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I have a question, I apologize if it seems trivial but I just want to make sure this is correct or be corrected if not. I have a function

$$f(r,x) = \dfrac{r}{x}$$

Where $r$ - is total revenue of purchases and $x$ is the number of customers who made a purchase on a website. Here is my question with an example:

Example: Say, I have $r = \$300$ from 10 customers, so $$f(300, 10) = 30$$

Now, I find that of my 10 customers two of them are purchasing from separate accounts. Ex: User1 is purchasing also as User 8 and User 2 is also purchasing as User 10. I have the same revenue but a different cohort size of 8 now.

$$f(300, 8) = 37.5$$.

I want to determine how much of an increase there is from $f(300, 8)$ to $f(300, 10)$.

My questions are:

  1. Can I just perform a percent increase? $$\dfrac{(37.5 - 30)}{30}*100 = 25\%$$

  2. If this is incorrect what is the correct way of comparing? This may seem trivial to some but I believe I'm being thrown off by the change in the denominator.

Thank you for any help or comments!

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    $\begingroup$ I think you may want to rephrase :$f(300,10)$ to $f(300,8)$ as that is what you calculated in #1. $\endgroup$
    – VCG
    Aug 25, 2016 at 20:17

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Percent increase sounds fine to me. If you shrink the denominator of a fraction by $20\%$, that's:

  • equivalent to multiplying the denominator by $.8$
  • which is equivalent to multiplying the overall number by $\frac{1}{.8} = 1.25$.
  • which is an increase of 25%

More algebra:

Going from $\frac{300}{10}$ to $\frac{300}{8}$ is

$$\begin{align*} \frac{37.5-30}{30} &= \frac{\frac{300}{8} - \frac{300}{10}}{\frac{300}{10}}\\ &= \left( \frac{\frac{10}{8} - 1}{1} \right) \\ &= .25 \end{align*} $$

In variables:

$$\begin{align*} \frac{\frac{x}{\alpha y} - \frac{x}{y}}{\frac{x}{y}} &= \frac{1}{\alpha} - 1 \\ \end{align*} $$

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