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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?

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2 Answers 2

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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 $x_1/y_1$.

BTW if you tell us what the underlying goal is we might be able to give more insight or suggest alternatives to you. As your question stands this is about all that we can say.

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    $\begingroup$ Thanks. I was able to figure out my end goal, and this was quite helpful. $\endgroup$
    – taurus
    Aug 11, 2020 at 3:01
  • $\begingroup$ Glad to help. It would be nice if you updated your question and tell how you reached your end goal, so that other people who read your post can learn more too. $\endgroup$ Aug 11, 2020 at 3:08
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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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