AMC 10 · 2006 · #5
Easy mode Grade 5Doug and Dave shared a pizza cut into 8 equal slices. A plain pizza costs 8 dollars. Dave wanted anchovies on half the pizza, and adding anchovies to that half (its 4 slices) cost an extra 2 dollars. Dave ate all 4 anchovy slices and 1 plain slice. Doug ate the other 3 plain slices. Each one pays only for the slices he ate. How many more dollars did Dave pay than Doug?
Pick an answer.
AMC 10 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A pizza has $8$ equal slices. A plain pizza costs $8$ dollars, and adding anchovies to one half (the $4$ slices on that side) costs an extra $2$ dollars. Dave ate all $4$ anchovy slices and $1$ plain slice; Doug ate the other $3$ plain slices. Each person pays only for the slices he ate. Find how many more dollars Dave paid than Doug.
Givens: The pizza has $8$ equal slices; Plain pizza costs $8$ dollars total; Anchovies on one half (its $4$ slices) cost an extra $2$ dollars; Dave ate all $4$ anchovy slices plus $1$ plain slice; Doug ate the remaining $3$ plain slices; Each pays for exactly what he ate; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$
Unknowns: How many more dollars Dave paid than Doug
Understand
Restated: A pizza has $8$ equal slices. A plain pizza costs $8$ dollars, and adding anchovies to one half (the $4$ slices on that side) costs an extra $2$ dollars. Dave ate all $4$ anchovy slices and $1$ plain slice; Doug ate the other $3$ plain slices. Each person pays only for the slices he ate. Find how many more dollars Dave paid than Doug.
Givens: The pizza has $8$ equal slices; Plain pizza costs $8$ dollars total; Anchovies on one half (its $4$ slices) cost an extra $2$ dollars; Dave ate all $4$ anchovy slices plus $1$ plain slice; Doug ate the remaining $3$ plain slices; Each pays for exactly what he ate; Answer choices: (A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) $5$
Plan
Primary tool: #8 Analyze the Units
Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities
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.
Execute — Answer: D
5.NF.B.3 Step 1 Price one plain slice
- The plain pizza costs $8$ dollars and is cut into $8$ equal slices, so the plain part of every slice costs the same share.
- Dividing the plain cost equally, each slice carries $8 \div 8 = 1$ dollar of plain cost.
💡 Equal slices split the plain bill evenly, so each one is worth its fair share of the $\$8$.
5.NF.B.3 Step 2 Price one anchovy slice
- The extra $2$ dollars for anchovies only covers the half that has them — that is $4$ slices.
- Spread over those $4$ slices, the anchovy topping adds $2 \div 4 = 0.50$ dollar each.
- So an anchovy slice costs the $\$1$ plain share plus the $\$0.50$ topping, or $\$1.50$; a plain slice stays at $\$1$.
💡 The topping charge belongs only to the slices that actually got anchovies, so split it just among those four.
5.NBT.B.7 Step 3 Add up what Dave ate
- Dave ate all $4$ anchovy slices and $1$ plain slice.
- The anchovy slices cost $4 \times \$1.50 = \$6$, and the plain slice costs $\$1$. Dave's total is $\$6 + \$1 = \$7$.
💡 Charge Dave for each slice he ate at that slice's own price, then total it.
4.OA.A.3 Step 4 Add up Doug's share and compare
- Doug ate the remaining $3$ plain slices, so he paid $3 \times \$1 = \$3$.
- As a check, $\$7 + \$3 = \$10$ matches the full bill. Dave paid $\$7 - \$3 = \$4$ more than Doug, which is choice (D).
💡 Doug only ever ate plain slices, so his bill is just the plain price times how many he ate.
5.NF.B.3 The plain pizza costs $8$ dollars and is cut into $8$ equal slices, so the plain 5.NF.B.3 The extra $2$ dollars for anchovies only covers the half that has them — that is 5.NBT.B.7 Dave ate all $4$ anchovy slices and $1$ plain slice. The anchovy slices cost $4 4.OA.A.3 Doug ate the remaining $3$ plain slices, so he paid $3 \times \$1 = \$3$. As a c Review
Reasonableness: The two payments should rebuild the whole bill, and $\$7 + \$3 = \$10 = \$8 + \$2$, so nothing is lost or double-counted. Dave ate more slices and the pricier ones, so it makes sense he paid more; a gap of $\$4$ sits comfortably among the single-digit choices and is not near the impossible extremes.
Alternative: Skip per-slice prices and reason by the topping. If there were no anchovies, all $8$ slices cost $\$1$ each, so Dave's $5$ slices would be $\$5$ and Doug's $3$ slices $\$3$ — a $\$2$ gap. The anchovy charge is $\$2$ and Dave alone ate every anchovy slice, so he shoulders all of it, widening the gap by another $\$2$. Total gap $= \$2 + \$2 = \$4$, again (D).
CCSS standards used (min grade 5)
5.NF.B.3Interpret a fraction as division of the numerator by the denominator (Dividing the plain $\$8$ over $8$ slices and the $\$2$ topping over $4$ slices to get each slice's cost.)5.NBT.B.7Add, subtract, multiply, and divide decimals to hundredths (Computing $4 \times \$1.50 = \$6$ and adding the plain slice to reach Dave's $\$7$.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Totaling Doug's slices and subtracting to find how much more Dave paid.)
⭐ Find 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.
⭐ Find 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.
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