AMC 8 · 2004 · #3

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition

ratio-proportionfraction-arithmeticrate identify-subproblems ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Short solution 💡 2 insights

Problem

Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for $18$ people. If they shared, how many meals should they have ordered to have just enough food for the $12$ of them?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Twelve friends each ordered one meal at a restaurant, and the $12$ meals turned out to be enough food for $18$ people. If they had decided to share from the start, how many meals would have been just right for the $12$ of them?

Givens: $12$ meals contain enough food for $18$ people; We want the number of meals that contain enough food for exactly $12$ people; Answer choices: (A) $8$, (B) $9$, (C) $10$, (D) $15$, (E) $18$

Unknowns: The number of meals needed so that the food is just enough for $12$ people

Understand

Plan

Primary tool: #3 Set Up an Equation

Secondary: #4 Introduce a Variable

The portion size is the same in every meal, so meals and people form a fixed ratio: $12$ meals match $18$ people. That is the standard setup for Tool #3 (Set Up an Equation) — a proportion that says "meals over people stays constant." Tool #4 (Introduce a Variable) names the unknown number of meals $x$ so we can solve. The cleanest path is to find the per-person rate (meals per person) first, then multiply by $12$.

Execute — Answer: A

#4 Introduce a Variable 6.EE.A.2 Step 1

Name the unknown.
Let $x$ be the number of meals that would feed exactly $12$ people.

$$x = \text{meals needed for } 12 \text{ people}$$

💡 Giving the unknown a letter is the Grade 6 "write expressions with variables" move.

#3 Set Up an Equation 6.RP.A.2 Step 2

Find the per-person rate.
Since $12$ meals feed $18$ people, divide to see how much of a meal goes to one person.

$$\text{rate} = \dfrac{12 \text{ meals}}{18 \text{ people}} = \dfrac{2}{3} \text{ meal per person}$$

💡 Dividing the two quantities turns a $12$-to-$18$ ratio into the per-person share that every diner gets.

#3 Set Up an Equation 6.RP.A.3 Step 3

Apply the rate to $12$ people.
Each person needs $\tfrac{2}{3}$ of a meal, so multiply by $12$.

$$x = 12 \times \dfrac{2}{3} = \dfrac{24}{3} = 8 \text{ meals}$$

💡 Once you know the per-person share, scaling to any number of people is one multiplication.

#3 Set Up an Equation 6.RP.A.3 Step 4

Match the value to a choice.
$8$ meals is answer (A).

$$x = 8 \;\Rightarrow\; \textbf{(A)}$$

💡 The per-person rate method lands exactly on a listed choice when the problem is well posed.

[1] #4 6.EE.A.2 Name the unknown. Let $x$ be the number of meals that would feed exactly $12$ pe

[2] #3 6.RP.A.2 Find the per-person rate. Since $12$ meals feed $18$ people, divide to see how m

[3] #3 6.RP.A.3 Apply the rate to $12$ people. Each person needs $\tfrac{2}{3}$ of a meal, so mu

[4] #3 6.RP.A.3 Match the value to a choice. $8$ meals is answer (A).

Review

Reasonableness: Sanity check: $12$ people is $\tfrac{12}{18} = \tfrac{2}{3}$ of $18$ people, so the meals needed should be $\tfrac{2}{3}$ of $12$, which is $8$. The answer is smaller than the original $12$ meals, which makes sense — fewer people need fewer meals. Choice (E) $18$ confuses meals with people; (D) $15$ is the midpoint of $12$ and $18$, a tempting but wrong shortcut; (B) $9$ and (C) $10$ are too close to $12$ to match the $\tfrac{2}{3}$ scaling.

Alternative: Tool #3 with a direct proportion: write $\dfrac{12 \text{ meals}}{18 \text{ people}} = \dfrac{x \text{ meals}}{12 \text{ people}}$ and cross-multiply to get $18x = 12 \times 12 = 144$, so $x = 144 \div 18 = 8$. Same answer (A), no unit rate computed separately.

CCSS standards used (min grade 6)

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Naming the unknown number of meals with the variable $x$ so the proportion can be written algebraically.)
6.RP.A.2 Understand the concept of a unit rate associated with a ratio (Turning the given $12$ meals to $18$ people ratio into the unit rate $\tfrac{2}{3}$ meal per person.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Applying the $\tfrac{2}{3}$ meal per person rate to $12$ people to find that $8$ meals are needed.)

⭐ Sharing problems become easy once you pin down the per-person share — find it once, then multiply by however many people you have.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: A

Review

CCSS standards used (min grade 6)

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