Mathematics-Online problems: Solution to the problem of the (previous) week

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	Mathematics-Online problems:
	Solution to the problem of the (previous) week

Problem:

The bottles in the figure below are filled with an alcoholic liquid having different alcohol contents. The left bottle is filled to one third, the center bottle to two thirds and the right bottle is filled completely.

$\includegraphics[width=0.5\linewidth]{TdM_09_A3_bild1.eps}$

The contents of the bottles are mixed by completely filling up one of the partly filled bottles from one of the other two bottles. What are the possible alcohol contents of a full bottle after one or two acts of decanting?
What is the alcohol content when mixing the liquid of all three bottles?

Answer:

Possible alcohol contents after one or two acts of decanting:
< < < < <
Total alcohol content:

(All results in % and ascending order.)

Solution:

Let be the alcohol contents of the bottles filled to 1/3, 2/3, and completely. Then, after one act of decanting, there are three possibilities for the alcohol contents :

Filling up the bottle of 1/3 content from the bottle of 2/3 content (or vice versa):

$\displaystyle z'_1 = \frac{1}{3}x + \frac{2}{3}y,\quad z'_2 = z\,.$

Filling up the bottle of 1/3 content from the full bottle:

$\displaystyle z' = \frac{1}{3}x + \frac{2}{3}z,\quad x' = z,\quad y' = y\,.$

Filling up the bottle of 2/3 content from the completely full bottle:

$\displaystyle z' = \frac{2}{3}y + \frac{1}{3}z,\quad x' = x,\quad y' = z\,.$

The process of decanting can be illustrated in a tree diagram.

$\includegraphics{TdM_09_A3_bild2}$

The possible alcohol contents of full bottles,

$\displaystyle 15,\,19,\,21,\,27,\,29,\,33,$
are underlined.
If we mix two full bottles obtained by decanting, the total alcohol content is

$\displaystyle \frac{1}{2}\cdot 27+\frac{1}{2}\cdot 21 = 24\,.$

[problem of the week]