Problem 15 · 2009 AMC 8
Medium
Ratios, Rates & Proportions
limiting-ingredient
A recipe that makes 5 servings of hot chocolate requires 2 squares of chocolate, 1/4 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 she maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate she can make?
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Answer: D — 8¾ servings.
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Hint 1 of 2
You can only make as much as your SCARCEST ingredient allows — like a chain breaking at its weakest link. Having extra of everything else doesn't help once one runs out.
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Hint 2 of 2
For each ingredient ask: how many full recipes does my supply cover? (supply ÷ amount-per-recipe). The smallest of those numbers is the limit; then turn recipes into servings.
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Approach: find the limiting (scarcest) ingredient
- Recipes each supply covers: chocolate 5 ÷ 2 = 2.5; sugar 2 ÷ ¼ = 8; milk 7 ÷ 4 = 1.75. (Water is unlimited.)
- The smallest is milk at 1.75 recipes — milk runs out first, so it caps everything.
- Each recipe makes 5 servings: 1.75 × 5 = 35/4 = 8¾ servings.
- Why this transfers: any "how much can I make / how many can I build" problem is a weakest-link question — compute the limit from each resource and take the minimum. Don't average them or use the most plentiful one.
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