2009 AIME II Problems/Problem 1

Problem

Before starting to paint, Bill had ounces of blue paint, ounces of red paint, and 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.

Solution

After the pink stripe is drawn, all three colors will be used equally so the pink stripe must bring the amount of red and white paint down to ounces each. Say is the fraction of the pink paint that is red paint and is the size of each stripe. Then equations can be written: and . The second equation becomes and substituting the first equation into this one we get so . The amount of each color left over at the end is thus and .