AMC 8 · 2009 · #15
Grade 6 rate-ratioProblem
A recipe that makes servings of hot chocolate requires squares of chocolate, cup sugar, cup water and cups milk. Jordan has squares of chocolate, cups of sugar, lots of water, and 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.
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.