AMC 8 · 2012 · #6

Grade 4 geometry-2d

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

area-rectanglesperimeter area-differenceidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticarea-rectangles

📏 Short solution 💡 2 insights

Problem

A rectangular photograph that is 8 inches tall and 10 inches wide sits inside a frame whose border is 2 inches wide on every side. Find the area, in square inches, of just the border (the frame region, not including the photo).

Pick an answer.

(A)

$hspace{.05in}36$

(B)

$hspace{.05in}40$

(C)

$hspace{.05in}64$

(D)

$hspace{.05in}72$

(E)

$hspace{.05in}88$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The border is a ring — an awkward shape to measure directly. Tool #7 (Identify Subproblems) turns it into two easy rectangles: find the area of the big outer rectangle (photo + border), find the area of the inner photo, then subtract. Tool #1 (Draw a Diagram) is the supporting move: sketch the photo inside the frame and label every side, so the 2-inch border gets added to BOTH ends of each dimension (not just one), which is the most common mistake on this problem.

1STEP 1

Sketch the photo inside the frame and add the 2-inch border to BOTH ends of each side — top and bottom, left and right.

outer height = 2 + 8 + 2 = 12 in, outer width = 2 + 10 + 2 = 14 in

2STEP 2

Use area = length × width to find the outer rectangle (photo + border).

outer area = 12 × 14 = 168 in²

3STEP 3

Do the same for the inner rectangle — the photo itself.

photo area = 8 × 10 = 80 in²

4STEP 4

Subtract the photo's area from the outer area; what remains is exactly the border ring.

border area = 168 - 80 = 88 in² → (E)

Answer

hspace{.05in}88

Sanity check the size: an 8 × 10 photo has area 80. A border 2 inches wide on every side adds roughly the perimeter (2 × (8+10) = 36) times the border width (2), which is about 72, plus the four corner squares (4 × 2 × 2 = 16), totaling 88. That matches answer (E) exactly and is in the right ballpark — bigger than 72, smaller than the photo itself.

💡Key takeaway

A border around a rectangle is just "big rectangle minus the photo inside" — Grade 4 area facts are all you need!

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