Ask The Wizard #388
In this image, how much profit is the money changing business making?
For the benefit of others, let me explain the way to read the table. I assume this table is for how much the money exchanger is selling and buying foreign currencies relative to the Great British Pound (£), since that currency isn't listed. Let's look at the US dollar ($) numbers, for example. It is saying they will sell you £1 for $1.5085 and will sell you $1.2658 for £1 .
Assuming they make the same profit either way, the expected return, for lack of a better term, is the square root of the ratio of the smaller number to the larger number. In this case sqrt(1.2658/1.5085) = 91.60%. The difference from 100% is the money changer's profit, or the "house edge" to using a gambling term, which in this example is 8.40%.
Currency Exchange Expected Return and House Edge
There is a straight water pipe (blue) in the vicinity of points A and B. Point A is 2 miles from the closest point on the pipe. Point B is 3 miles from the closest point on the pipe. The two points along the pipe that mark the closest points to A and B are 5 miles apart. It is desired to lay two new pipes (red), linking A and B to the water-bearing pipe, having only one point of contact with the water pipe with the two new pipes going directly to A and B. In other words, the new pipes must form a V shape. What is the least distance of pipe required?
Here is my solution (PDF)
There is a straight water pipe (blue) in the vicinity of points A and B. Point A is 2 miles from the closest point on the pipe. Point B is 3 miles from the closest point on the pipe. These two points along the pipe that mark the closest points to A and B are 5 miles apart. It is desired to provide water to points A and B by laying new pipes linking points A and B to anywhere along the blue pipe. These new pipes may be in any form you wish. What is the least distance of new pipe needed?
The new pipes should form a Y shape, with one end leading to the existing water pipe and the other two ends leading to the two houses.
The point in (or in some cases out) of a triangle that minimizes the sum of the distances to each vertex is called the Fermat Point. I won't get into how to find it, but a property is that the lines it makes to the three vertexes form three 120° angles.
For more information, see the Wikipedia entry on the Fermat Point.
Here is my solution (PDF).