Cyclic Transfer of Brine (Salty Water) - Part 2
Three beakers contain different amounts of brine with different concentrations. The first beaker has mL of pure water (salt concentration is g / L ). Second beaker has mL of brine with a salt concentration of g / L , while the third beaker has mL of brine with a salt concentration of g / L. Now, mL is transferred from the first beaker to the second beaker, and the second beaker is shaken well, then mL is transferred from the second beaker to the third beaker, and the third beaker is shaken well, and finally, mL is transferred from the third beaker to the first beaker, and the first beaker is shaken well. This whole sequence of three transfers is repeated a large number of times. At the end, what will be concentration of brine (in g / L) in each of the three beakers ? If the answer is , for coprime positive integers and , then find .
The answer is 227.
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1 solution
The first beaker has 0 g of salt and 1 0 0 0 mL of liquid, the second beaker has 1 0 0 0 5 0 0 ⋅ 1 0 0 = 5 0 g of salt and 5 0 0 mL of liquid, and the third beaker has 1 0 0 0 2 5 0 ⋅ 2 0 = 5 g of salt and 2 5 0 mL of liquid, for a total of 0 + 5 0 + 5 = 5 5 g of salt and 1 0 0 0 + 5 0 0 + 2 5 0 = 1 7 5 0 mL of liquid.
Since after the whole sequence each beaker will have the same concentration, the concentration for each beaker is the same as the total concentration, which will be 1 . 7 5 5 5 = 7 2 2 0 g/L . Therefore, p = 2 2 0 , q = 7 , and p + q = 2 2 7 .