I’m trying to solve a question for X: Let X be the required number of actions to perform today in order to create Y results by the end-of-the-year, based on a Z% multiplier. Example: 100 actions today on November 1st creates 9 results by December 31st, or 100 actions on February 17th might create 47 results. Basically I have a formula of X today * Z(%) = Y at EOY
My problem is that in my real life example, X really should be uniformly distributed across all the remaining days in the year, i.e. not occurring all on today. Thus, if I’m solving for X because I have a desired Y value, I need X to actually be much larger than my formula results in, because all the X actions will be spread out. So instead of 100 actions occurring today, it would need to more say 150 actions (or whatever the correct formula would result in) are needed between now & EOY based on uniform distribution to generate Y results at EOY. As each day goes by, an X produced the next day generates slightly less Y until it’s effectively 0 at EOY.
My question is: what’s the right formula to use here? I feel like this is similar to the idea of present value with interest calculations, but I’m just blanking. I assume I would have some kind of sum-based formula going from days from today to EOY, but I just can’t seem to write my question the right way in Google or StackExchange to steer me. Any advice?
EDIT for more context Sorry about not giving the best context on the question. Here's an example that might use this hypothetical formula I'm looking for:
- A generic action, say a person buys a computer, generates another computer resale within 180 days (call it a replacement or upgrade), 48% of the time. So X sales that occurred today would generate X * 0.48 sales between now and 180 days from now
- If the end of the calendar year (Dec 31st) is 90 days away, with a uniform distribution of those re-sales, I'd expect X * 0.48 * 90/180 = X * 0.24 sales between now & EOY
- Continuing that pattern, X sales on Dec 1st (30 days out from EOY) might generate X * 0.48 * 30/180 sales between then and EOY
What this all assumes though is that there are X actions (or sales in this example) all on one date, e.g. today or Dec 1st in the above examples. What I'm trying to solve for is kinda the reverse: If I want Y results/resales between now and the EOY, what is the total number of X actions that need to occur, uniformly distributed across each day between now and the EOY? It's something like: (X * 0.48 * Days from today til EOY) + (X * 0.48 * Days from tomorrow til EOY) + ... (X * 0.48 * N days starting from today and approaching 0 at EOY) = Y Results by EOY. I feel like that's some kind of Sum from the max days to 0 kind of equation. I have a target Y "Results by EOY" value and would want to solve for X.
T = resale time period (180 days)
P = resale percent over time period T (48%)
t = time period of interest (90 days)
r = Pt/T is the portion of X that will be sold after time t has passed (so rX is total sales over the time period) (24%)
Y = expected sales at end of year
X = sales today
$Y = XPt/T $
so $X = YT/(Pt) = Y/r$