AMC 8 · 2004 · #3

Grade 6 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

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

📏 Short solution 💡 2 insights

Problem

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?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Set Up an Equation

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.

1STEP 1

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

x = meals needed for 12 people

2STEP 2

Since 12 meals feed 18 people, each person eats 12 ÷ 18 = $\frac{2}{3}$ of a meal.

rate = (12 meals)/(18 people) =

\frac{2}{3}

meal per person

3STEP 3

Each person needs $\frac{2}{3}$ of a meal, so x = 12 × $\frac{2}{3}$ = 8 meals.

x = 12 ×

\frac{2}{3}

\frac{24}{3}

= 8 meals

4STEP 4

So 8 meals matches choice (A).

x = 8 → (A)

Answer

Sanity check: 12 people is

\frac{12}{18}

\frac{2}{3}

of 18 people, so the meals needed should be

\frac{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

\frac{2}{3}

scaling.

💡Key takeaway

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

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