AMC 10 · 2024 · #16

Grade 8 geometry-2dalgebra

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Multiplicative Rate or Series

similar-figuresarea-rectanglesratio-proportionexponents convert-to-algebraidentify-subproblems ↑ Prerequisites: area-rectanglessimilar-figuresexponents

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

An outer rectangle is tiled by smaller rectangles that are all similar to the outer one. Each small rectangle is labeled with its area. Find the length AB of the outer rectangle's top side.

Pick an answer.

(A)

$4+4\sqrt5$

(B)

$10\sqrt2$

(C)

$5+5\sqrt5$

(D)

$10\sqrt[4]{8}$

(E)

AMC 10 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Convert to Algebra

Eight different areas with one shared shape is a textbook trigger for Tool #13 (Convert to Algebra): name the short and long side of a unit-area rectangle x and y, and every other rectangle's sides become x√(A) and y√(A). Once everything is in (x, y), the figure becomes one equation (top edge = bottom edge), which fixes the ratio $\frac{y}{x}$ . Tool #7 (Identify Subproblems) then splits the work into clean stages — (A) get the ratio from the side-equality, (B) get x and y from the area constraint xy=1, (C) plug back into the top-edge sum for AB. Each stage is short on its own.

1STEP 1

Name the unit rectangle's sides x (short) and y (long), so xy = 1; then any area-A rectangle has sides x√(A) and y√(A).

xy = 1, sides of area-A rectangle = (x√(A), y√(A))

2STEP 2

Along AB the area-32 rectangle lies long-side horizontal and the 25, 49 short-side, giving AB = 4√(2) y + 12x.

AB = y√(32) + x√(25) + x√(49) = 4√(2) y + 12x

3STEP 3

The bottom edge crosses 36, 16 short-side (6x, 4x) and 8, 18 long-side (2√(2) y, 3√(2) y), giving 10x + 5√(2) y.

bottom = 6x + 4x + 2√(2) y + 3√(2) y = 10x + 5√(2) y

4STEP 4

Setting top = bottom, 4√(2) y + 12x = 10x + 5√(2) y collapses to 2x = √(2) y, so $\frac{y}{x}$ = √(2).

4√(2) y + 12x = 10x + 5√(2) y → 2x = √(2) y →

\frac{y}{x}

= √(2)

5STEP 5

Combine y = x√(2) with xy = 1: x²√(2) = 1, so x = 2⁻¹/4 and y = 2¹/4.

x(x√(2)) = 1 → x² =

\frac{1}{√(2)}

→ x =

\frac{1}{sqrt[4]{2}}

, y = x√(2) = sqrt[4]{2}

6STEP 6

Substitute x, y into AB = 4√(2) y + 12x: both terms become 2³/4 multiples, giving 10 · 2³/4 = 10sqrt[4]{8} → (D).

AB = 4√(2) (2¹/4) + 12 (2⁻¹/4) = 4 · 2³/4 + 6 · 2³/4 = 10 · 2³/4 = 10sqrt[4]{8} → (D)

Answer

10sqrt[4]{8}

The ratio k = √(2) matches the famous A4-paper ratio, which is exactly the ratio that lets a rectangle tile copies of itself similarly — a strong sanity signal. Numerically 10sqrt[4]{8} ≈ 10 · 1.6818 ≈ 16.82, which sits between the other choices (B) 10√(2) ≈ 14.1 and (E) 20. Also, computing the bottom edge with the same x, y gives 10x + 5√(2) y = 10 · 2⁻¹/4 + 5√(2)· 2¹/4 = 6 · 2³/4 + 4 · 2³/4 = 10 · 2³/4 — the same number, confirming the equation was consistent.

💡Key takeaway

Similar rectangles all share one side ratio k. Naming the unit rectangle's sides x and y turns every length in the figure into a multiple of x or y, and the equality of opposite edges of the outer rectangle gives one clean equation that pins down k = √(2) — the same ratio as A4 paper.

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