AMC 8 · 2009 · #15

Grade 6 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up

ratio-proportionfraction-multiplicationrate identify-subproblemsratio-proportion ↑ Prerequisites: fraction-arithmeticratio-proportion

📏 Medium solution 💡 3 insights

Problem

A hot chocolate recipe makes 5 servings and uses 2 squares of chocolate,

\frac{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?

Pick an answer.

(A)

$5 \frac18$

(B)

$6\frac14$

(C)

$7\frac12$

(D)

$8 \frac34$

(E)

$9\frac78$

AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

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.

1STEP 1

Water is unlimited, so it can never be the limit — only chocolate, sugar, and milk can run out.

ingredients to check = {chocolate, sugar, milk}

2STEP 2

Chocolate: the recipe is 2 squares per 5 servings, so Jordan's 5 squares stretch to 12.5 servings.

5 ×

\frac{5}{2}

\frac{25}{2}

= 12.5 servings

3STEP 3

Sugar: the recipe uses $\frac{1}{4}$ cup per 5 servings, so Jordan's 2 cups cover a huge 40 servings.

2 × 20 = 40 servings

4STEP 4

Milk: the recipe uses 4 cups per 5 servings, so Jordan's 7 cups reach only 8 $\frac{3}{4}$ servings.

7 ×

\frac{5}{4}

\frac{35}{4}

= 8

\frac{3}{4}

servings

5STEP 5

Milk gives the fewest servings, so it runs out first and caps the batch at 8 $\frac{3}{4}$ — answer (D).

min (12.5, 40, 8

\frac{3}{4}

) = 8

\frac{3}{4}

→ (D)

Answer

\frac{3}{4}

Jordan has 2.5 recipes' worth of chocolate (

\frac{5}{2}

), 8 recipes' worth of sugar (

\frac{2}{\frac{1}{4}}

), and 1.75 recipes' worth of milk (

\frac{7}{4}

). The smallest scale factor is 1.75, and 1.75 × 5 = 8.75 = 8

\frac{3}{4}

servings. This matches (D) and makes sense — milk is the only ingredient he is short on relative to the recipe.

💡Key takeaway

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.

More like this