In order to understand this correctly, we need to think carefully about what all the numbers here mean. Let us go through them all.
You have a population of 230 million adults in the US that are not in institutions (and presumably not in the military). They are your total pool of possible workers.
You have 154.1 million adults in your labor force. You correctly did the math to determine that number. These are people who are either A) working or B) wanting to work but unable to find a job.
If you subtract 154.1 million from 230 million, you get 75.9 million adults who are not part of the labor force. These people are not working, but are not looking for paid work either. They are most likely to be either retired people or parents staying home with children. Keep in mind that these people are not working but are also not unemployed.
If you have 85 million who are not working, and 75.9 million who are not working and not unemployed, the difference between those numbers must be the number of unemployed people. So, 85 – 75.9 gets you an answer of 9.1 million unemployed people.
To find your unemployment rate, you divide the number of unemployed people by the number of people in the work force. 9.1/154.1 = 5.9%
This tells you that your unemployment rate in this scenario is 5.9%.
The major thing to remember here is that not everyone who is not working is unemployed. You are only unemployed if you want to work but cannot find work. Another way to put this is that you have to be in the workforce to be unemployed.