2009 AIME II Problem 1

Question

Before starting to paint, Bill had

130

ounces of blue paint,

164

ounces of red paint, and

188

ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stripe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.

Answer

Answer: 114

Concepts:linear equation algebraic manipulation

Difficulty rating: 1750

Small hint:

Equal leftovers mean the red used exceeds the blue used by

164 - 130,

and similarly for white

Big hint:

Those excesses all went into the pink stripe, so one stripe’s worth of paint is

34 + 58

ounces

Solution:

Say each stripe used

s

ounces of paint. Blue was used only on the blue stripe, so

s

ounces of blue were used. Since the three leftovers are equal and the colors started

34

and

58

ounces apart, red use exceeded blue use by

164 - 130 = 34

ounces and white use exceeded blue use by

188 - 130 = 58

ounces. That extra red and white is exactly the pink stripe, so

s = 34 + 58 = 92.

Bill therefore had

130 - 92 = 38

ounces of each color left, for a total of

3 \cdot 38 = 114

ounces.