Dr. Sheperd, a wonderful prof who is now retired from SDSU, showed me this neat trick (that I think that he came up with himself!) to get all the solutions to equal the same exact volume in each test tube. Simply use this equation & simplify using algebra (Ah Hah- Something you learned from high school math is applicable!):
X/ (X + Y) = Z
Well, I'm not trying to be a spoiler but it's hardly a trick. This equation is how it works, purely. The diluted factor represent the concentration of the broth in its diluted solution, simple as that.
I haven't come across that kind of exercise before but I assume it requires us to do 2 things. (1) dilute the given solution to a required stock solution (2x10^7 copies/ul) and (2) find a diluted factor to make a serial dilution. The first part should be easy for you, i suppose. Second part is somehow troublesome. So here the trick (real trick) that would help if you are not a fan of algebra. This trick is extremely helpful if you work in a lab and want to diluted some solutions to a concentration needed. This trick works with any kind of concentration (%, mol/ml. ug/ml, copies/ml, cells/ml...)
Lets say you have a solution that have A concentration, and you want to dilute it to B concentration (A>B of course). The solvent denoted as C which is generally water (but you can dilute with less concentrated solution too).
Draw a diagram as my attachment:
You could see that the result come up instantly (feel free to double check it using algebra lol). It's too simple to explain, yet too hard to prove.
Back to your problem, I was using your problem as example. As you continue to do it on and on with next dilutions, dilution factor 1/10 (ratio 1/9) is repeated in pattern.
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Somebody may feel hard to believe if this trick works but it does, check it if you want to. In routine work, it would save us lots of time, especially in labs where we have to dilute solution almost every days.
It also come in handy if you want to concentrate a very diluted solution to a more concentrated solution. For example, if you accidentally diluted a DNA solution to 2x10^3 copies/ul instead of 2x10^4 copies/ul, and you have some stock (2x10^7) left. In that case, A would be 2x10^7, B=2x10^4, C=2x10^3.