Things I know:
- Need to make $\$165,000$ by the end of December to break even.
- Made $\$48,000$ so far this year (Jan-Apr).
- Made $\$12,500$ in April.
How would I figure out what my consistent growth would need to be to hit my break even point?
Doing a guess and check method I came up with $11.060176\%$. This means
- May = $\$13,882.52$
- June = $\$15,417.95$
- July = $\$17,123.21$
- August = $\$19,017.06$
- September = $\$21,120.38$
- October = $\$23,456.34$
- November = $\$26,050.65$
- December = $\$28,931.89$
I know the answer for this specific scenario is $11.060176\%$ growth month over month, but that was with a lot of guess and check. How would you do this in a formulaic way.
Thanks in advance.
Best Answer
It is unclear what problem you are trying to solve: do you want
To solve 2:
What you have is the standard geometric series $$br+br^2+br^3+br^{n}=br\frac{r^{n}-1}{r-1}$$ where $b$ is your zeroth value ($12500$ in Apr), $r$ is what you're solving for, and $n$ is the total number of time periods you have (May-Dec, so 8).
You need a total of $165000$ from these 8 months, so $$12500r\frac{r^8-1}{r-1}=165000\quad\Rightarrow\quad r^9-14.2r+13.2=0$$ which unfortunately isn't directly solvable in closed form, but will have a unique solution $r>1$.
You can pull out much faster numerical methods to solve this (Newton's method, etc). WolframAlpha gets $11.60175\%$ as expected.