2006 AMC 10A Problem 5

P1-10~2 minPrint

Linear Equations & Word Problems Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

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?

Show solution without answering

Key insight

Give each slice its own price: every slice costs 1 dollar, and the 2-dollar anchovy charge is shared only by the 4 anchovy slices.

Solution

"Each paid for what he had eaten," so first find the price of each slice.

Plain part. The plain pizza costs $8$ dollars for $8$ slices, so every slice carries $1$ dollar.
Anchovy part. The extra $2$ dollars is only for the anchovy half, the $4$ anchovy slices. So those $4$ slices together cost $4 + 2 = 6$ dollars.
Dave ate the $4$ anchovy slices ( $6$ dollars) and $1$ plain slice ( $1$ dollar). He pays $7$ dollars.
Doug ate the other $3$ plain slices. He pays $3$ dollars.
Check: $7 + 3 = 10$ , which is the whole bill, $8 + 2$ . Good.

Dave paid $7 - 3 = 4$ dollars more.

The answer is $\boxed{\textbf{(D)}\ 4}$ .

Why this works

When people pay for what they eat, attach every cost to the slice it belongs to, including extra charges for toppings. Then each person's bill is just the sum of his slices. Checking that the two bills add up to the total catches most mistakes.

Alternative approach

Start from the total. The whole bill is $10$ dollars. Doug ate only plain slices, $3$ of them at $1$ dollar each, so he pays $3$ . Dave pays the rest, $10 - 3 = 7$ . The difference is $4$ .

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

The plain pizza is $8$ dollars for $8$ slices, so each slice carries $1$ dollar of base cost. The $2$ -dollar anchovy charge belongs entirely to the four anchovy slices.

Dave ate the four anchovy slices, worth $4 + 2 = 6$ dollars, plus one plain slice worth $1$ dollar: he pays $7$ dollars.

Doug ate the other three plain slices: $3$ dollars.

Check: $7 + 3 = 10 = 8 + 2$ , the whole bill. The difference is $7 - 3 = 4$ .

The answer is $\boxed{\textbf{(D)}\ 4}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Prealgebra, Chapter 13: Data and Statistics (§13.1)