AMC 10 · 2017 · #3

Easy mode Grade 4

Problem

Tamara's garden has $6$ flower beds, set in $3$ rows of $2$ . Each bed is $6$ feet long and $2$ feet wide. Between the beds, and all around the outside, there are walkways exactly $1$ foot wide, as shown. What is the total area of the walkways, in square feet?

Pick an answer.

(A)

(B)

(C)

(D)

120

(E)

150

AMC 10 2017 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.

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Toolkit + CCSS Solution

Understand

Restated: A garden holds six identical $6$-foot by $2$-foot flower beds, set in $3$ rows of $2$. Every bed is separated from its neighbors and bordered along the outside by walkways exactly $1$ foot wide. Find the total area, in square feet, covered by the walkways.

Givens: Each flower bed measures $6$ feet by $2$ feet; The beds form a grid of $3$ rows and $2$ columns (six beds total); Walkways are $1$ foot wide and run between the beds and all around the outside; Answer choices: (A) $72$, (B) $78$, (C) $90$, (D) $120$, (E) $150$

Unknowns: The total area of the walkways, in square feet

Understand

Plan

Primary tool: #16 Change Focus / Count the Complement

Secondary: #1 Draw a Diagram, #7 Identify Subproblems

The walkways form an awkward plus-sign-and-border shape that is hard to measure directly. But the walkways are exactly whatever is left over once the beds are removed from the big outer rectangle. So Tool #16 (Count the Complement) turns a hard area into an easy subtraction: garden area minus bed area. Tool #1 (Draw a Diagram) lets us read off the outer rectangle's width and height by lining up beds and $1$-foot strips, and Tool #7 (Identify Subproblems) splits the work into three clean pieces: outer area, total bed area, then the difference.

Execute — Answer: B

#1 Draw a Diagram 2.MD.B.5 Step 1

Find the garden's width

Scan one row left to right.
It holds $2$ beds, each $6$ feet wide, plus the $1$-foot walkway strips: one on the far left, one between the two beds, and one on the far right.
That is $3$ strips of $1$ foot.
So the total width is $2$ beds plus $3$ strips.

$$2 \times 6 + 3 \times 1 = 12 + 3 = 15 \text{ ft}$$

💡 Lay the beds and the gaps end to end and just add up their widths.

#1 Draw a Diagram 2.MD.B.5 Step 2

Find the garden's height

Scan one column bottom to top.
It holds $3$ beds, each $2$ feet tall, plus $1$-foot walkway strips above, below, and between the rows: that is $4$ strips of $1$ foot.
So the total height is $3$ beds plus $4$ strips.

$$3 \times 2 + 4 \times 1 = 6 + 4 = 10 \text{ ft}$$

💡 Three beds stacked with a strip above, below, and in each gap means four strips, not three.

#16 Change Focus / Count the Complement 4.MD.A.3 Step 3

Area of the whole garden

Now treat the garden as one big rectangle.
Its area is width times height.

$$15 \times 10 = 150 \text{ ft}^2$$

💡 The outer boundary is a plain rectangle, so its area is just length times width.

#7 Identify Subproblems 4.MD.A.3 Step 4

Area of all six beds

Each bed is a $6$ by $2$ rectangle, so one bed covers $12$ square feet.
There are six identical beds.

$$6 \times (6 \times 2) = 6 \times 12 = 72 \text{ ft}^2$$

💡 Identical pieces let you find one area and multiply by how many there are.

#16 Change Focus / Count the Complement 3.NBT.A.2 Step 5

Subtract to get the walkways

The garden is made of exactly two things: beds and walkways.
So the walkway area is the whole garden minus all the beds.
Subtracting gives $150 - 72 = 78$ square feet, which is choice (B).

$$150 - 72 = 78 \;\Rightarrow\; \textbf{(B)}$$

💡 Whatever the beds do not cover must be walkway, so one subtraction finishes it.

[1] #1 2.MD.B.5 Scan one row left to right. It holds $2$ beds, each $6$ feet wide, plus the $1$-

[2] #1 2.MD.B.5 Scan one column bottom to top. It holds $3$ beds, each $2$ feet tall, plus $1$-f

[3] #16 4.MD.A.3 Now treat the garden as one big rectangle. Its area is width times height.

[4] #7 4.MD.A.3 Each bed is a $6$ by $2$ rectangle, so one bed covers $12$ square feet. There ar

[5] #16 3.NBT.A.2 The garden is made of exactly two things: beds and walkways. So the walkway area

Review

Reasonableness: The walkway area $78$ is a bit more than the bed area $72$, and together $78 + 72 = 150$ rebuilds the full garden, so nothing is lost or double-counted. The answer sits comfortably between $72$ (all-bed) and $150$ (all-garden), exactly where a walkway-only total should land.

Alternative: Add up the walkways directly. The three vertical $1$-foot-wide lanes run the full $10$-foot height: $3 \times (1 \times 10) = 30$. The four horizontal $1$-foot-tall lanes, where not already counted, span the two $6$-foot beds: $4 \times (1 \times 12) = 48$. Total $30 + 48 = 78$, matching (B).

CCSS standards used (min grade 4)

2.MD.B.5 Solve word problems involving lengths using same units (Adding bed widths and $1$-foot walkway strips to find the garden's total width and height.)
4.MD.A.3 Apply area and perimeter formulas for rectangles in real-world problems (Computing the area of the outer garden rectangle and the area of each $6$-by-$2$ flower bed.)
3.NBT.A.2 Fluently add and subtract within 1000 (Subtracting the total bed area from the garden area to isolate the walkway area.)

⭐ When a leftover shape is awkward to measure, find the whole and subtract the easy parts you can see.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Find the garden's width

Find the garden's height

Area of the whole garden

Area of all six beds

Subtract to get the walkways

Review

CCSS standards used (min grade 4)

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