AMC 8 · 1999 · #19

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

rateunit-conversionmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: ratemulti-digit-arithmeticunit-conversion

📏 Medium solution 💡 3 insights

Problem

There are 108 students who each eat about 2 cookies, so Hansel and Gretel must supply 216 cookies. One full recipe makes a pan of 15 cookies and uses 3 tablespoons of butter. Only full recipes are allowed. With 8 tablespoons per stick of butter, how many whole sticks of butter must be bought?

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 Break into Subproblems

One sentence asks for sticks of butter, but four short jobs are hiding inside it — the classic signal for Tool #7 (Break into Subproblems). The chain is: (i) total cookies needed, (ii) number of pans (round up to a whole pan), (iii) total tablespoons of butter, (iv) number of sticks (round up to a whole stick). Each subproblem is a single arithmetic move, and the two rounding-up steps come from the real-world constraints "full recipes only" and "whole sticks only." No algebra is required; tracking the units carries the whole solution.

1STEP 1

Subproblem 1: multiply the 108 students by 2 cookies each to get 216 cookies needed.

108 × 2 = 216 cookies

2STEP 2

Subproblem 2: 216 ÷ 15 = 14.4, but full recipes only — round up to 15 pans (14 pans give just 210).

216 ÷ 15 = 14.4 ⟶ 15 pans

3STEP 3

Subproblem 3: each of the 15 pans uses 3 tablespoons, so 15 × 3 = 45 tablespoons of butter.

15 × 3 = 45 tablespoons

4STEP 4

Subproblem 4: 45 ÷ 8 = 5.625, and partial sticks can't be bought — round up to 6 sticks (5 sticks give only 40).

45 ÷ 8 = 5.625 ⟶ 6 sticks → (B)

Answer

Cross-check both round-up steps. Pans: 15 pans give 15 × 15 = 225 cookies, which is ≥ 216, and 14 pans give only 210 ＜ 216 — so 15 is the smallest legal pan count. Sticks: 6 sticks give 6 × 8 = 48 tablespoons, which is ≥ 45, and 5 sticks give only 40 ＜ 45 — so 6 is the smallest legal stick count. Choices (A) 5 (not enough butter), (C) 7, (D) 8, (E) 9 (more than needed) are all ruled out. Final units check: students → cookies → pans → tablespoons → sticks — each conversion uses the right rate, so the answer (B) 6 is consistent.

💡Key takeaway

Word problems with real-life "whole units only" constraints become a chain of small jobs. Multiply, divide, round UP whenever a partial unit is not allowed — that turns a wordy AMC 8 problem into four short Grade 4-6 arithmetic steps.

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