AMC 8 · 2020 · #5

Grade 4 rate-ratio

Archetype Arithmetic Expression Decomposition

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

📏 Short solution 💡 2 insights

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Problem

Three fourths of a pitcher is filled with pineapple juice. The pitcher is emptied by pouring an equal amount of juice into each of $5$ cups. What percent of the total capacity of the pitcher did each cup receive?

$\textbf{(A) }5 \qquad \textbf{(B) }10 \qquad \textbf{(C) }15 \qquad \textbf{(D) }20 \qquad \textbf{(E) }25$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: A pitcher is $\tfrac{3}{4}$ 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?

Givens: The pitcher is filled to $\tfrac{3}{4}$ of its total capacity with juice; All the juice is poured into $5$ cups, each receiving an equal amount; Answer choices: (A) $5$, (B) $10$, (C) $15$, (D) $20$, (E) $25$ (all in percent)

Unknowns: The amount of juice in each cup, expressed as a percent of the pitcher's total capacity

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities

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.

Execute — Answer: C

#7 Identify Subproblems 4.NF.A.1 Step 1

Subproblem 1: express the juice in the pitcher as a percent of the pitcher's total capacity.
Rewrite $\tfrac{3}{4}$ with denominator $100$ by multiplying top and bottom by $25$.

$$\tfrac{3}{4} = \tfrac{3 \times 25}{4 \times 25} = \tfrac{75}{100}$$

💡 Rewriting a fraction with an equivalent denominator is the Grade 4 fraction-equivalence move ($\tfrac{3}{4} = \tfrac{75}{100}$).

#7 Identify Subproblems 4.NF.C.6 Step 2

Read the equivalent fraction $\tfrac{75}{100}$ as a percent.
"Percent" literally means "per one hundred", so $\tfrac{75}{100} = 75\%$.
The pitcher holds juice equal to $75\%$ of its capacity.

$$\tfrac{75}{100} = 75\%$$

💡 Reading a fraction with denominator $100$ as a percent is the Grade 4 decimal/percent notation skill.

#7 Identify Subproblems 3.OA.C.7 Step 3

Subproblem 2: share the $75\%$ of juice equally among $5$ cups.
That is a plain whole-number division: $75 \div 5$.

$$75\% \div 5 = 15\%$$

💡 Dividing $75$ by $5$ within $100$ is a Grade 3 fluency fact ($5 \times 15 = 75$).

#3 Eliminate Possibilities 3.OA.C.7 Step 4

Verify by elimination against the answer choices.
Each cup's percent, multiplied by $5$, must equal the total juice percent $75\%$.
Test the choices: (A) $5 \times 5 = 25$; (B) $10 \times 5 = 50$; (C) $15 \times 5 = 75$ ✓; (D) $20 \times 5 = 100$; (E) $25 \times 5 = 125$.
Only choice (C) hits $75\%$, confirming the answer.

$$15 \times 5 = 75 \;\Rightarrow\; \textbf{(C)}$$

💡 Testing each choice against the constraint "$5$ cups must add back to $75\%$" is a Grade 3 multiplication fluency check.

[1] #7 4.NF.A.1 Subproblem 1: express the juice in the pitcher as a percent of the pitcher's tot

[2] #7 4.NF.C.6 Read the equivalent fraction $\tfrac{75}{100}$ as a percent. "Percent" literally

[3] #7 3.OA.C.7 Subproblem 2: share the $75\%$ of juice equally among $5$ cups. That is a plain

[4] #3 3.OA.C.7 Verify by elimination against the answer choices. Each cup's percent, multiplied

Review

Reasonableness: 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: $5 \times 15\% = 75\%$, matching the $\tfrac{3}{4}$ we started with.

Alternative: Tool #9 (Solve an Easier Related Problem): pretend the pitcher holds exactly $\$75$ instead of $75\%$ of its capacity. Splitting $\$75$ among $5$ cups gives $\$15$ per cup, so each cup gets $15$ of the same units the whole pitcher was measured in — i.e. $15\%$ of the pitcher. The dollars-and-cents version exposes the structure without any percent machinery.

CCSS standards used (min grade 4)

4.NF.A.1 Explain why a fraction is equivalent to another fraction (Rewriting $\tfrac{3}{4}$ as the equivalent fraction $\tfrac{75}{100}$ by multiplying numerator and denominator by $25$.)
4.NF.C.6 Use decimal notation for fractions with denominators 10 or 100 (Reading $\tfrac{75}{100}$ as $75\%$ ("per one hundred") so the juice can be expressed directly as a percent of pitcher capacity.)
3.OA.C.7 Fluently multiply and divide within 100 (Dividing $75 \div 5 = 15$ to share the juice equally among the $5$ cups, and verifying with $15 \times 5 = 75$.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 4)

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