I am working on a personal project of which I have a motor that rotates at 100 RPMs and I would like to achieve the final gear in the sequence to rotate at 0.0694444 RPMs. Currently I have 10 gears, labeled 0 - 9. gears 0 and 9 are on their own separate axles, and gear 1 - 8 are on shared axles in the following pairs (1,2),(3,4),(5,6),(7,8). Below is a Google Drive link for picture of gear placement.
Google Drive link of Gear Placement
Using the formula: S1 * T1 = S2 * T2, where S1 is the RPM of the Drive gear, and T1 is the number of teeth of the drive gear, S2 is the RPM of the driven gear, and T2 is the number of teeth of the driven gear, I have been able to find the RPMs of gears 1 - 9, however I can't get the exact number of rotation needed to achieve the rate of 0.069444 for gear #9. The closest I have gotten is 0.069447302 RPMs, which is the same as saying I got 99.99582% out of a 100 %, when I need 100%. Below is a Google Drive link to a screen shot of an excel spreadsheet showing the automated calculations and values.
Google Drive link of Gear Results
On the excel, I have "Drive" next to gears 2, 4, 6, 8. I am calling those "Drive" gears because they are driving the connected gears, and they rotate at the same RPMs as the connected gears on their respected axle.
To recap my question. How many gears, and how many teeth per gear are needed to achieve the rate of 0.069444 RPMs for the final gear in the sequence with a motor that rotates at 100 RPMs?
I'm sorry of my question is a bit of a mess, or if I maybe unclear in some things. I am brand new in gear mechanics as of 2 days ago. I'll try to explain better if needed.
You can write: $$S_0T_0=S_1T_1\\S_2=S_1\\S_2T_2=S_3T_3\\...\\S_7=S_8\\S_8T_8=S_9T_9$$ When you write $S_9$ in terms of $S_0$ you get $$S_9=S_0\frac{T_0T_2T_4T_6T_8}{T_1T_3T_5T_7T_9}$$ As mentioned in the comments, the final RPM is $5/72$, so the fraction in the above equation has to be $5/7200$. You notice that gear 1 has 44 teeth, so $T_1=4\cdot 11$. Since there is no $11$ in the factors of the numerator, and $7200$ is not divisible by $11$, your fraction cannot ever be exactly $5/7200$.