AMC 8 · 1999 · #17

Grade 5 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

ratemulti-digit-arithmeticunit-conversion identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticrate

📏 Medium solution 💡 3 insights

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Problem

108 students each eat on average 2 cookies. The cookie recipe makes 15 cookies per pan and uses 2 eggs per pan. Only full recipes are allowed. Eggs are sold by the half-dozen (a half-dozen = 6 eggs). How many half-dozens of eggs must Walter buy so that there are enough cookies for everyone?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 1999 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The question hides four small steps behind one sentence — "how many half-dozens of eggs?" — so Tool #7 (Identify Subproblems) lets us walk the chain one link at a time: cookies needed → pans needed → eggs needed → half-dozens needed. Tool #8 (Analyze the Units) is the natural partner because the recipe gives us conversion rates (15 cookies/pan, 2 eggs/pan, 6 eggs/half-dozen); tracking units tells us exactly which way to multiply or divide, and warns us at the two stages where "only whole units allowed" forces a round-up.

1STEP 1

108 students each eating 2 cookies means the bakers need 216 cookies in all.

108 students × 2 cookies/student = 216 cookies

2STEP 2

216 cookies at 15 per pan is 14.4 pans; partial recipes are banned, so round up to 15 pans.

\frac{216 cookies}{15 cookies/pan}

= 14.4 pans → ⌈ 14.4 ⌉ = 15 pans

3STEP 3

Each pan needs 2 eggs, so 15 pans use 30 eggs.

15 pans × 2 eggs/pan = 30 eggs

4STEP 4

A half-dozen is 6 eggs, and 30 splits evenly into 5 half-dozens — answer (C).

\frac{30 eggs}{6 eggs/half-dozen}

= 5 half-dozens → (C)

Answer

Spot-check the round-up. 14 pans would yield 14 × 15 = 210 cookies — short of the 216 needed — so 15 pans is indeed the smallest legal pan count. Then 15 pans need 30 eggs, and 30 = 5 × 6 so 5 half-dozens hits the count exactly with none left over. Trying nearby answer choices: (A) 1 half-dozen = 6 eggs only 3 pans only 45 cookies — nowhere near 216; (B) 2 half-dozens = 12 eggs only 6 pans only 90 cookies — still short; (D) 7 half-dozens = 42 eggs makes up to 21 pans, more than enough but wasteful; (E) 15 is wildly excessive. Only (C) 5 is both sufficient and minimal.

💡Key takeaway

Chain the units: 216 cookies need ⌈ $\frac{216}{15}$ ⌉ = 15 pans, 15 pans need 30 eggs, and 30 eggs are exactly 5 half-dozens — answer (C).

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