AMC 10 · 2010 · #2

Grade 6 geometry-2d

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

ratio-proportionarea-rectangleslinear-equations-one-var convert-to-algebra ↑ Prerequisites: ratio-proportion

📏 Medium solution 💡 2 insights 📊 Diagram

Problem

Four identical small squares and one rectangle are fitted together, with no gaps or overlaps, to build one large square. The four squares sit side by side in a single row across the top, spanning the full width of the large square, and the rectangle fills everything below them. Find how many times as large the rectangle's length is compared with its width.

Pick an answer.

(A)

$\ \dfrac{5}{4}$

(B)

$\ \dfrac{4}{3}$

(C)

$\ \dfrac{3}{2}$

(D)

$\ 2$

(E)

$\ 3$

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The figure has no numbers, so give the small square a side length s. Once s is named, the large square's side and the rectangle's two sides all become simple multiples of s, and the requested comparison turns into a ratio of those multiples.

1STEP 1

Name the small square's side

Let s stand for the side of one small square; the four are identical, so every one has side s.

s = side of one small square

2STEP 2

Read the large square from the four squares

The four squares fill the top row edge to edge, so the big square's side is four of them: 4s.

4 × s = 4s

3STEP 3

Find the rectangle's length and width

The rectangle takes all that is left below, so its length is 4s and its width is 4s - s = 3s.

length = 4s, width = 4s - s = 3s

4STEP 4

Compare length to width

"How many times as large" means length divided by width: 4s/3s = 4/3 once s cancels — choice (B).

4s/3s = 4/3

Answer

4/3

In the picture the rectangle is a little wider than it is tall, so its length should be a bit more than its width, not double or triple it. The value 4/3≈1.33 matches that gentle stretch, while choices like 2 or 3 would make the rectangle look far shorter than it does. So (B) is sensible.

💡Key takeaway

Give the little square a name like s, and every other length in the picture becomes an easy multiple of it.

Name the small square's side
Read the large square from the four squares
Find the rectangle's length and width
Compare length to width

Name the small square's side

Read the large square from the four squares

Find the rectangle's length and width

Compare length to width

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