AMC 8 · 2009 · #15
Grade 6 rate-ratioProblem
A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, 41 cup sugar, 1 cup water and 4 cups milk. Jordan has 5 squares of chocolate, 2 cups of sugar, lots of water, and 7 cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?
Pick an answer.
AMC 8 2009 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.
Toolkit + CCSS Solution
Understand
Restated: A hot chocolate recipe makes $5$ servings and uses $2$ squares of chocolate, $\tfrac{1}{4}$ cup sugar, $1$ cup water, and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, unlimited water, and $7$ cups of milk. Keeping the same ingredient ratio, what is the largest number of servings he can make?
Givens: Recipe for $5$ servings: $2$ chocolate, $\tfrac{1}{4}$ cup sugar, $1$ cup water, $4$ cups milk; Jordan has: $5$ chocolate, $2$ cups sugar, lots of water, $7$ cups milk; He must keep the same ratio of ingredients; Answer choices: (A) $5\tfrac{1}{8}$, (B) $6\tfrac{1}{4}$, (C) $7\tfrac{1}{2}$, (D) $8\tfrac{3}{4}$, (E) $9\tfrac{7}{8}$
Unknowns: The greatest number of servings Jordan can make
Understand
Restated: A hot chocolate recipe makes $5$ servings and uses $2$ squares of chocolate, $\tfrac{1}{4}$ cup sugar, $1$ cup water, and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, unlimited water, and $7$ cups of milk. Keeping the same ingredient ratio, what is the largest number of servings he can make?
Givens: Recipe for $5$ servings: $2$ chocolate, $\tfrac{1}{4}$ cup sugar, $1$ cup water, $4$ cups milk; Jordan has: $5$ chocolate, $2$ cups sugar, lots of water, $7$ cups milk; He must keep the same ratio of ingredients; Answer choices: (A) $5\tfrac{1}{8}$, (B) $6\tfrac{1}{4}$, (C) $7\tfrac{1}{2}$, (D) $8\tfrac{3}{4}$, (E) $9\tfrac{7}{8}$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #3 Eliminate Possibilities, #8 Analyze the Units
There are three ingredients to worry about (chocolate, sugar, milk — water is unlimited so tool #3 lets us cross it off immediately). Tool #7 turns the one big question into three clean subproblems: "how many servings does each ingredient by itself allow?" Each subproblem is a simple rate (tool #8 — servings per square, per cup) scaled up. The final answer is the smallest of the three: that ingredient runs out first and caps the batch.
Execute — Answer: D
6.RP.A.3 Step 1 - Cross off the ingredients that cannot limit the batch.
- Water is unlimited, so it never runs out.
- We only need to check chocolate, sugar, and milk.
💡 Eliminating the unlimited ingredient up front (tool #3) trims the problem from four cases to three.
5.NF.B.4 Step 2 - Chocolate subproblem.
- The recipe uses $2$ squares per $5$ servings, so $1$ square makes $\tfrac{5}{2}$ servings.
- Jordan's $5$ squares allow:
💡 Servings-per-square is a Grade 5 fraction-times-whole calculation.
5.NF.B.7 Step 3 - Sugar subproblem.
- The recipe uses $\tfrac{1}{4}$ cup per $5$ servings, so $1$ cup makes $5 \div \tfrac{1}{4} = 20$ servings.
- Jordan's $2$ cups allow:
💡 Dividing by the unit fraction $\tfrac{1}{4}$ multiplies by $4$ — a Grade 5 fraction-division move.
5.NF.B.4 Step 4 - Milk subproblem.
- The recipe uses $4$ cups per $5$ servings, so $1$ cup makes $\tfrac{5}{4}$ servings.
- Jordan's $7$ cups allow:
💡 Same servings-per-cup rate move as the chocolate step, just with milk's ratio.
6.RP.A.3 Step 5 - Pick the smallest of the three capacities.
- Chocolate allows $12.5$, sugar allows $40$, milk allows $8\tfrac{3}{4}$.
- Milk runs out first, so it caps the batch.
💡 Choosing the minimum across subproblems is the standard "limiting ingredient" finish — the bottleneck wins.
6.RP.A.3 Cross off the ingredients that cannot limit the batch. Water is unlimited, so it 5.NF.B.4 Chocolate subproblem. The recipe uses $2$ squares per $5$ servings, so $1$ squar 5.NF.B.7 Sugar subproblem. The recipe uses $\tfrac{1}{4}$ cup per $5$ servings, so $1$ cu 5.NF.B.4 Milk subproblem. The recipe uses $4$ cups per $5$ servings, so $1$ cup makes $\t 6.RP.A.3 Pick the smallest of the three capacities. Chocolate allows $12.5$, sugar allows Review
Reasonableness: Jordan has $2.5$ recipes' worth of chocolate ($\tfrac{5}{2}$), $8$ recipes' worth of sugar ($\tfrac{2}{1/4}$), and $1.75$ recipes' worth of milk ($\tfrac{7}{4}$). The smallest scale factor is $1.75$, and $1.75 \times 5 = 8.75 = 8\tfrac{3}{4}$ servings. This matches (D) and makes sense — milk is the only ingredient he is short on relative to the recipe.
Alternative: Tool #6 (Guess and Check) on the choices. The largest plausible answer is (E) $9\tfrac{7}{8}$; check if $9\tfrac{7}{8}$ servings is reachable. Milk needed $= \tfrac{4}{5} \times 9\tfrac{7}{8} = \tfrac{4 \times 79}{5 \times 8} = \tfrac{79}{10} = 7.9$ cups — more than $7$. Fails. Try (D) $8\tfrac{3}{4}$: milk needed $= \tfrac{4}{5} \times \tfrac{35}{4} = 7$ cups — exactly what he has. (D) works; (E) doesn't.
CCSS standards used (min grade 6)
5.NF.B.4Multiply a fraction or whole number by a fraction (Scaling the servings-per-square ($\tfrac{5}{2}$) and servings-per-cup-of-milk ($\tfrac{5}{4}$) rates by Jordan's amounts to get $12.5$ and $8\tfrac{3}{4}$.)5.NF.B.7Divide unit fractions by whole numbers and whole numbers by unit fractions (Computing $5 \div \tfrac{1}{4} = 20$ servings per cup of sugar.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Treating each ingredient's limit as a rate (servings per unit), and comparing across ingredients to pick the bottleneck.)
⭐ This AMC 8 problem just needs the Grade 6 rate idea you already know: figure out how far each ingredient stretches, then the smallest one decides the answer.
⭐ This AMC 8 problem just needs the Grade 6 rate idea you already know: figure out how far each ingredient stretches, then the smallest one decides the answer.
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