares — a $4$-by-$4$ tile grid that just happens to be rotated $45^\circ$.
  • Call the area of one small tile $1$ unit.
$$\text{total tiles} = 4 \times 4 = 16$$

💡 Grade 3 area-by-tiling: when a shape is covered by identical unit tiles, its area is just the number of tiles.

#13 Count Possibilities 3.OA.D.8 Step 2
  • Count the gray tiles.
  • The central gray region is a $2 \times 2$ block of tiles, so it contains $4$ tiles.
  • The small gray diamond on the far left is exactly $1$ tile, and the small gray diamond on the far right is also exactly $1$ tile.
$$\text{gray tiles} = 4 + 1 + 1 = 6$$

💡 Grade 3 multi-step word problem: add the count from each gray piece to get the total gray count.

#13 Count Possibilities 3.OA.D.8 Step 3
  • Find the white tile count by subtracting.
  • Every tile is either gray or white, so the white tiles are whatever is left after taking out the gray ones.
$$\text{white tiles} = 16 - 6 = 10$$

💡 Subtract part from whole — the classic "the rest" move.

#13 Count Possibilities 6.RP.A.1 Step 4
  • Write the ratio of gray area to white area, then simplify.
  • Each tile has the same area, so the ratio of areas is the same as the ratio of tile counts.
  • Divide both sides by the GCF $2$ to put it in lowest terms.
$$\text{gray} : \text{white} = 6 : 10 = 3 : 5 \;\Rightarrow\; \textbf{(D)}$$

💡 Ratios shrink by dividing both parts by the same number, just like fractions.

[1] #16 3.MD.C.6 Re-see the figure as a tilted $4 \times 4$ grid. The lines drawn inside the big
[2] #13 3.OA.D.8 Count the gray tiles. The central gray region is a $2 \times 2$ block of tiles,
[3] #13 3.OA.D.8 Find the white tile count by subtracting. Every tile is either gray or white, so
[4] #13 6.RP.A.1 Write the ratio of gray area to white area, then simplify. Each tile has the sam

Review

Reasonableness: Quick check: gray $+$ white $= 6 + 10 = 16$ tiles, which matches the $4 \times 4 = 16$ 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.

Alternative: Tool #9 (Solve an Easier Problem): give each small tile area $1$ and compute the actual areas instead of working with counts. Gray area $= 6$, white area $= 10$, ratio $= 6/10 = 3/5$, so the answer is $3:5 = $ (D). Same answer, just with numbers instead of pure counts.

CCSS standards used (min grade 6)

  • 3.MD.C.6 Measure areas by counting unit squares (Treating each small tilted square as one unit of area so the whole diamond's area becomes a count of $16$ tiles.)
  • 3.OA.D.8 Solve two-step word problems using the four operations (Adding $4 + 1 + 1$ to count gray tiles, then subtracting from $16$ to count white tiles.)
  • 6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship (Writing gray-to-white as the ratio $6:10$ and simplifying to $3:5$.)

⭐ 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.

⭐ 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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