AMC 10 · 2007 · #1

Grade 4 geometry-2d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

area-rectanglessurface-areamulti-digit-arithmetic identify-subproblems ↑ Prerequisites: area-rectangles

📏 Short solution 💡 1 insight

Problem

A house has 3 identical bedrooms, each 12 feet long, 10 feet wide, and 8 feet high. Only the four walls of each bedroom get painted, and in every bedroom 60 square feet of doorways and windows are left unpainted. Find the total painted wall area, in square feet.

Pick an answer.

(A)

$\ 678$

(B)

$\ 768$

(C)

$\ 786$

(D)

$\ 867$

(E)

$\ 876$

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The total is built from several smaller pieces — the area of one bedroom's four walls, the unpainted doorways/windows, and the fact that there are three bedrooms — so Tool #7 (Identify Subproblems) solves it one layer at a time: one wall, one bedroom, then all three. Tool #1 (Draw a Diagram) keeps the box straight so that only the four vertical walls (two 12×8 and two 10×8) are counted and the floor and ceiling are correctly left out.

1STEP 1

Find the area of the four walls

Two opposite walls are 12×8=96 sq ft each, the other two are 10×8=80 sq ft each, and the floor and ceiling are not walls.

12×8=96, 10×8=80

2STEP 2

Add up one bedroom's walls

Add all four walls: 96+96+80+80=352, so one bedroom has 352 square feet of wall before any openings.

2×96+2×80=192+160=352

3STEP 3

Subtract the doorways and windows

Take off the 60 sq ft of doorways and windows: 352-60=292 square feet painted in each bedroom.

352-60=292

4STEP 4

Multiply by the three bedrooms

Three identical bedrooms at 292 sq ft each give 292×3=876 square feet in total, which is choice (E).

292×3=876 → (E)

Answer

876

A quick sanity check: one bedroom's walls total 352 sq ft, and 60 sq ft of that is not painted, leaving roughly 290 sq ft per room. Three rooms is about 3×290≈870 sq ft, right next to 876. The answer must also be a multiple of 3 (three identical rooms), and 876=3×292 passes, while none of the trap choices are as clean. The height 8 appears in every wall, so 876 being a bit under 900 fits three medium rooms.

💡Key takeaway

Break a room into its four rectangular walls, add them up, take out the openings, then multiply by the number of identical rooms.

Find the area of the four walls
Add up one bedroom's walls
Subtract the doorways and windows
Multiply by the three bedrooms

Find the area of the four walls

Add up one bedroom's walls

Subtract the doorways and windows

Multiply by the three bedrooms

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