AMC 10 · 2008 · #2

Grade 6 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Axis-Aligned Figure Area from Coordinates

area-rectanglesratio-proportionpercentage convert-to-algebra ↑ Prerequisites: area-rectangles

📏 Medium solution 💡 1 insight

Problem

A square sits inside a rectangle. The rectangle's width is twice the square's side, and the rectangle's length is twice its own width. Find what percent of the rectangle's area lies inside the square.

Pick an answer.

(A)

$\ 12.5$

(B)

$\ 25$

(C)

$\ 50$

(D)

$\ 75$

(E)

$\ 87.5$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

Because only ratios are given, the shape's size is free — so pick a name for the one length everything is compared to. Tool #4 (Introduce a Variable): let the square's side be s and write every other length in terms of s. Tool #1 (Draw a Diagram) makes the two ratios concrete and confirms the square fits inside. Once both areas are in terms of s, the s cancels and Tool #3 (Eliminate Possibilities) confirms the single matching choice.

1STEP 1

Draw it and name the side

Sketch it and let the square's side be $s$ . The width is twice the side, so the width is $2s$ .

side = s, width = 2s

2STEP 2

Use the second ratio for the length

The length is twice the width, and the width is $2s$ , so the length is $4s$ .

length = 2 × width = 2 × 2s = 4s

3STEP 3

Find both areas

Rectangle area is $4s$ times $2s$ , which is $8s^2$ ; the square's area is $s$ times $s$ , which is $s^2$ .

A_rect = 4s × 2s = 8s², A_sq = s²

4STEP 4

Turn the fraction into a percent

The ratio $\frac{s^2}{8s^2}$ cancels to $\frac{1}{8}$ , which is 12.5 percent — choice (A).

s²/8s² = 1/8 = 12.5% → (A)

Answer

12.5

The rectangle is 2 × 4 = 8 little squares each the size of the drawn square, so the square is exactly 1 of those 8 parts, i.e. 1/8 = 12.5%. That matches. A value like 25% would mean the rectangle held only 4 such squares, but the width alone already spans 2 squares and the length spans 4, giving 8 — so 12.5% is the sensible answer.

💡Key takeaway

When a problem gives only ratios, name the smallest length and build the rest from it — the letter cancels at the end, so the width of 2 and length of 4 make 8 squares, and one of them is 12.5%.

Draw it and name the side
Use the second ratio for the length
Find both areas
Turn the fraction into a percent

Draw it and name the side

Use the second ratio for the length

Find both areas

Turn the fraction into a percent

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