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 is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Water $= 4 \times$ sugar (in cups); Sugar $= 2 \times$ lemon juice (in cups); Lemon juice used $= 3$ cups; Answer choices: (A) $6$, (B) $8$, (C) $12$, (D) $18$, (E) $24$

Unknowns: The number of cups of water needed

Understand

Restated: 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?

Givens: Water $= 4 \times$ sugar (in cups); Sugar $= 2 \times$ lemon juice (in cups); Lemon juice used $= 3$ cups; Answer choices: (A) $6$, (B) $8$, (C) $12$, (D) $18$, (E) $24$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units, #1 Draw a Diagram

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 $\to$ sugar $\to$ 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.

Execute — Answer: E

#7 Identify Subproblems 4.OA.A.2 Step 1

Subproblem 1 — find the sugar.
The recipe says sugar is twice the lemon juice.
Multiply the $3$ cups of lemon juice by $2$.

$$\text{sugar} = 2 \times 3 = 6 \text{ cups}$$

💡 "Twice as much" is a multiplicative comparison — the Grade 4 standard for "X times as many" word problems.

#7 Identify Subproblems 4.OA.A.2 Step 2

Subproblem 2 — find the water.
The recipe says water is $4$ times the sugar.
Use the $6$ cups of sugar from Subproblem 1 and multiply by $4$.

$$\text{water} = 4 \times 6 = 24 \text{ cups}$$

💡 Same "X times as much" comparison again — feed Subproblem 1's answer straight into Subproblem 2.

#8 Analyze the Units 4.OA.A.2 Step 3

Unit check.
Lemon juice was in cups, the comparison rules are unitless multipliers, so sugar and water also come out in cups.
The answer matches the units the question asks for.
The result $24$ matches choice (E).

$$3 \text{ cups} \xrightarrow{\times 2} 6 \text{ cups} \xrightarrow{\times 4} 24 \text{ cups} \;\Rightarrow\; \textbf{(E)}$$

💡 Tracking the unit "cups" through each multiplication confirms we did not accidentally mix recipes for two different ingredients.

[1] #7 4.OA.A.2 Subproblem 1 — find the sugar. The recipe says sugar is twice the lemon juice. M

[2] #7 4.OA.A.2 Subproblem 2 — find the water. The recipe says water is $4$ times the sugar. Use

[3] #8 4.OA.A.2 Unit check. Lemon juice was in cups, the comparison rules are unitless multiplie

Review

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

Alternative: Tool #1 (Draw a Diagram) — sketch a tape diagram with $1$ block for lemon juice, $2$ blocks for sugar, and $8$ blocks for water. If one block is $3$ cups (the given lemon-juice amount), then water $= 8 \times 3 = 24$ cups. This skips the intermediate "sugar" calculation and reads the answer right off the diagram.

CCSS standards used (min grade 4)

4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison (Translating "twice as much sugar as lemon juice" into $2 \times 3 = 6$ and "$4$ times as much water as sugar" into $4 \times 6 = 24$, then tracking the cup unit through both multiplications.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 4)

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