Competition · AMC preparation · step 4 of 4
AMC 8 · 2012 · #4
Grade 4 rate-ratioPick an answer.
AMC 8 2012 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Split the pizza into 12
Draw the pizza as 12 equal wedges; since all are the same size, one wedge is of the whole.
Splitting a whole into b equal parts and calling one part 1/b is the Grade 3 definition of a unit fraction.
3.NF.A.1Draw A DiagramSplit Peter's share in two
Split Peter's total into two pieces — the full slice and the shared slice — then handle each and add.
Breaking what Peter ate into the "full" part and the "shared" part is exactly the Tool #7 subproblems move.
3.NF.A.1Identify SubproblemsHalve the shared slice
The shared slice is also , and Peter eats half of it: × = .
Multiplying a fraction by a whole-number partition (here, halving a slice) is the Grade 4 fraction-times-number idea.
The half-slice Peter takes from the shared slice is 1/24 of the whole pizza.
▸ Why?
One whole slice is 1/12 of the pizza, so eating half of that slice means taking half of that 1/12.
▸ Why?
The pizza is cut into 12 equal slices, so a single slice is one of 12 equal parts whose copies rebuild the whole pizza, which is what 1/12 names.
▸ Why?
Cutting that one slice into 2 equal halves is the same as halving every slice, which turns the 12 equal slices into 24 equal pieces of the pizza, so one half-slice is one of 24 equal parts.
▸ Why?
Halving each of the 12 slices makes 12 groups of 2 small pieces, and 12 groups of 2 is 12 × 2 = 24 pieces in all.
▸ Why?
All 24 small pieces are the same size and together fill the pizza with no gaps or overlaps, so each one is one of 24 equal parts of the whole, which is 1/24.
Add Peter's two pieces
Add the two pieces: rewrite as , so + = .
Rewriting one fraction so both denominators match, then adding numerators, is the Grade 4 "add fractions" technique.
4.NF.B.3Identify SubproblemsSimplify the fraction
Simplify by dividing top and bottom by 3 to get — choice (C).
Dividing numerator and denominator by the same number to get an equivalent fraction is the Grade 4 equivalent-fraction rule.
4.NF.A.1Draw A DiagramThis AMC 8 problem only needs Grade 4 fraction skills — picturing slices and adding simple fractions — that you already know!
- Split the pizza into 12
- Split Peter's share in two
- Halve the shared slice
- Add Peter's two pieces
- Simplify the fraction
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