 So we'll begin with an example that serves more to illustrate the point than really to challenge your algebraic manipulation skills. We have two machines here and a very simple task. Machine A requires two seconds to complete this task. Machine B requires ten seconds. So we have execution times. So we'll be more interested in using the execution time part of the equation than the performance side of the equation. And we'll just say that our relative performance is, well, we'll put Machine A in for time one and Machine B in for time two. So we'll get ten seconds divided by two seconds. Our seconds will cancel. Ten divided by two is five. So we have a number here and this tells us something useful actually. One is its ratio. So we have no more units. If you try to solve one of these equations and you get something out with units, you've done something wrong. So we shouldn't expect to see units and we get a number. Well, we should expect to get a positive number and we have two positive times. But this number can be more or less than one. It can be large. It can be small. But one thing we'll notice is that if we get a number that is equal to one, that means that both of those two times were the same. So both of those machines were formed just as well. If we get a number smaller than one, though, that then means that one of those machines was worse than the other. If we get a number larger than one, then we have a machine. One machine is better than the other. What we want to know is which one is which. So as you can tell, Machine One runs this task in two seconds. Two seconds is clearly much, much better than taking ten seconds to run the same task. So what we're saying here is that Machine One runs this task five times faster than Machine Two. If you want to think of this in terms of a runner, a runner managed to take two seconds to run the hundred meter dash, whereas some normal person took a whole ten seconds. This person that managed the same time of two seconds was clearly much, much faster than the person that took ten seconds. So fast machine completes this task five times faster than the slower machine. So we can say that Machine A is five times faster than Machine B. There's nothing else terribly interesting about this, just that we realize Machine A is much, much faster than Machine B. But we can flip this problem around and ask, well, how much faster is Machine B than Machine A? So if we do this again, well, we'd say in this case, Machine B is number one. So Machine B takes ten seconds. So I'll put my ten seconds on the bottom and I'll put my two seconds on the top. Again, my seconds will cancel. Two divided by ten gives me one-fifth or 0.2. So this time I have a number that's clearly less than one. And that's telling me that this machine here is much slower than this machine. Well, that's pretty obvious. Ten seconds is clearly more time than two seconds. So it makes sense the Machine B gets a lower performance or relative to Machine A. So we could say that Machine B is 0.2 times as fast as Machine A. Now, since we've compared the same two machines, these statements need to be equivalent. And they are. In this case, we're saying that Machine A is five times faster than Machine B. Here we're saying that Machine B is one-fifth as fast as Machine A. And these are equivalent. You can also notice that these are reciprocals. We've inverted our fraction and the corresponding result is five versus one divided by five. So when you're doing one of these problems, if you get a result out that clearly does not make sense, Machine A was clearly faster than Machine B and you got a number saying 0.2. It's obvious you have your equation inverted. You can really just go back and take the reciprocal of your result and get the correct result out.