Competition · AMC preparation · step 4 of 4
AMC 8 · 2020 · #5
Grade 4 rate-ratioPick an answer.
AMC 8 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The question hides two clean subproblems: (1) what percent of the pitcher is actually juice, and (2) once that juice is split into 5 equal parts, how big is one part? Tool #7 (Identify Subproblems) names these two sub-questions and solves them one at a time. Because this is multiple choice, Tool #3 (Eliminate Possibilities) gives us a fast cross-check: 5 times the correct answer must equal the total juice percent (75%), so only one choice survives.
Convert the fraction to hundredths
Rewrite the juice fraction with denominator 100 (times 25 top and bottom) to get .
Rewriting a fraction with an equivalent denominator is the Grade 4 fraction-equivalence move (3/4 = 75/100).
4.NF.A.1Identify SubproblemsRead it as a percent
Read as a percent — it means per hundred — so the pitcher holds 75% of its capacity in juice.
Reading a fraction with denominator 100 as a percent is the Grade 4 decimal/percent notation skill.
4.NF.C.6Identify SubproblemsSplit 75 percent among 5 cups
Share that 75% equally among the 5 cups: 75 ÷ 5 = 15% per cup.
Dividing 75 by 5 within 100 is a Grade 3 fluency fact (5 × 15 = 75).
Pouring the juice, which fills 75% of the pitcher, equally into the 5 cups leaves each cup holding 15% of the pitcher's capacity.
▸ Why?
Sharing the 75 equally among 5 cups is exactly dividing 75 by 5, and that division gives 15.
▸ Why?
Dividing 75 by 5 asks how big one part is when 5 equal parts together make 75.
▸ Why?
Five equal parts of 15 stacked together count up to 75, so 15 is the size of one part.
▸ Why?
Recovering the part size from the total and the number of parts is the reverse of building the total from equal groups, and division is that reverse of multiplication.
▸ Why?
Each cup's share is still measured against the same whole pitcher, so a share of 15 counts as 15% of the pitcher — the same unit the 75% used.
▸ Why?
The 75% of juice is cut into 5 cup-shares with nothing spilled and nothing left behind, so those shares are just the parts that add back to the 75% whole.
Check by multiplying back
Check by elimination: 15 × 5 = 75 matches the total juice, so only (C) survives.
Testing each choice against the constraint "5 cups must add back to 75%" is a Grade 3 multiplication fluency check.
3.OA.C.7Eliminate PossibilitiesThis AMC 8 problem only needs Grade 4 fraction-to-percent thinking ( = = 75%) plus a Grade 3 division fact you already know!
- Convert the fraction to hundredths
- Read it as a percent
- Split 75 percent among 5 cups
- Check by multiplying back
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