AMC 10 · 2006 · #5
Grade 5 arithmeticPick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The whole question is really about dollars-per-slice, so Tool #8 (Analyze the Units) drives it: find the cost of one plain slice and one anchovy slice. Tool #7 (Identify Subproblems) then handles each eater separately — add up Dave's slices, add up Doug's slices. Tool #3 (Eliminate Possibilities) keeps the arithmetic honest: the two payments must still add to the $10 total, which flags any slip.
Price one plain slice
The plain $8 splits evenly over the 8 equal slices, so every slice carries $1 of plain cost.
Equal slices split the plain bill evenly, so each one is worth its fair share of the $8.
5.NF.B.3Analyze The UnitsPrice one anchovy slice
The extra $2 rides on the 4 anchovy slices alone, adding $0.50 each, so such a slice costs $1.50 and a plain one $1.
The topping charge belongs only to the slices that actually got anchovies, so split it just among those four.
The topping charge belongs only to the slices that got the topping, so it is split just among those.
▸ Why?
Splitting a charge among equal slices gives each one an equal share of it.
▸ Why?
The topping is bought once for a fixed set of slices, so its price attaches to that set alone.
Add up what Dave ate
Dave's four anchovy slices come to 4 × $1.50 = $6, and his plain slice adds $1, so Dave pays $7.
Charge Dave for each slice he ate at that slice's own price, then total it.
5.NBT.B.7Identify SubproblemsAdd up Doug's share and compare
Doug's 3 plain slices cost $3, and $7 + $3 = $10 checks out, so Dave paid $4 more — choice (D).
Doug only ever ate plain slices, so his bill is just the plain price times how many he ate.
4.OA.A.3Identify SubproblemsFind the price of each kind of slice first, put the anchovy charge only on the slices that got anchovies, then charge each person for exactly what he ate.
- Price one plain slice
- Price one anchovy slice
- Add up what Dave ate
- Add up Doug's share and compare