 Additional uses of the method of excess and deficit come from the following idea. So again, let's take a look at that problem from their way. And remember, we solved it by setting up our contribution rates and finding the excess or deficit. And as a system of equations, if we multiplied our first equation by 16 and the second by 11, then added, we were able to get the ratio of the cost to the number of persons. And what this means is the following. If we divide the sum of the cross products by the sum of the excess and deficit, we get the cost per person. And the important idea here is that this cost per person is what would make the excess and deficit 0. So let's consider the following problem. A vine grows downward at 7 inches per day from the top of the 90-inch high wall, while a melon grows upward at 10 inches per day, how long before they meet. To solve this as an excess and deficit problem, we have to figure out what the contribution rates are and what the excess and deficit corresponds to. And so one way we can do that is if our contribution rates are the days that the plants have grown and our excess and deficit is based on the distance apart, then we might be able to construct the following table. So let's consider that we have the number of days we've grown, the length of the vine and the melon, and the distance apart. So if they both grow for one day, the vine has length 7 and the melon has height 10, and so they are 73 inches apart. If they grow for two days, that's 14 inches for the length of the vine and 20 inches for the height of the melon, they're 56 inches apart. And while we could solve this problem as a double deficit problem, let's go ahead and find an actual excess and deficit. So if we keep growing for a couple more days, we see that after five days, the two plants are 5 inches apart, while after six days, the two plants have actually grown past each other by 12 inches. And so there's our excess and our deficit. And so we can set up our table. After five days, we have a deficit of 5, and after six days, we have an excess of 12. We have the sum of the cross-products. We have the sum of the excess and deficit. And the cost per person, this ratio between the sum of the cross-products and the sum of the excess and deficit, is going to be 90 divided by 17. That's 5 and 5 17ths. And the thing to remember here is that this is what's going to make the excess and deficit zero. So in the context of this problem, it's how long it takes before the two plants meet. This approach allows us to apply the method of excess and deficit to many other types of problems. Some of these problems look like but aren't quite exponential in nature. So for example, one problem, there is a wall 5 chi thick, two rats tunnel from opposite sides. On the first day, both rats tunnel 1 chi. Each day thereafter, the bigger rat doubles its rate, while the smaller rat halves its rate. And so, how many days until the two rats meet and how far has each tunneled? So again, we can think about our contribution rates as the number of days the rats have tunneled and the excess and deficit can be measured by how far apart the rats are. So on day one, both rats have tunneled at a rate of 1 chi per day. Both of them have tunneled 1 chi. And so, all together, they've tunneled 2 chi, which puts them 3 chi apart. On the second day, the bigger rat doubles its tunneling rate to 2, the smaller rat halves it to 1 half. So the bigger rat has tunneled a total distance of 1 plus 2, 3, while the smaller rat has tunneled a total distance of 1 plus 1 half, 1 and 1 half. And so, all together, they've tunneled a total distance of 4 and 1 half, and that puts them half a chi apart. That's our deficit. If we let them dig another day, on day three, the bigger rat has tunneled at a rate of 4 and gone a total distance of 7 while the smaller rat has tunneled at a rate of 1 quarter and gone a total distance of 1 and 3 quarters. So the total distance they've tunneled, 8 and 3 quarters, but since the wall is only 5 chi thick, that means they've actually gone past each other by 3 and 3 quarters, which is our excess. And so again, we'll find the sum of the cross products, the sum of the excess and deficit, the cost per day that's 9 divided by 4 and 1 quarter gives us, and this is the amount that makes the excess and deficit zero. It's the time to completion. Liao Hui also used the method of excess and deficit in a way we would recognize today as the method of double false position. We'll illustrate Liao Hui's approach by trying to find the side of a square with an area of 300. And again, we want to think about our excess and deficit and our contribution rates. And in this particular case, the contribution rates are the lengths. So let's consider a square with a side length of, oh, how about 10 has area 10 squared? That's 100. And since we want our area to be 300, that's actually a deficit of 200. And so a contribution rate of 10 gives us a deficit of 200. Now, if we take a bigger square, let's say a square with a side of 20, the area is 20 squared 400. And again, our target is 300. So that gives us an excess of 100. And so we can set up our table. We find the sum of the cross products. We find the sum of the excess and deficit and our cost per length. That quotient, 5,000 divided by 300, is going to be what makes the excess and deficit equal to zero. Well, not quite. We do observe that our result, 16 and 2 3rd squared, is 277 and 7 9th, which is not quite 300. But we can get a better answer by being closer to the actual values. Now, since we know that 16 is a little bit short, we might go to a square with a side length of 17. So a square with a side length of 17 has area 17 squared 289. That's a deficit of 11. If we go up to the next larger square, a square with a side length of 18 has area 18 squared 324. And that's an excess of 24. So again, we have our excess and deficit. We find the sum of the cross products. We find the sum of the excess and deficit. And again, that cost per length is going to make the excess and deficit zero. And to check our answer, we know that 17 and 11 35th squared is 2 99 and 961 12 25ths. And that gives us a very good approximation to the side of the square with area 300.