AMC 8 · 2012 · #4

Easy mode Grade 4

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Problem

Picture a pizza cut into $12$ equal slices.

Peter eats one whole slice. Then he takes one more slice and splits it evenly with his brother Paul.

What fraction of the whole pizza did Peter eat?

Pick an answer.

(A)

$~rac{1}{24}$

(B)

$~rac{1}{12}$

(C)

$~rac{1}{8}$

(D)

$~rac{1}{6}$

(E)

$~rac{1}{4}$

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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?

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

#1 Draw a Diagram 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.

$$1 \text{ slice} = \tfrac{1}{12} \text{ of the pizza}$$

💡 Splitting a whole into $b$ equal parts and calling one part $\tfrac{1}{b}$ is the Grade 3 definition of a unit fraction.

#7 Identify Subproblems 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.

$$\text{Peter} = (\text{full slice}) + (\text{half of a slice})$$

💡 Breaking what Peter ate into the "full" part and the "shared" part is exactly the Tool #7 subproblems move.

#1 Draw a Diagram 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 $\tfrac{1}{2} \times \tfrac{1}{12} = \tfrac{1}{24}$.

$$\tfrac{1}{2} \times \tfrac{1}{12} = \tfrac{1}{24}$$

💡 Multiplying a fraction by a whole-number partition (here, halving a slice) is the Grade 4 fraction-times-number idea.

#7 Identify Subproblems 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 $\tfrac{2}{24} + \tfrac{1}{24} = \tfrac{3}{24}$.

$$\tfrac{1}{12} + \tfrac{1}{24} = \tfrac{2}{24} + \tfrac{1}{24} = \tfrac{3}{24}$$

💡 Rewriting one fraction so both denominators match, then adding numerators, is the Grade 4 "add fractions" technique.

#1 Draw a Diagram 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).

$$\tfrac{3}{24} = \tfrac{3 \div 3}{24 \div 3} = \tfrac{1}{8} \;\Rightarrow\; \textbf{(C)}$$

💡 Dividing numerator and denominator by the same number to get an equivalent fraction is the Grade 4 equivalent-fraction rule.

[1] #1 3.NF.A.1 Picture the pizza as a circle cut into $12$ equal wedges. Because all wedges are

[2] #7 3.NF.A.1 Split Peter's eating into two subproblems: (a) the one full slice he ate alone,

[3] #1 4.NF.B.4 The shared slice is itself $\tfrac{1}{12}$ of the pizza, and Peter eats half of

[4] #7 4.NF.B.3 Add Peter's two pieces. To add $\tfrac{1}{12}$ and $\tfrac{1}{24}$, rewrite $\tf

[5] #1 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.1 Understand 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.4 Apply 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.3 Add 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.1 Explain 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!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 4)

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