AMC 8 · 2020 · #1

Grade 4 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

ratio-proportionmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

Luka's lemonade recipe says: water is 4 times the sugar, and sugar is twice the lemon juice. If he pours 3 cups of lemon juice, how many cups of water does the recipe call for?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

We are not given the amount of water directly, but we are given lemon juice and two "X times as much as Y" rules that link the three ingredients in a chain: lemon juice → sugar → water. Tool #7 (Identify Subproblems) is a perfect fit: solve "how much sugar?" first, then use that answer to solve "how much water?". Tool #8 (Analyze the Units) is a safety net — everything is in cups, so we just confirm the unit stays consistent and we are not accidentally multiplying by ratios that should be applied to lemon juice. A tape-diagram sketch (Tool #1) makes the 1 : 2 : 8 ratio of lemon juice : sugar : water visually obvious if a student wants to see it.

1STEP 1

Subproblem 1 — the sugar: sugar is twice the lemon juice, so multiply 3 cups by 2 to get 6 cups.

sugar = 2 × 3 = 6 cups

2STEP 2

Subproblem 2 — the water: water is 4 times the sugar, so multiply the 6 cups by 4 to get 24 cups.

water = 4 × 6 = 24 cups

3STEP 3

Unit check: the multipliers are unitless, so cups stay cups all the way — 24 cups matches choice (E).

3 cups → 6 cups → 24 cups → (E)

Answer

Water should be the largest of the three quantities because of all three ingredients it is multiplied by the biggest factor in the chain (1 → 2 → 8 times the lemon juice). 24 cups of water for 3 cups of lemon juice is exactly 8 × — that matches the combined multiplier 4 × 2 = 8. The answer also picks the largest choice (E), which lines up with the intuition that water dominates a lemonade recipe.

💡Key takeaway

This AMC 8 problem only needs Grade 4 "X times as many" multiplicative comparison you already know!

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