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

At Central Middle School, the 108 students who take the AMC 8 meet in the evening to talk about food and eat an average of two cookies apiece. Hansel and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists these items: $1\frac{1}{2}$ cups flour, $2$ eggs, $3$ tablespoons butter, $\frac{3}{4}$ cups sugar, and $1$ package of chocolate drops. They will make full recipes, not partial recipes.

Hansel and Gretel must make enough pans of cookies to supply 216 cookies. There are 8 tablespoons in a stick of butter. How many sticks of butter will be needed? (Some butter may be leftover, of course.)

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: $108$ students eat $2$ cookies each, so $216$ cookies are needed; One full recipe (one pan) makes $15$ cookies and uses $3$ tablespoons of butter; Only full recipes can be made (no partial pans); $1$ stick of butter $= 8$ tablespoons; Answer choices: (A) $5$, (B) $6$, (C) $7$, (D) $8$, (E) $9$

Unknowns: The number of whole sticks of butter Hansel and Gretel must buy

Understand

Plan

Primary tool: #7 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.

Execute — Answer: B

#7 Break into Subproblems 4.OA.A.3 Step 1

Subproblem 1: find the total number of cookies needed.
Each of the $108$ students eats $2$ cookies on average.

$$108 \times 2 = 216 \text{ cookies}$$

💡 "Average of $2$ apiece" times the number of students gives the total — a Grade 4 multistep word-problem setup.

#7 Break into Subproblems 6.NS.B.2 Step 2

Subproblem 2: find the number of pans needed.
One pan makes $15$ cookies, so divide the total cookies by $15$.
The exact quotient is $14.4$, but only full recipes are allowed, so round UP to the next whole pan; $14$ pans would give only $14 \times 15 = 210 < 216$ cookies.

$$216 \div 15 = 14.4 \;\longrightarrow\; 15 \text{ pans}$$

💡 Real-world "must be at least this many, no halves allowed" forces a round-up, not the usual nearest-whole-number rounding.

#7 Break into Subproblems 5.NBT.B.5 Step 3

Subproblem 3: find the total tablespoons of butter.
Each of the $15$ pans uses $3$ tablespoons.

$$15 \times 3 = 45 \text{ tablespoons}$$

💡 Same multiplicative step as subproblem 1: amount per unit times number of units.

#7 Break into Subproblems 6.NS.B.2 Step 4

Subproblem 4: convert tablespoons to sticks.
One stick is $8$ tablespoons, so divide $45$ by $8$.
The exact quotient is $5.625$, and you cannot buy $0.625$ of a stick, so round UP again; $5$ sticks would give only $5 \times 8 = 40 < 45$ tablespoons.

$$45 \div 8 = 5.625 \;\longrightarrow\; 6 \text{ sticks} \;\Rightarrow\; \textbf{(B)}$$

💡 Same "at least this many, whole units only" constraint as the pans step — round up, even when the fractional part is small.

[1] #7 4.OA.A.3 Subproblem 1: find the total number of cookies needed. Each of the $108$ student

[2] #7 6.NS.B.2 Subproblem 2: find the number of pans needed. One pan makes $15$ cookies, so div

[3] #7 5.NBT.B.5 Subproblem 3: find the total tablespoons of butter. Each of the $15$ pans uses $

[4] #7 6.NS.B.2 Subproblem 4: convert tablespoons to sticks. One stick is $8$ tablespoons, so di

Review

Reasonableness: Cross-check both round-up steps. Pans: $15$ pans give $15 \times 15 = 225$ cookies, which is $\ge 216$, and $14$ pans give only $210 < 216$ — so $15$ is the smallest legal pan count. Sticks: $6$ sticks give $6 \times 8 = 48$ tablespoons, which is $\ge 45$, 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 $\to$ cookies $\to$ pans $\to$ tablespoons $\to$ sticks — each conversion uses the right rate, so the answer (B) $6$ is consistent.

Alternative: Tool #8 (Analyze Units): chain the conversions in a single line, $216 \text{ cookies} \times \dfrac{1 \text{ pan}}{15 \text{ cookies}} \times \dfrac{3 \text{ tbsp}}{1 \text{ pan}} \times \dfrac{1 \text{ stick}}{8 \text{ tbsp}} = \dfrac{216 \times 3}{15 \times 8} = \dfrac{648}{120} = 5.4 \text{ sticks}$. This is the "continuous" answer that ignores the full-recipe rule. Add it back: pans must round up from $14.4$ to $15$, which bumps butter from $5.4$ to $\tfrac{15 \times 3}{8} = 5.625$ sticks, then sticks round up to $6$ — answer (B).

CCSS standards used (min grade 6)

4.OA.A.3 Solve multistep word problems with whole numbers using the four operations (Chaining four arithmetic subproblems (cookies $\to$ pans $\to$ tablespoons $\to$ sticks) to answer one real-world question.)
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm (Computing $108 \times 2 = 216$ for total cookies and $15 \times 3 = 45$ for total tablespoons of butter.)
6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm (Dividing $216 \div 15 = 14.4$ to find pans and $45 \div 8 = 5.625$ to find sticks, then rounding each up to the next whole number to satisfy the real-world constraints.)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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