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
.
See Also
2009 AIME II (Problems • Answer Key • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |