AMC 10 · 2007 · #1
Grade 4 geometry-2dIsabella's house has 3 bedrooms. Each bedroom is 12 feet long, 10 feet wide, and 8 feet high. Isabella must paint the walls of all the bedrooms. Doorways and windows, which will not be painted, occupy 60 square feet in each bedroom. How many square feet of walls must be painted?
Pick an answer.
AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: Each bedroom is a box $12$ ft long, $10$ ft wide, and $8$ ft high; There are $3$ identical bedrooms; Only walls are painted — not the floor or ceiling; $60$ square feet of doorways and windows in each bedroom are not painted; Answer choices: (A) $678$, (B) $768$, (C) $786$, (D) $867$, (E) $876$
Unknowns: The total number of square feet of wall that must be painted across all $3$ bedrooms
Understand
Restated: 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.
Givens: Each bedroom is a box $12$ ft long, $10$ ft wide, and $8$ ft high; There are $3$ identical bedrooms; Only walls are painted — not the floor or ceiling; $60$ square feet of doorways and windows in each bedroom are not painted; Answer choices: (A) $678$, (B) $768$, (C) $786$, (D) $867$, (E) $876$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #1 Draw a Diagram
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 $12\times8$ and two $10\times8$) are counted and the floor and ceiling are correctly left out.
Execute — Answer: E
4.MD.A.3 Step 1 Find the area of the four walls
- Each bedroom is a box.
- The four walls are the vertical faces.
- Two opposite walls run along the length: each is $12$ ft wide and $8$ ft high, so each is $12\times8=96$ sq ft.
- The other two walls run along the width: each is $10$ ft wide and $8$ ft high, so each is $10\times8=80$ sq ft.
- The floor and ceiling are not walls, so they are skipped.
💡 A wall is just a rectangle, and its area is width times height.
4.MD.A.3 Step 2 Add up one bedroom's walls
- There are two walls of $96$ sq ft and two walls of $80$ sq ft.
- Add all four: $96+96+80+80=352$.
- So one bedroom has $352$ square feet of wall before removing any openings.
💡 Opposite walls come in matching pairs, so double each size and add the two pairs.
4.NBT.B.4 Step 3 Subtract the doorways and windows
- In each bedroom, $60$ square feet of doorways and windows will not be painted.
- Take that off the wall total: $352-60=292$.
- So each bedroom needs $292$ square feet of paint.
💡 Openings are holes in the wall, so their area is removed from what gets painted.
4.NBT.B.5 Step 4 Multiply by the three bedrooms
- All three bedrooms are identical, each needing $292$ square feet.
- Multiply by $3$: $292\times3=876$.
- That is the total painted wall area, which matches choice (E).
- Choice (A) $678$ and (D) $867$ come from scrambling these digits, and (B) $768$ and (C) $786$ come from arithmetic slips along the way.
💡 Three identical rooms means three equal amounts, so multiply one room's area by three.
4.MD.A.3 Each bedroom is a box. The four walls are the vertical faces. Two opposite walls 4.MD.A.3 There are two walls of $96$ sq ft and two walls of $80$ sq ft. Add all four: $96 4.NBT.B.4 In each bedroom, $60$ square feet of doorways and windows will not be painted. T 4.NBT.B.5 All three bedrooms are identical, each needing $292$ square feet. Multiply by $3 Review
Reasonableness: 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 $3\times290\approx870$ sq ft, right next to $876$. The answer must also be a multiple of $3$ (three identical rooms), and $876=3\times292$ 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.
Alternative: Handle all three bedrooms at once by scaling first. The painted area of one bedroom is $2(12\cdot8)+2(10\cdot8)-60=292$; multiply the whole expression by $3$ to get $3\times292=876$. Or factor the height out early: each room's wall area is $8\times(2\cdot12+2\cdot10)=8\times44=352$ (perimeter times height), then subtract $60$ and multiply by $3$.
CCSS standards used (min grade 4)
4.MD.A.3Apply the area formula for a rectangle in real world and mathematical problems (Finding each wall's area as width times height ($12\times8$ and $10\times8$) and totaling one bedroom's four walls to $352$ sq ft.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers using the standard algorithm (Subtracting the $60$ sq ft of doorways and windows: $352-60=292$.)4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number (Scaling one bedroom's $292$ sq ft to all three bedrooms: $292\times3=876$.)
⭐ Break a room into its four rectangular walls, add them up, take out the openings, then multiply by the number of identical rooms.
⭐ Break a room into its four rectangular walls, add them up, take out the openings, then multiply by the number of identical rooms.
More like this
Same archetype — closest grade level first.