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AMC 8 · 2020 · #3

Grade 4 arithmeticgeometry-2d
Archetype Arithmetic Expression Decomposition
area-rectanglesratemulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: area-rectanglesmulti-digit-arithmetic
📏 Short solution 💡 2 insights
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Problem

Carrie has a rectangular garden that measures 66 feet by 88 feet. She plants the entire garden with strawberry plants. Carrie is able to plant 44 strawberry plants per square foot, and she harvests an average of 1010 strawberries per plant. How many strawberries can she expect to harvest?

(A) 560(B) 960(C) 1120(D) 1920(E) 3840\textbf{(A) }560 \qquad \textbf{(B) }960 \qquad \textbf{(C) }1120 \qquad \textbf{(D) }1920 \qquad \textbf{(E) }3840

Pick an answer.

(A)
560
(B)
960
(C)
1120
(D)
1920
(E)
3840
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Toolkit + CCSS Solution

Understand

Restated: Carrie has a rectangular strawberry garden measuring $6$ feet by $8$ feet. She fits $4$ plants in every square foot, and each plant gives an average of $10$ strawberries. How many strawberries total can she expect to harvest from the whole garden?

Givens: Garden shape: rectangle, $6$ ft $\times$ $8$ ft; Planting density: $4$ plants per square foot; Average yield: $10$ strawberries per plant; Answer choices: (A) $560$, (B) $960$, (C) $1120$, (D) $1920$, (E) $3840$

Unknowns: The total expected number of strawberries harvested from the whole garden

Understand

Restated: Carrie has a rectangular strawberry garden measuring $6$ feet by $8$ feet. She fits $4$ plants in every square foot, and each plant gives an average of $10$ strawberries. How many strawberries total can she expect to harvest from the whole garden?

Givens: Garden shape: rectangle, $6$ ft $\times$ $8$ ft; Planting density: $4$ plants per square foot; Average yield: $10$ strawberries per plant; Answer choices: (A) $560$, (B) $960$, (C) $1120$, (D) $1920$, (E) $3840$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units

The question "how many strawberries" hides three smaller questions stacked on top of each other: (1) how much area is the garden? (2) how many plants fit in that area? (3) how many strawberries do those plants give? Tool #7 (Identify Subproblems) makes that ladder explicit so we never multiply the wrong pair of numbers. Tool #8 (Analyze the Units) is the safety check at each rung — square feet $\times$ plants/sq ft cancels to plants, and plants $\times$ strawberries/plant cancels to strawberries — so the units guarantee we have the right quantity at every step.

Execute — Answer: D

#7 Identify Subproblems 3.MD.C.7 Step 1
  • Subproblem 1: find the garden's area.
  • The garden is a rectangle, so area $=$ length $\times$ width.
$$6 \text{ ft} \times 8 \text{ ft} = 48 \text{ sq ft}$$

💡 Grade 3 students already know that a rectangle's area is just length times width, with units of square feet.

#8 Analyze the Units 4.NBT.B.5 Step 2
  • Subproblem 2: turn area into total plants.
  • Each square foot holds $4$ plants, so multiply the area by the planting density.
$$48 \text{ sq ft} \times 4 \tfrac{\text{plants}}{\text{sq ft}} = 192 \text{ plants}$$

💡 Multiplying $48 \times 4$ is a Grade 4 multi-digit $\times$ one-digit calculation, and the units "sq ft" cancel to leave "plants".

#8 Analyze the Units 4.OA.A.3 Step 3
  • Subproblem 3: turn plants into total strawberries.
  • Each plant yields $10$ strawberries on average, so multiply the plant count by $10$.
$$192 \text{ plants} \times 10 \tfrac{\text{strawberries}}{\text{plant}} = 1920 \text{ strawberries} \;\Rightarrow\; \textbf{(D)}$$

💡 Chaining three multiplications to answer one real-world question is exactly the Grade 4 multi-step word-problem standard.

[1] #7 3.MD.C.7 Subproblem 1: find the garden's area. The garden is a rectangle, so area $=$ len
[2] #8 4.NBT.B.5 Subproblem 2: turn area into total plants. Each square foot holds $4$ plants, so
[3] #8 4.OA.A.3 Subproblem 3: turn plants into total strawberries. Each plant yields $10$ strawb

Review

Reasonableness: The whole calculation is one long multiplication, $6 \times 8 \times 4 \times 10 = 1920$. We can re-bracket it as $(6 \times 8) \times (4 \times 10) = 48 \times 40 = 1920$ to double-check — same answer. The magnitude also makes sense: a small backyard plot ($48$ sq ft) packed with $192$ plants and a generous $10$ berries each should land in the low thousands of berries, exactly where $1920$ sits. Choices like $560$ or $3840$ would mean we forgot a factor or doubled one.

Alternative: Tool #8 (Analyze the Units) on its own: write the whole chain as one big fraction, $6 \text{ ft} \times 8 \text{ ft} \times 4 \tfrac{\text{plants}}{\text{ft}^2} \times 10 \tfrac{\text{berries}}{\text{plant}}$. The ft$^2$ cancels with sq ft and "plants" cancels with "plants", leaving just "berries" — and the numbers multiply to $1920$. The unit-cancellation alone tells us we set up the right product.

CCSS standards used (min grade 4)

  • 3.MD.C.7 Relate area to multiplication and addition operations (Finding the area of the rectangular garden as $6 \times 8 = 48$ square feet.)
  • 4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number (Computing $48 \times 4 = 192$ to turn the garden's area into a total plant count.)
  • 4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Chaining area, plants-per-area, and berries-per-plant into a single multi-step word problem and finishing with $192 \times 10 = 1920$.)

⭐ This AMC 8 problem only needs Grade 4 multi-step multiplication word problems you already know!

⭐ This AMC 8 problem only needs Grade 4 multi-step multiplication word problems you already know!

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