AMC 8 · 2012 · #4
Grade 4 rate-ratioProblem
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: A pizza is cut into $12$ equal slices. Peter eats one whole slice and then splits another slice equally with his brother Paul. What fraction of the whole pizza did Peter eat?
Givens: Pizza is divided into $12$ equal slices; Peter ate $1$ whole slice; Peter shared $1$ more slice equally with Paul (so Peter got half of that slice); Answer choices: (A) $\tfrac{1}{24}$, (B) $\tfrac{1}{12}$, (C) $\tfrac{1}{8}$, (D) $\tfrac{1}{6}$, (E) $\tfrac{1}{4}$
Unknowns: The fraction of the whole pizza that Peter alone ate
Understand
Restated: A pizza is cut into $12$ equal slices. Peter eats one whole slice and then splits another slice equally with his brother Paul. What fraction of the whole pizza did Peter eat?
Givens: Pizza is divided into $12$ equal slices; Peter ate $1$ whole slice; Peter shared $1$ more slice equally with Paul (so Peter got half of that slice); Answer choices: (A) $\tfrac{1}{24}$, (B) $\tfrac{1}{12}$, (C) $\tfrac{1}{8}$, (D) $\tfrac{1}{6}$, (E) $\tfrac{1}{4}$
Plan
Primary tool: #1 Draw a Diagram
Secondary: #7 Identify Subproblems
Draw the $12$-slice pizza and shade exactly what Peter ate — one whole slice plus half of a second slice. Tool #1 (Draw a Diagram) turns the words into a picture so the part-of-whole question becomes a counting question. Tool #7 (Identify Subproblems) splits Peter's intake into two clean pieces — the full slice and the half slice — and we add the two fractions at the end.
Execute — Answer: C
3.NF.A.1 Step 1 - Picture the pizza as a circle cut into $12$ equal wedges.
- Because all wedges are the same size, one wedge is $\tfrac{1}{12}$ of the whole pizza.
💡 Splitting a whole into $b$ equal parts and calling one part $\tfrac{1}{b}$ is the Grade 3 definition of a unit fraction.
3.NF.A.1 Step 2 - Split Peter's eating into two subproblems: (a) the one full slice he ate alone, and (b) the second slice he shared with Paul.
- Handle each piece separately, then add.
💡 Breaking what Peter ate into the "full" part and the "shared" part is exactly the Tool #7 subproblems move.
4.NF.B.4 Step 3 - The shared slice is itself $\tfrac{1}{12}$ of the pizza, and Peter eats half of it.
- Half of $\tfrac{1}{12}$ is $\tfrac{1}{2} \times \tfrac{1}{12} = \tfrac{1}{24}$.
💡 Multiplying a fraction by a whole-number partition (here, halving a slice) is the Grade 4 fraction-times-number idea.
4.NF.B.3 Step 4 - Add Peter's two pieces.
- To add $\tfrac{1}{12}$ and $\tfrac{1}{24}$, rewrite $\tfrac{1}{12}$ with denominator $24$: $\tfrac{1}{12} = \tfrac{2}{24}$.
- Then $\tfrac{2}{24} + \tfrac{1}{24} = \tfrac{3}{24}$.
💡 Rewriting one fraction so both denominators match, then adding numerators, is the Grade 4 "add fractions" technique.
4.NF.A.1 Step 5 - Simplify $\tfrac{3}{24}$ by dividing top and bottom by $3$ to get the equivalent fraction $\tfrac{1}{8}$.
- That matches choice (C).
💡 Dividing numerator and denominator by the same number to get an equivalent fraction is the Grade 4 equivalent-fraction rule.
3.NF.A.1 Picture the pizza as a circle cut into $12$ equal wedges. Because all wedges are 3.NF.A.1 Split Peter's eating into two subproblems: (a) the one full slice he ate alone, 4.NF.B.4 The shared slice is itself $\tfrac{1}{12}$ of the pizza, and Peter eats half of 4.NF.B.3 Add Peter's two pieces. To add $\tfrac{1}{12}$ and $\tfrac{1}{24}$, rewrite $\tf 4.NF.A.1 Simplify $\tfrac{3}{24}$ by dividing top and bottom by $3$ to get the equivalent Review
Reasonableness: Peter ate $1$ slice plus half a slice, which is $1.5$ slices out of $12$ — clearly less than $\tfrac{1}{4}$ of the pizza (that would be $3$ slices) and more than $\tfrac{1}{12}$ (one slice). $\tfrac{1}{8}$ sits between those two, so the size is sensible. As a check: $\tfrac{1}{8} \times 12 = 1.5$ slices, exactly what Peter ate.
Alternative: Tool #2 (Make a Systematic List) using "half-slices" as the unit: cut every slice in half, so the pizza has $24$ half-slices. Peter ate $2$ half-slices (the full slice) plus $1$ half-slice (the shared one) $= 3$ half-slices. Fraction $= \tfrac{3}{24} = \tfrac{1}{8}$ — same answer (C), no fraction addition needed.
CCSS standards used (min grade 4)
3.NF.A.1Understand a unit fraction $\tfrac{1}{b}$ as one part of a whole partitioned into $b$ equal parts (Recognizing that one slice of the $12$-slice pizza is $\tfrac{1}{12}$ of the whole, and decomposing Peter's intake into a full slice plus a half slice.)4.NF.B.4Apply and extend understanding of multiplication to multiply a fraction by a whole number (Computing half of one slice as $\tfrac{1}{2} \times \tfrac{1}{12} = \tfrac{1}{24}$ of the whole pizza.)4.NF.B.3Add and subtract fractions with like denominators by joining and separating parts (Adding Peter's two pieces $\tfrac{1}{12} + \tfrac{1}{24}$ after rewriting them with a common denominator ($\tfrac{2}{24} + \tfrac{1}{24} = \tfrac{3}{24}$).)4.NF.A.1Explain equivalence of fractions and generate equivalent fractions (Simplifying $\tfrac{3}{24}$ to $\tfrac{1}{8}$ by dividing numerator and denominator by $3$, and recognizing $\tfrac{1}{12} = \tfrac{2}{24}$ to find a common denominator.)
⭐ This AMC 8 problem only needs Grade 4 fraction skills — picturing slices and adding simple fractions — that you already know!
⭐ This AMC 8 problem only needs Grade 4 fraction skills — picturing slices and adding simple fractions — that you already know!
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