AMC 10 · 2006 · #5

Grade 5 number-theory

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

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

📏 Short solution 💡 2 insights

Problem

Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$ dollars, and there was an additional cost of $2$ dollars for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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.

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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

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

#8 Analyze the Units 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.

$\dfrac{\$8}{8\text{ slices}} = \$1 \text{ per slice}$

💡 Equal slices split the plain bill evenly, so each one is worth its fair share of the $\$8$.

#8 Analyze the Units 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 $\$1$.

$\dfrac{\$2}{4\text{ anchovy slices}} = \$0.50;\quad \$1 + \$0.50 = \$1.50 \text{ per anchovy slice}$

💡 The topping charge belongs only to the slices that actually got anchovies, so split it just among those four.

#7 Identify Subproblems 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 $\$6 + \$1 = \$7$.

$4 \times \$1.50 + \$1 = \$6 + \$1 = \$7$

💡 Charge Dave for each slice he ate at that slice's own price, then total it.

#7 Identify Subproblems 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).

$3 \times \$1 = \$3;\quad \$7 - \$3 = \$4 \;\Rightarrow\; \textbf{(D)}$

💡 Doug only ever ate plain slices, so his bill is just the plain price times how many he ate.

[1] #8 5.NF.B.3 The plain pizza costs $8$ dollars and is cut into $8$ equal slices, so the plain

[2] #8 5.NF.B.3 The extra $2$ dollars for anchovies only covers the half that has them — that is

[3] #7 5.NBT.B.7 Dave ate all $4$ anchovy slices and $1$ plain slice. The anchovy slices cost $4

[4] #7 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.3 Interpret 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.7 Add, 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.3 Solve 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.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Price one plain slice

Price one anchovy slice

Add up what Dave ate

Add up Doug's share and compare

Review

CCSS standards used (min grade 5)

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