AMC 8 · 2015 · #1

Grade 5 geometry-2d

Archetype Arithmetic Expression Decomposition

area-rectanglesunit-conversion dimensional-analysis ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

Onkon wants to cover his room's floor with his favourite red carpet. How many square yards of red carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.)

$\textbf{(A) }12\qquad\textbf{(B) }36\qquad\textbf{(C) }108\qquad\textbf{(D) }324\qquad \textbf{(E) }972$

Pick an answer.

(A)

(B)

(C)

108

(D)

324

(E)

972

View mode:

Toolkit + CCSS Solution

Understand

Restated: Onkon's room floor is a rectangle that measures $12$ feet long and $9$ feet wide. He wants to cover the whole floor with red carpet, but carpet is sold in square yards, not square feet. Using the fact that $3$ feet $= 1$ yard, how many square yards of carpet does he need?

Givens: Floor is a rectangle: $12$ feet long, $9$ feet wide; Unit conversion: $3$ feet $= 1$ yard; Answer choices: (A) $12$, (B) $36$, (C) $108$, (D) $324$, (E) $972$ (square yards)

Unknowns: The floor's area in square yards

Understand

Givens: Floor is a rectangle: $12$ feet long, $9$ feet wide; Unit conversion: $3$ feet $= 1$ yard; Answer choices: (A) $12$, (B) $36$, (C) $108$, (D) $324$, (E) $972$ (square yards)

Plan

Primary tool: #8 Analyze the Units

Secondary: #1 Draw a Diagram

The trap of this problem is units: the dimensions come in feet but the answer must be in square yards. Tool #8 (Analyze the Units) keeps the bookkeeping honest — convert each side from feet to yards first ($12$ ft $= 4$ yd, $9$ ft $= 3$ yd), then multiply to get area in square yards. Tool #1 (Draw a Diagram) helps younger solvers see why the answer is $12$ and not $108$: sketching the $4$-yd by $3$-yd rectangle and tiling it with $1$-yd squares shows exactly $12$ tiles, making the unit conversion concrete instead of abstract.

Execute — Answer: A

#8 Analyze the Units 5.MD.A.1 Step 1

Convert each side from feet to yards using $3$ feet $= 1$ yard.
Divide each length by $3$.

$$12 \text{ ft} \div 3 \tfrac{\text{ft}}{\text{yd}} = 4 \text{ yd}, \quad 9 \text{ ft} \div 3 \tfrac{\text{ft}}{\text{yd}} = 3 \text{ yd}$$

💡 Dividing feet by (feet per yard) cancels the "feet" unit and leaves "yards" — a Grade 5 measurement conversion.

#1 Draw a Diagram 3.MD.C.7 Step 2

Sketch the floor as a $4 \text{ yd} \times 3 \text{ yd}$ rectangle and split it into $1 \text{ yd} \times 1 \text{ yd}$ squares.
The grid has $4$ columns and $3$ rows of unit squares.

$$4 \text{ columns} \times 3 \text{ rows} = 12 \text{ unit squares}$$

💡 Tiling a rectangle with unit squares and counting them is the Grade 3 visual definition of area.

#8 Analyze the Units 4.MD.A.3 Step 3

Apply the rectangle area formula with both sides already in yards.
The product gives area directly in square yards.

$$\text{Area} = 4 \text{ yd} \times 3 \text{ yd} = 12 \text{ yd}^2 \;\Rightarrow\; \textbf{(A)}$$

💡 Multiplying yards by yards produces square yards — the units track the geometry exactly.

[1] #8 5.MD.A.1 Convert each side from feet to yards using $3$ feet $= 1$ yard. Divide each leng

[2] #1 3.MD.C.7 Sketch the floor as a $4 \text{ yd} \times 3 \text{ yd}$ rectangle and split it

[3] #8 4.MD.A.3 Apply the rectangle area formula with both sides already in yards. The product g

Review

Reasonableness: Cross-check by computing in square feet first: $12 \times 9 = 108$ square feet. Since $1$ square yard $= 3 \text{ ft} \times 3 \text{ ft} = 9$ square feet, the floor is $108 \div 9 = 12$ square yards. Same answer, $\textbf{(A)}$. The trap answer $108$ appears as choice (C), which is the result if a solver forgets to convert. The other choices ($36$, $324$, $972$) come from dividing or multiplying by $3$ instead of $9$ — clear sign that the units must be tracked carefully.

Alternative: Tool #1 (Draw a Diagram) alone solves the problem: draw the $4 \text{ yd} \times 3 \text{ yd} = 12$ feet by $9$ feet rectangle, slice it into $3 \text{ ft} \times 3 \text{ ft}$ squares (each $= 1$ square yard). The grid has $4 \times 3 = 12$ such squares, so the answer is $12$ square yards. No formula needed beyond "count the tiles."

CCSS standards used (min grade 5)

3.MD.C.7 Relate area to the operations of multiplication and addition (Visualizing the floor as a grid of $1$-yard unit squares and counting $4 \times 3 = 12$ tiles to get the area.)
4.MD.A.3 Apply the area and perimeter formulas for rectangles in real-world and mathematical problems (Using $\text{Area} = \text{length} \times \text{width}$ to compute $4 \text{ yd} \times 3 \text{ yd} = 12$ square yards.)
5.MD.A.1 Convert among different-sized standard measurement units within a given system (Converting $12$ ft to $4$ yd and $9$ ft to $3$ yd using the given relationship $3$ ft $= 1$ yd before computing the area.)

⭐ This AMC 8 problem only needs Grade 5 unit conversion plus the Grade 4 rectangle area formula you already know!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 5)

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