Solve Paint Can Problem: Red-White Content & Same Shade of Pink

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There are two 1-gallon (4 quarts) cans. One contains 4qt of white paint, and
the other contains 3qt of red paint. You pour 1 qt of white paint into the red, mix it, and then pour 1 qt of the mixture back into the can of white paint. What is the
red-white content of each can now? If you continually repeat
the process, when will the two cans be the same shade of pink?
 

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bulldog106 said:
There are two 1-gallon (4 quarts) cans. One contains 4qt of white paint, and
the other contains 3qt of red paint. You pour 1 qt of white paint into the red, mix it, and then pour 1 qt of the mixture back into the can of white paint. What is the
red-white content of each can now? If you continually repeat
the process, when will the two cans be the same shade of pink?

You start out with white-to-red ratios of:
Can 1: 4:0
Can 2: 0:3

After 1qt of Can 1 is added to Can 2:
Can 1: 3:0
Can 2: 1:3

After 1 qt of Can 2 is added to Can 1:
Can 1: 3.25:0.75
Can 2: 0.75:2.25

Do this process 10 times total, and you get approximately:
Can 1: 2.291150649:1.708849351 ~ 1.340756
Can 2: 1.708849351:1.291150649 ~ 1.323509

Do it 20 times and you get approximately:
Can 1: 2.285731526:1.714268474 ~ 1.333356799
Can 2: 1.714268474:1.285731526 ~ 1.333302047

So, you slowly get closer and closer to the ultimate:
Can 1: 16/7 white paint, 12/7 red paint = 4:3 ratio
Can 2: 12/7 white paint, 9/7 red paint = 4:3 ratio

But you never quite get there, mathematically speaking.

DaveE