AMC 8 · 2019 · #2

Grade 4 geometry-2d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Rectilinear Frame / Box Border

area-rectanglesspatial-visualization identify-subproblemsphysical-representation ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic

📏 Medium solution 💡 2 insights 📊 Diagram

📘 View easy version →

Problem

Three identical small rectangles are arranged to make one big rectangle ABCD. The figure shows two of the small rectangles stacked on the left (each lying with its long side horizontal) and one standing on the right with its long side vertical. The shorter side of each small rectangle is 5 feet. Find the area, in square feet, of the big rectangle ABCD.

Pick an answer.

(A)

(B)

(C)

100

(D)

125

(E)

150

AMC 8 2019 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

The figure is the whole story, so Tool #1 (Draw a Diagram) is the natural entry: label every short side s and every long side l on the picture, and the constraint that the two columns must reach the same height of ABCD pops out as the equation 2s = l. From there Tool #7 (Identify Subproblems) splits the work into three clean pieces — (1) deduce l from s, (2) compute the width and height of ABCD, (3) multiply for the area — instead of trying to attack the area in one move. Reaching for algebra (Tool #13) would be overkill: a single labeled drawing makes the relationship visible.

1STEP 1

The left column is two short sides (s + s = 2s) tall; the right column is one long side (l) tall. Same height, so 2s = l.

2s = l

2STEP 2

Substitute s = 5 into 2s = l, so each small rectangle's long side is l = 10 ft.

l = 2 × 5 = 10 ft

3STEP 3

ABCD is l = 10 ft tall; its width is a long side plus a short side, 10 + 5 = 15 ft. So ABCD measures 10 by 15.

height = 10 ft, width = 10 + 5 = 15 ft

4STEP 4

Multiply ABCD's sides: area = length × width = 15 × 10 = 150 square feet → (E).

area = 15 × 10 = 150 sq ft → (E)

Answer

150

Cross-check by counting copies: each small rectangle has area s × l = 5 × 10 = 50 sq ft, and the big rectangle is made of exactly 3 of them, so the total area must be 3 × 50 = 150 sq ft. This matches the formula answer and lands on choice (E). The magnitudes (a 15 × 10 ft floor area) also make physical sense for 'rectangle made of three small rectangles'.

💡Key takeaway

This AMC 8 problem only needs the Grade 4 rectangle area formula (length times width) you already know!

More like this