Competition · AMC preparation · step 4 of 4
AMC 8 · 2009 · #15
Grade 6 rate-ratioPick an answer.
AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Pick which ingredients matter
Water is unlimited, so it can never be the limit — only chocolate, sugar, and milk can run out.
Eliminating the unlimited ingredient up front (tool #3) trims the problem from four cases to three.
6.RP.A.3Eliminate PossibilitiesCheck the chocolate
Chocolate: the recipe is 2 squares per 5 servings, so Jordan's 5 squares stretch to 12.5 servings.
Servings-per-square is a Grade 5 fraction-times-whole calculation.
5.NF.B.4Identify SubproblemsCheck the sugar
Sugar: the recipe uses cup per 5 servings, so Jordan's 2 cups cover a huge 40 servings.
Dividing by the unit fraction 1/4 multiplies by 4 — a Grade 5 fraction-division move.
5.NF.B.7Identify SubproblemsCheck the milk
Milk: the recipe uses 4 cups per 5 servings, so Jordan's 7 cups reach only 8 servings.
Same servings-per-cup rate move as the chocolate step, just with milk's ratio.
5.NF.B.4Identify SubproblemsTake the smallest amount
Milk gives the fewest servings, so it runs out first and caps the batch at 8 — answer (D).
Choosing the minimum across subproblems is the standard "limiting ingredient" finish — the bottleneck wins.
The most servings Jordan can make is the smallest of the three serving-counts that his chocolate, his sugar, and his milk each allow on their own.
▸ Why?
Each ingredient on its own allows only a fixed number of servings, set by how much of that ingredient one serving takes.
▸ Why?
Making n servings uses the same per-serving portion once for each serving, so the amount an ingredient uses is its per-serving portion counted n times.
▸ Why?
So the servings an ingredient allows is how many whole per-serving portions fit inside what Jordan has, found by dividing his amount by that portion, which undoes the repeated multiplication.
▸ Why?
A real batch uses chocolate, sugar, and milk at the same time, so it can be no larger than the ingredient that allows the fewest servings, because that ingredient runs out first.
▸ Why?
Every added serving puts one more per-serving portion onto each ingredient's running total, so the ingredient holding the fewest portions empties before the others and cannot supply the next serving.
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.
- Pick which ingredients matter
- Check the chocolate
- Check the sugar
- Check the milk
- Take the smallest amount
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