The free space equation

The idea is: if you have some resource which is partially used, and you increase the total capacity, how much does your free capacity grow in relation to your increase in total capacity? ChatGPT couldn't give me a name for this idea, so I'm calling it the free space equation.

For example, if you are thinking of adding a second hard disk the same size as your current one, and your current one is 80% used, then after adding the second disk your free space will increase from 20% to 60%. You've doubled your total capacity, but tripled your free space.

Or, if your car takes up 90% of the width of your driveway, you can double the amount of free space around your car by increasing the width of your driveway by only 10%.

Once you start noticing it, this comes up all over the place.

A king-size bed is only ~11% wider than a double bed, but if you and your partner are using up 80% of the width of your double bed, then changing to a king-size bed gets you about 50% more free space.

If you're spending 95% of your earnings, then increasing your earnings by 10% gets you 3x as much disposable income (modulo tax).

If you expect a journey to take 1 hour, and you only allow yourself 1 hour and 5 minutes, then you expect to use ~92% of the time available. If you give yourself an extra 5 minutes then you increase the total time dedicated to the journey by ~8%, but you double your margin of error. You're much less likely to be late.

This chart shows the ratio of marginal increase in free space to the proportional increase in total size:

The function is y = 1 / (1 - x), where x is your current used proportion (on a scale of 0 to 1), and y is the ratio of free space increase to total size increase.

You can see that you get a larger percentage increase in the size of your free space if your free space is a smaller percentage of the total space.

Pretty uncontroversial, but it's a useful way to think about the relationship between free space and total space.