2 part percentage (%) problem

Question

Your company has a product called Version 1. At each upgrade or improvement the product is given a designator of .1, .2, etc, until Version 1.9 which, at the next improvement, the product is then called Version 2 and the process repeats with each improvement as before (V 2.1, V 2.2, V 2.3, etc).

That pattern repeats after each .9 designator. Thus the improvement after V 2.9 is called Version 3, after V 3.9 is Version 4 with no limit to how many versions can be created.

At each version upgrade, the product improves by 1% (one percent) making it 1% better, thus Version 1.1 is 1% better than Version 1, Version 1.2 is 1% better than Version 1.1, and so on.

Part One: How much better, percentage wise, is Version 9.9 compared to Version 1?

Part Two: What Version number would we arrive at when your company product is at least 900% better than Version 1?

Answer 1

Okay, remember this point - if the second is 1% better than the first, and the third is 1% better than the second, then the third is 2.01 percent better than the first.  (1.01 * 1.01 = 1.0201)

 

Now, to avoid having to do any excess carrying around of version numbers, I'm going to renumber the versions. Version 1.0 is now called Release 0, and the version N improvements after Release 0 is Release N.  (Divide N by 10 and add 1.0 to get back to the original version system) 

 

Version 9.9 is now Release 89.  Release 89 is (1.01) ^ 89 better than Release 0.  I make that somewhere around 242% as good, which means 142% better - use your own calculator to get the exact answer.  

 

To get your last answer, you're going to need to take the log to the base 1.01 of the number you're seeking (900 percent improvement over 100, which is 1000 percent or 10.  I make that out somewhere in the Release 230s, which would translate back to Version 24.R for some R.  Your calculator may vary.