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There is an algorithm called "Mischungskreuz" (German for "x of mixing") that is sometimes taught as a shortcut to figure out the following problem:

You have two solutions that contain a solute at different concentrations $c_1$ and $c_2$. At what ratio $V_1/V_2$ do you have to mix them so that the mixture has the desired concentration $c_m$?

For example, let's say you want to make a 22% solution from a 35% and a 15% solution. You write the desired concentration in the center and the available concentrations at the left ends of the "x", and get the ratio of volumes on the right side of the "x" as shown below:

enter image description here

So in this case, mix 7 parts of 35% with 13 parts of 15% solution to get the desired 22%.

Source: https://de.wikipedia.org/wiki/Mischungskreuz

My questions are: Is this method taught outside of Germany, and is there a non-German (maybe English?) name for it?

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  • $\begingroup$ Cute. I never saw that before. $\endgroup$
    – MaxW
    Mar 23, 2019 at 22:24
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    $\begingroup$ Very interesting! I have not seen it any English textbook so far. I am an analytical chemist. Most English books teach the dilution formula or mass balance as CiVi=CfVf. The German mixing cross (if this translation is better of Mischungskreuz) is a short cut to solve two problems. If you check Wörterbuch der Chemie / Dictionary of Chemistry: Deutsch/Englisch - English, it also calls it the dilution formula. books.google.com/… $\endgroup$
    – AChem
    Mar 23, 2019 at 22:25
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    $\begingroup$ I repostt my comment, originally to the answer below, for maximum visibility: I do have an important comment and I am not sure if I should write it in form of answer. From a physical chemical point of view this rules is simply wrong. A fact that likely explain why isn't teached or widespread. Of course by contingencies, though wrong, it can be useful, for instance for preparing cleaning or conditioning solutions, of for purposes with little or no physical chemical and analytical relevance. This likely explains why it was(is?) familiar to those working in dairy or sugar industries. $\endgroup$
    – Alchimista
    Mar 25, 2019 at 8:59

1 Answer 1

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This is a so-called "Pearson's square" or "Box method" of balancing ratios, originally used extensively in dairy industry (at least since 1920s judging from Google Books search). Earlier the similar approach has been used in sugar industry by using "Cobenz diagrams" aka spider diagrams.

Widely popularized in Soviet books for analytical chemistry at least since 1940s (probably adapted from the German literature as many other tech novelties of that time were), also used in current Russian literature by the names "Метод креста" ("Cross method"); "Конверт Пирсона" ("Pearson's envelope") or "Диагональная схема правила смешения" ("Diagonal mixing rule scheme").

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    $\begingroup$ Very interesting and thanks for sharing this info. I have always been in favor of learning another language besides English for scientific purposes. English is my second language. I am writing one article for the Journal of Chemical Education on the utility of foreign languages in literature search. Your point provides another motivation to finish that article soon. $\endgroup$
    – AChem
    Mar 23, 2019 at 23:05
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    $\begingroup$ I remember learning it in the Czech chemistry high school as "křížové pravidlo", What literally means "the cross rule". But the truth is, Czech chemistry was influenced by Germany, as German language was like a Latin in chemistry in late 19th/early 20th century. $\endgroup$
    – Poutnik
    Mar 24, 2019 at 7:22
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    $\begingroup$ I do have an important comment and I am not sure if I should write it in form of answer. From a physical chemical point of view this rules is simply wrong. A fact that likely explain why isn't teached or widespread. Of course by contingencies is wrong but can be useful, for instance for preparing cleaning or conditioning solutions, of for purposes with little or no physical chemical relevance. Again, this likely explain why it was (is?) familiar to those working in dairy or sugar industries. $\endgroup$
    – Alchimista
    Mar 24, 2019 at 10:18

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