# Recover Original Ratio from Two Percent Changes

I have two variables $$x$$ and $$y$$ which are measured at two time steps $$0$$ and $$1$$. Available to me is the percent change from the first timestep to the second, relative to the first i.e. $$\frac{x_1 - x_0}{x_0}$$ and $$\frac{y_1 - y_0}{y_0}$$. I would like to recover the ratio $$\frac{x_1}{y_1}$$. I've tried adding 1 = $$\frac{x_0}{x_0} = \frac{y_0}{y_0}$$ to the two percent changes and then dividing them but this only gets me to $$\frac{x_1}{y_1} \cdot \frac{y_0}{x_0}$$. It seems to me that my problem has 2 equations and 4 unknowns, even though 2 of the unknowns don't appear in the desired ratio, making it have infinite solutions. Is that correct? Otherwise, is there another manipulation I can make to get my desired ratio?

Yes, you cannot get a unique value for $$x_1/y_1$$ from the information you have. Suppose $$x$$ goes up by 10%, from 100 to 110. Then $$y$$ could go up from 100 to 110 (+10%), or it could instead go up from 1000 to 1100 (also +10%), but these have different values of $$x_1/y_1$$.
As you have hinted at in your question, you are effectively being provided with $$\frac{x_1}{x_0}$$ and $$\frac{y_1}{y_0}$$, since you can just add one to the original terms. Let's say $$x_1 = 2$$, $$x_0 = 3$$, $$y_1 = 4$$ and $$y_0 = 5$$. Now consider $$x_1' = 4$$, $$x_0' = 6$$, $$y_1' = 4$$ and $$y_0' = 5$$. You will observe that the percent changes haven't changed, but the ratio $$\frac{x_1}{y_1}$$ has. In other words, there are an infinite number of solutions to the ratio.
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