Competition · AMC preparation · step 4 of 4
AMC 8 · 2020 · #1
Grade 4 rate-ratioPick an answer.
AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Find the sugar
Subproblem 1 — the sugar: sugar is twice the lemon juice, so multiply 3 cups by 2 to get 6 cups.
"Twice as much" is a multiplicative comparison — the Grade 4 standard for "X times as many" word problems.
4.OA.A.2Identify SubproblemsFind the water
Subproblem 2 — the water: water is 4 times the sugar, so multiply the 6 cups by 4 to get 24 cups.
Same "X times as much" comparison again — feed Subproblem 1's answer straight into Subproblem 2.
The cups of water the recipe needs equal the cups of lemon juice multiplied by 2 (to get sugar) and then by 4.
▸ Why?
Water is reached in two linked multiplication steps — first the lemon juice becomes sugar, then that sugar becomes water — because the recipe ties sugar to lemon juice and water to sugar.
▸ Why?
"Twice as much sugar as lemon juice" means the sugar is 2 equal copies of the lemon-juice amount, and making equal copies is exactly what multiplying by 2 does.
▸ Why?
"Four times as much water as sugar" means the water is 4 equal copies of the sugar amount, and making equal copies is exactly what multiplying by 4 does.
▸ Why?
The sugar amount found in the first step is the very same sugar the water rule multiplies, so the second step may use the first step's result in place of the word "sugar".
Check the units
Unit check: the multipliers are unitless, so cups stay cups all the way — 24 cups matches choice (E).
Tracking the unit "cups" through each multiplication confirms we did not accidentally mix recipes for two different ingredients.
4.OA.A.2Analyze The UnitsThis AMC 8 problem only needs Grade 4 "X times as many" multiplicative comparison you already know!
- Find the sugar
- Find the water
- Check the units
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