AMC 8 · 2020 · #5

Grade 4 rate-ratio

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

fraction-arithmeticpercentagefraction-decimal-conversion identify-subproblems ↑ Prerequisites: fraction-arithmeticpercentage

📏 Short solution 💡 2 insights

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Problem

A pitcher is

\frac{3}{4}

full of pineapple juice. All of that juice is shared equally among 5 cups. What percent of the pitcher's total capacity does the juice in each cup represent?

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

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.

1STEP 1

Rewrite the juice fraction $\frac{3}{4}$ with denominator 100 (times 25 top and bottom) to get $\frac{75}{100}$ .

\frac{3}{4}

\frac{3 × 25}{4 × 25}

\frac{75}{100}

2STEP 2

Read $\frac{75}{100}$ as a percent — it means per hundred — so the pitcher holds 75% of its capacity in juice.

\frac{75}{100}

= 75%

3STEP 3

Share that 75% equally among the 5 cups: 75 ÷ 5 = 15% per cup.

75% ÷ 5 = 15%

4STEP 4

Check by elimination: 15 × 5 = 75 matches the total juice, so only (C) survives.

15 × 5 = 75 → (C)

Answer

If the pitcher were 100% full, splitting it among 5 cups would give each cup 20%. Our pitcher is only 75% full, so each cup should get a bit less than 20% — and 15% fits that picture exactly. The cups also account for the juice: 5 × 15% = 75%, matching the

\frac{3}{4}

we started with.

💡Key takeaway

This AMC 8 problem only needs Grade 4 fraction-to-percent thinking ( $\frac{3}{4}$ = $\frac{75}{100}$ = 75%) plus a Grade 3 division fact you already know!

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