AMC 10 · 2010 · #3

Grade 6 geometry-2d

Archetype Overlap / Complement Counting └ Sub-archetype Geometric Area via Inclusion-Exclusion or Area Balance

area-rectanglespercentageratio-proportion convert-to-algebraspatial-visualization ↑ Prerequisites: area-rectangles

📏 Long solution 💡 3 insights 📊 Diagram

Problem

A rectangle and a square overlap, and the shared region is a known share of each. Find the ratio of the rectangle's sides.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Introduce a Variable

Nothing in the problem carries a number of centimetres, so Tool #4 (Introduce a Variable) names the square's side a and works with everything else as a multiple of a. Before any algebra, Tool #1 (Draw a Diagram) settles the one fact the whole problem turns on: the overlap is a band that runs the full width of the square and the full height of the rectangle. That fact is what the percentages get attached to — on their own, 50% and 20% only say the rectangle's area is 2/5 of the square's area, which does not pin down a side ratio. With the band shape in hand, Tool #13 (Convert to Algebra) turns each percentage into one length relation, and Tool #6 (Guess and Check) closes the loop by building an actual rectangle and square with those measurements to confirm such a picture really exists.

1STEP 1

See what the overlap really is

The overlap is a band spanning the square's full width.

overlap = (square's full width) × AD

2STEP 2

Name the square's side

Naming the square's side writes every area.

[EFGH] = a², [ABCD] = AB × AD, overlap = a × AD

3STEP 3

The 20% fact fixes AD

One percentage fixes the rectangle's short side.

a · AD = 1/5a² → AD = a/5

4STEP 4

The 50% fact fixes AB

The other fixes its long side.

a · AD = 1/2(AB · AD) → a = 1/2AB → AB = 2a

5STEP 5

Divide the two sides

Dividing gives the ratio 10.

AB/AD = 2a/a/5 = 2a · 5/a = 10

6STEP 6

Build one and check it

Building an example confirms 10, choice (E).

a=10: [EFGH]=100, [ABCD]=20 · 2=40, overlap=10 · 2=20; 20/40=50%, 20/100=20%, AB/AD=10

Answer

Two quick sanity tests. First, size: the shared band is half the rectangle but only a fifth of the square, so the rectangle's area is 0.2/0.5 = 2/5 of the square's area — a small area — while the rectangle is longer than the square is wide. A shape that is long but low on area has to be very thin, so a large side ratio is expected, and 10 is the largest choice offered. Second, the explicit 20 × 2 rectangle against the 10 × 10 square in the last step satisfies both percentage conditions exactly, so 10 is not just forced but actually achieved. Note also where the argument's weight sits: the two percentages by themselves only give [ABCD] = 2/5[EFGH], which fixes an area, not a shape. It is the band geometry — the overlap spanning the square's full width and the rectangle's full height — that converts those areas into the two length facts AD = a/5 and AB = 2a.

💡Key takeaway

The shaded band is the same piece seen twice — once as half the rectangle, once as a fifth of the square — and each view hands you one side length in terms of the square's side.

See what the overlap really is
Name the square's side
The 20% fact fixes AD
The 50% fact fixes AB
Divide the two sides
Build one and check it