 So in this last question we have to go from a population density in the country to an average distance between people. Now normally when you say density you tend to mean a certain amount of something per unit volume. So I might talk about the density of water as being a thousand kilograms per cubic meter. But that would be a very unusual choice to talk about a population density because then you have to worry about how high people were stacked. Normally when you talk about a population density you're talking about the number of people in the country. And so you tend to talk about the number of people for a given unit of area. So it's people per unit area that describes your population density not people per unit volume. Alright so if we have a certain number of people per unit area how does the average distance between people scale? Well we already know that the area scales as L squared. What that means of course is that the length scales as the square root of the area. So if you take the square root of both sides there'll be some pre-factors in here they have to take the square root of as well. But the length will be proportional to the square root of the area times the square root of those pre-factors. So then we realize that if the area has gone down by a factor of 10 to the minus 6 then the length must have gone down by the square root of that. And since the square root of 10 to the minus 6 is 10 to the minus 3 what that means is that people are a thousand times further apart in the country than they are in the city. I figured out which way around put that just from common sense. Obviously if the density goes down then the distance between people on average is going to go up. Now just to remind you all of these things we could answer very quickly and the way you do it is you spot whether you need to convert between a length in an area a length in a volume or a volume in an area. So in the first case we saw that the amount of ink was basically proportional to the amount of area so if we double the point size we have to quadruple the ink. In the second case we saw that the paint was a matter of area but the wood was a matter of volume and so we knew we needed 10,000 times as much paint but a million times more wood. And for the population density we had to spot the population density was something per unit area and so that the length was going to go down as the square root of that. And once you get somewhat comfortable with these kinds of scaling arguments it becomes much easier to estimate things that are outside your experience by relating them to things that are in your experience.