AMC 8 · 2011 · #3

Grade 6 geometry-2d

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

area-rectanglesratio-proportionpattern-recognition area-differenceidentify-subproblems ↑ Prerequisites: area-rectanglesratio-proportion

📏 Medium solution 💡 3 insights 📊 Diagram

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Problem

A 5 × 5 square of tiles has 8 black tiles forming a middle-ring border and 17 white tiles (the 1 × 1 center plus the outer 5 × 5 ring). Add one more layer of black tiles all the way around. What is the ratio of black tiles to white tiles in the new figure?

Pick an answer.

(A)

8:17

(B)

25:49

(C)

36:25

(D)

32:17

(E)

36:17

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

How to solve

Strategy Draw a Diagram

The picture is the whole problem. Tool #1 (Draw a Diagram) lets us see the figure as nested squares — a 1 × 1 white center inside a 3 × 3 black ring inside a 5 × 5 white ring — and then mentally wrap a 7 × 7 black ring around it. Tool #7 (Identify Subproblems) splits the counting into two clean pieces: (a) how many black tiles are in the new outer ring (a difference of two square areas), and (b) what are the new totals once we add that ring to the old counts. The white count never changes, so the only real work is counting the new black tiles.

1STEP 1

See the figure as nested squares, so 3² - 1² = 8 black and 17 white — matching the given counts.

3² - 1² = 8 black, 1 + (5² - 3²) = 17 white

2STEP 2

A one-tile border grows the side by 2, so the figure becomes a 7 × 7 square of 49 tiles.

5 + 2 = 7, 7² = 49

3STEP 3

The new ring is the 7 × 7 minus the 5 × 5, so it holds 7² - 5² = 24 black tiles.

7² - 5² = 49 - 25 = 24

4STEP 4

Add old and new black: 8 + 24 = 32; white stays 17. Check: 32 + 17 = 49 = 7 × 7. ✓

black = 8 + 24 = 32, white = 17, 32 + 17 = 49 ✓

5STEP 5

Since 17 is prime and can't divide 32, black : white = 32 : 17 is already in lowest terms — choice (D).

black : white = 32 : 17 → (D)

Answer

32:17

The new black ring (24 tiles) is much bigger than the original black ring (8 tiles), so the black count should jump well past the white count. We get 32 ＞ 17, which matches. The total 32 + 17 = 49 = 7² confirms no tile was lost or double-counted. Choices that put black below white (A) or use only the original 25-square (B) or swap the colors (C) all fail this sanity check; only (D) 32 : 17 is consistent.

💡Key takeaway

This AMC 8 problem only needs Grade 6 ratio language built on Grade 3 area thinking — "big square minus small square" — that you already know!