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Is there any way to knew or at least estimate the absolute proportion change from a percentage change? I am studying the corrosion of glass, and have several sets of chemical compound percentages, such as these, taken from the core and the surface of a Roman shard:

$$ \begin{array}{lrcr} \hline \text{Element} & \text{Core} && \text{Surface}\\ \hline \mathrm{Na} & 19.39\% & \to & 1.04\%\\ \mathrm{Mg} & 0.43\% & \to & 0.51\%\\ \mathrm{Al} & 1.8\% & \to & 2.25\%\\ \mathrm{Si} & 68.15\% & \to & 83.89\%\\ \mathrm{S} & 0.49\% & \to & 0.64\%\\ \mathrm{Cl} & 1.34\% & \to & 1.61\%\\ \mathrm{K} & 0.42\% & \to & 0.42\%\\ \mathrm{Ca} & 6.66\% & \to & 8.18\%\\ \mathrm{Fe} & 0.4\% & \to & 0.44\%\\ \mathrm{Sb} & 1.07\% & \to & 1.45\%\\ \hline \mathrm{Total} & 100\% & \to & 100\%\\ \hline \end{array} $$

As glass naturally degrades, it loses alkali, such as $\mathrm{Na}$, $\mathrm{Ca}$, etc. (seen in the difference between the two sets). However, some elements like $\mathrm{Al}$ retain a comparatively stable absolute amount, while their percentage changes significantly. Obviously, no material was added to the glass, but some percentages increased. So is there a calculation that could tell that, for instance "Sodium decreased twofold" or "Silica stayed the same" using just percents? The absolute amounts of each element only decrease with time, however I do not know how to quantify this loss.

As an example of what I mean, say there are $100$ coloured balls:

$$ \begin{array}{lrr} \hline \text{Color} & \text{Amount} & \text{Percentage}\\ \hline \text{Yellow} & 25 & 25\%\\ \text{Red} & 41 & 41\%\\ \text{Green} & 24 & 24\%\\ \text{Blue} & 10 & 10\%\\ \hline \end{array} $$

We take some of them away, now there are $78$ left:

$$ \begin{array}{lrr} \hline \text{Color} & \text{Amount} & \text{Percentage}\\ \hline \text{Yellow} & 10 & 13\%\\ \text{Red} & 36 & 46\%\\ \text{Green} & 22 & 28\%\\ \text{Blue} & 10 & 13\%\\ \hline \end{array} $$

Again, some of them, like the blue, were not taken, but the percentage changed, while the others, like red, were taken, but their percentage actually increased. However, in the case of glass, the absolute numbers are unknown. Therefore, is there any way to say, for instance, that $3/5$ of the yellow balls were taken or that the blue were not taken etc., while only having the percent values of this set?

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Let's consider two examples for a second:

  1. We've got $5$ blue balls and $3$ red balls. We take away $2$ blue ones and leave red ones the same, then blue has $50\%$ and red has $50\%$.

  2. We've got $5$ blue balls and $3$ red balls. We take away $4$ blue balls and $2$ red balls, then blue has $50\%$ and red has $50\%$.

The results of these examples are the same, but what happens differs wildly. This makes it impossible.

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  • $\begingroup$ That's quite disappointing, but would it still be possible to solve this if we were to assume that one element (say aluminum) does not change (or changes very little) in absolute terms? For instance, if it increases twofold, it would mean that half the other material was lost and therefore it's maybe viable to calculate this loss somehow... $\endgroup$ – A.Melinis Apr 12 '17 at 14:33
  • $\begingroup$ Yeah, if we assume one element to remain the exact same, we can calculate all of them. Example: 2 white, 4 red and 4 blue. Then the percentages would be 20%, 40% and 40%. If we assume White to remain the same, and blue/red get changed to 1 and 2 respectively, we'd end up with: 2 white, 1 red, 2 blue, with percentages 40%, 20%, 40%. Because we know for certain that white stayed the same, we can "normalise" for the change and say that each particle is now worth twice as much as it was. (so one ball is worth 20% instead of 10%). This allows us to exactly determine the changes. $\endgroup$ – Mitchell Faas Apr 12 '17 at 17:13
  • $\begingroup$ Thank you! That should do it! I also assume that this change should be normalised for the element that increased the most (percentage wise). For instance if we assume that white remained the same and increased twofold, but some other element increased threefold, it means that white in actuality did not remain the same, while the other element did... In any case, cheers! $\endgroup$ – A.Melinis Apr 23 '17 at 19:11

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