AMC 8 · 1999 · #17
Grade 5 rate-ratioProblem
At Central Middle School the 108 students who take the AMC 8 meet in the evening to talk about problems and eat an average of two cookies apiece. Walter and Gretel are baking Bonnie's Best Bar Cookies this year. Their recipe, which makes a pan of 15 cookies, lists this items: cups of flour, eggs, tablespoons butter, cups sugar, and package of chocolate drops. They will make only full recipes, not partial recipes.
Walter can buy eggs by the half-dozen. How many half-dozens should he buy to make enough cookies? (Some eggs and some cookies may be left over.)
Pick an answer.
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
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) $15$
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
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.
💡 Average cookies per student times number of students gives total cookies — the unit "students" cancels, leaving "cookies".
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.
💡 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.
4.OA.A.2 Step 3 - Convert pans into eggs.
- Each pan needs $2$ eggs.
💡 Pans times eggs-per-pan gives total eggs — the unit "pans" cancels.
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.
💡 Dividing eggs by eggs-per-half-dozen leaves "half-dozens" — and $30$ splits evenly into $6$s.
5.NBT.B.5 Find the total number of cookies needed. With $108$ students eating an average o 5.NF.B.3 Convert cookies into pans. Each pan produces $15$ cookies, so divide. Because on 4.OA.A.2 Convert pans into eggs. Each pan needs $2$ eggs. 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.2Multiply or divide to solve word problems involving multiplicative comparison (Multiplying $15$ pans by $2$ eggs/pan to get $30$ eggs.)5.NBT.B.5Fluently multiply multi-digit whole numbers using the standard algorithm (Computing $108 \times 2 = 216$ total cookies.)5.NBT.B.6Find 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.3Interpret 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).
⭐ 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).
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