AMC 8 · 1999 · #17

Easy mode Grade 5

Problem

At Central Middle School, $108$ students come to an evening meeting about the AMC 8. Each student eats $2$ cookies on average.

Walter and Gretel are baking the cookies. Their recipe makes one pan of $15$ cookies, and each pan uses $2$ eggs.

They are only allowed to bake whole recipes — no half pans. So they need to bake enough whole pans to give every student $2$ cookies. (It is okay if some cookies are left over.)

Walter buys eggs in half-dozens. One half-dozen is $6$ eggs.

How many half-dozens of eggs does Walter need to buy?

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: $108$ 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?

Givens: Students $= 108$, cookies per student (average) $= 2$; One pan yields $15$ cookies and uses $2$ eggs; Only full recipes — pans must be a whole number, round up; Eggs sold by the half-dozen ($1$ half-dozen $= 6$ eggs); Answer choices: (A) $1$, (B) $2$, (C) $5$, (D) $7$, (E) $15$

Unknowns: The smallest number of half-dozens of eggs Walter must buy

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units

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 $\to$ pans needed $\to$ eggs needed $\to$ half-dozens needed. Tool #8 (Analyze the Units) is the natural partner because the recipe gives us conversion rates ($15 \text{ cookies/pan}$, $2 \text{ eggs/pan}$, $6 \text{ eggs/half-dozen}$); 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.

Execute — Answer: C

#8 Analyze the Units 5.NBT.B.5 Step 1

Find the total number of cookies needed.
With $108$ students eating an average of $2$ cookies each, the bakers must produce at least the product.

$$108 \text{ students} \times 2 \;\dfrac{\text{cookies}}{\text{student}} = 216 \text{ cookies}$$

💡 Average cookies per student times number of students gives total cookies — the unit "students" cancels, leaving "cookies".

#7 Identify Subproblems 5.NF.B.3 Step 2

Convert cookies into pans.
Each pan produces $15$ cookies, so divide.
Because only full recipes are allowed, round any fraction up — $14$ pans give only $210$ cookies, which is not enough.

$$\dfrac{216 \text{ cookies}}{15 \;\text{cookies/pan}} = 14.4 \text{ pans} \;\Rightarrow\; \lceil 14.4 \rceil = 15 \text{ pans}$$

💡 Dividing cookies by cookies-per-pan leaves "pans" as the unit. When the result isn't a whole number, round up so the count is enough.

#7 Identify Subproblems 4.OA.A.2 Step 3

Convert pans into eggs.
Each pan needs $2$ eggs.

$$15 \text{ pans} \times 2 \;\dfrac{\text{eggs}}{\text{pan}} = 30 \text{ eggs}$$

💡 Pans times eggs-per-pan gives total eggs — the unit "pans" cancels.

#8 Analyze the Units 5.NBT.B.6 Step 4

Convert eggs into half-dozens.
A half-dozen contains $6$ eggs, and $30$ is a clean multiple of $6$, so no rounding is needed.

$$\dfrac{30 \text{ eggs}}{6 \;\text{eggs/half-dozen}} = 5 \text{ half-dozens} \;\Rightarrow\; \textbf{(C)}$$

💡 Dividing eggs by eggs-per-half-dozen leaves "half-dozens" — and $30$ splits evenly into $6$s.

[1] #8 5.NBT.B.5 Find the total number of cookies needed. With $108$ students eating an average o

[2] #7 5.NF.B.3 Convert cookies into pans. Each pan produces $15$ cookies, so divide. Because on

[3] #7 4.OA.A.2 Convert pans into eggs. Each pan needs $2$ eggs.

[4] #8 5.NBT.B.6 Convert eggs into half-dozens. A half-dozen contains $6$ eggs, and $30$ is a cle

Review

Reasonableness: Spot-check the round-up. $14$ pans would yield $14 \times 15 = 210$ cookies — short of the $216$ needed — so $15$ pans is indeed the smallest legal pan count. Then $15$ pans need $30$ eggs, and $30 = 5 \times 6$ 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.

Alternative: Tool #3 (Eliminate Possibilities). Each half-dozen $= 6$ eggs $= 3$ pans (since $2$ eggs per pan) $= 45$ cookies. So $n$ half-dozens make $45n$ cookies; we need $45n \geq 216$, i.e. $n \geq 4.8$, so $n = 5$. This collapses the chain into one rate ($45$ cookies per half-dozen) and a single ceiling step, landing on (C).

CCSS standards used (min grade 5)

4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison (Multiplying $15$ pans by $2$ eggs/pan to get $30$ eggs.)
5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm (Computing $108 \times 2 = 216$ total cookies.)
5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors (Dividing $30 \div 6 = 5$ to convert eggs into half-dozens.)
5.NF.B.3 Interpret a fraction as division of the numerator by the denominator and solve word problems involving division of whole numbers leading to answers in the form of fractions (Computing $216 \div 15 = 14.4$ pans and rounding up to $15$ because partial pans are not allowed.)

⭐ Chain the units: $216$ cookies need $\lceil 216/15 \rceil = 15$ pans, $15$ pans need $30$ eggs, and $30$ eggs are exactly $5$ half-dozens — answer (C).

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 5)

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