AMC 8 · 2012 · #4

Grade 4 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

fraction-arithmeticratio-proportion identify-subproblems ↑ Prerequisites: fraction-arithmetic

📏 Short solution 💡 2 insights

Problem

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?

Pick an answer.

(A)

$~\frac{1}{24}$

(B)

$~\frac{1}{12}$

(C)

$~\frac{1}{8}$

(D)

$~\frac{1}{6}$

(E)

$~\frac{1}{4}$

AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Draw a Diagram

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.

1STEP 1

Draw the pizza as 12 equal wedges; since all are the same size, one wedge is $\frac{1}{12}$ of the whole.

1 slice =

\frac{1}{12}

of the pizza

2STEP 2

Split Peter's total into two pieces — the full slice and the shared slice — then handle each and add.

Peter = (full slice) + (half of a slice)

3STEP 3

The shared slice is also $\frac{1}{12}$ , and Peter eats half of it: $\frac{1}{2}$ × $\frac{1}{12}$ = $\frac{1}{24}$ .

\frac{1}{2}

\frac{1}{12}

\frac{1}{24}

4STEP 4

Add the two pieces: rewrite $\frac{1}{12}$ as $\frac{2}{24}$ , so $\frac{2}{24}$ + $\frac{1}{24}$ = $\frac{3}{24}$ .

\frac{1}{12}

\frac{1}{24}

\frac{2}{24}

\frac{1}{24}

\frac{3}{24}

5STEP 5

Simplify $\frac{3}{24}$ by dividing top and bottom by 3 to get $\frac{1}{8}$ — choice (C).

\frac{3}{24}

\frac{3 ÷ 3}{24 ÷ 3}

\frac{1}{8}

→ (C)

Answer

\frac{1}{8}

Peter ate 1 slice plus half a slice, which is 1.5 slices out of 12 — clearly less than

\frac{1}{4}

of the pizza (that would be 3 slices) and more than

\frac{1}{12}

(one slice).

\frac{1}{8}

sits between those two, so the size is sensible. As a check:

\frac{1}{8}

× 12 = 1.5 slices, exactly what Peter ate.

💡Key takeaway

This AMC 8 problem only needs Grade 4 fraction skills — picturing slices and adding simple fractions — that you already know!

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