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.

答案

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.

2009 AIME II Problem 1

Solution:

Problem 1 in Other Years