Competition · AMC preparation · step 4 of 4

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
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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)
36
(B)
40
(C)
64
(D)
72
(E)
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

Find the outer dimensions

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

Find the outer area

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

outer area = 12 × 14 = 168 in²
3STEP 3

Find the photo area

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

photo area = 8 × 10 = 80 in²
4STEP 4

Subtract to get the border

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

border area = 168 - 80 = 88 in² → (E)
Answer
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!

  • Find the outer dimensions
  • Find the outer area
  • Find the photo area
  • Subtract to get the border

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