AMC 8 · 2012 · #1

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

ratio-proportionratemulti-digit-arithmetic easier-related-problemdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Short solution 💡 2 insights

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Problem

Rachelle uses 3 pounds of meat to make 8 hamburgers for her family. How many pounds of meat does she need to make 24 hamburgers for a neighbourhood picnic?

Pick an answer.

(A)

(B)

$6\dfrac{2}{3}$

(C)

$7\dfrac{1}{2}$

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: Rachelle uses $3$ pounds of meat for $8$ hamburgers. At the same rate, how many pounds of meat does she need for $24$ hamburgers?

Givens: $3$ pounds of meat $\to 8$ hamburgers; The recipe (pounds per hamburger) stays the same; Target output: $24$ hamburgers; Answer choices: (A) $6$, (B) $6\tfrac{2}{3}$, (C) $7\tfrac{1}{2}$, (D) $8$, (E) $9$ (pounds)

Unknowns: The number of pounds of meat needed for $24$ hamburgers

Understand

Restated: Rachelle uses $3$ pounds of meat for $8$ hamburgers. At the same rate, how many pounds of meat does she need for $24$ hamburgers?

Plan

Primary tool: #9 Solve an Easier Related Problem

Secondary: #8 Analyze the Units

The numbers $8$ and $24$ are friendly: $24$ is exactly $3$ batches of $8$. Tool #9 (Easier Related Problem) says — instead of fighting a proportion, recognize the smaller version we already solved ($3$ lb feeds $8$ burgers) and copy it three times. Tool #8 (Analyze the Units) keeps the bookkeeping honest: the rate is "pounds per hamburger", so multiplying that rate by a number of hamburgers gives pounds — exactly what the question asks for.

Execute — Answer: E

#9 Solve an Easier Related Problem 3.OA.A.3 Step 1

Compare the two batch sizes.
The big batch has $24$ hamburgers, the small batch has $8$.
Divide to see how many small batches fit inside the big one.

$$24 \div 8 = 3$$

💡 Asking "how many groups of $8$ make $24$?" is the Grade 3 "equal groups" idea — exactly the Easier Related Problem move.

#9 Solve an Easier Related Problem 4.OA.A.2 Step 2

The big batch is $3$ times the small batch.
Because the recipe stays the same, the meat must also be $3$ times as much.
This is multiplicative comparison: $3$ times as many burgers needs $3$ times as much meat.

$$\text{meat} = 3 \times 3 \text{ lb} = 9 \text{ lb}$$

💡 "$3$ times as much" is the Grade 4 multiplicative-comparison sentence — the same scaling factor on both sides.

#8 Analyze the Units 6.RP.A.3 Step 3

Check by units.
The rate is $\tfrac{3 \text{ lb}}{8 \text{ burgers}} = \tfrac{3}{8}$ lb per burger.
Multiplying that rate by $24$ burgers cancels "burgers" and leaves "pounds" — the unit the question asks for.

$$\tfrac{3 \text{ lb}}{8 \text{ burgers}} \times 24 \text{ burgers} = \tfrac{3 \times 24}{8} \text{ lb} = \tfrac{72}{8} \text{ lb} = 9 \text{ lb} \;\Rightarrow\; \textbf{(E)}$$

💡 Multiplying a unit rate (lb per burger) by a count of burgers to get pounds is the Grade 6 rate-reasoning idea.

[1] #9 3.OA.A.3 Compare the two batch sizes. The big batch has $24$ hamburgers, the small batch

[2] #9 4.OA.A.2 The big batch is $3$ times the small batch. Because the recipe stays the same, t

[3] #8 6.RP.A.3 Check by units. The rate is $\tfrac{3 \text{ lb}}{8 \text{ burgers}} = \tfrac{3}

Review

Reasonableness: Common sense: $3$ family-sized batches at $3$ pounds each is $9$ pounds total — that matches. The answer is bigger than $3$ lb (we need more burgers, so we need more meat) and a whole number (because $24$ is a whole multiple of $8$). Answer choices like $6\tfrac{2}{3}$ or $7\tfrac{1}{2}$ would only appear if $24$ weren't divisible by $8$, so they can be ruled out by structure alone.

Alternative: Tool #3 (Eliminate Possibilities): for each choice, divide pounds by burgers and see if the rate matches $\tfrac{3}{8}$ lb per burger. (E) gives $9/24 = 3/8$ ✓. (A) gives $6/24 = 1/4 \ne 3/8$. (B) $6\tfrac{2}{3}/24 = 20/72 = 5/18$. (C) $7.5/24 = 5/16$. (D) $8/24 = 1/3$. Only (E) preserves the recipe.

CCSS standards used (min grade 6)

3.OA.A.3 Use multiplication and division within $100$ to solve word problems in equal-groups situations (Recognizing that $24$ hamburgers is $3$ equal groups of $8$ hamburgers, i.e. $24 \div 8 = 3$.)
4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison (Concluding that "$3$ times as many burgers" needs "$3$ times as much meat", giving $3 \times 3 = 9$ pounds.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Confirming the answer with the unit rate $\tfrac{3}{8}$ lb per burger times $24$ burgers $= 9$ lb.)

⭐ When the new batch is just a whole-number multiple of the old batch, you don't need fancy proportions — multiply both sides by the same factor.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

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