 A rate is a ratio between two quantities. We use different words to describe rates. You can travel 50 feet per minute. You can dig one hole in two hours. You can answer five questions every eight minutes. Ratios can be expressed as rational expressions as long as we keep in mind units attached to quantities. For example, suppose Tom can paint a fence in three hours. Express this as a ratio and as a rational expression. So as a ratio, this is one fence in three hours. Well, that's not very exciting, but remember units attached to quantities. So we could also express as a ratio three hours for one fence. As a rational expression, this could be one fence divided by three hours. Also as a rational expression, three hours divided by one fence. We can solve ratio problems using a proportionality. This is the equality of two ratios where we express the ratios as rational expressions. For example, suppose Tom can paint a fence in three hours. How many fences can he paint in 15 hours? So proportionality claims two ratios are equal. This ratio of one fence in three hours can be expressed as one of the two rational expressions, one fence every three hours, or three hours per fence. We can use either one as long as we are consistent. So if X is the number of fences Tom paints, then the ratio X fences per 15 hours. It's got to be the same as, well, we have fences in the numerator hours in the denominator, so we'll use one fence per three hours as our equivalent rational expression. Then we can solve X 15s equals one third, and remember, you could eliminate the denominator by multiplying by it. In this case, we want to eliminate this denominator of 15, so let's multiply both sides by 15 and get our solution. So Tom can paint five fences in 15 hours. We can use proportionalities anytime we have a single rate, but sometimes there's more than one rate involved. If there's more than one rate involved, we need to combine them. And there are two ways of doing this. For example, suppose Tom can paint a fence in three hours while Ben can paint a fence in five hours. How long will it take both of them to paint a fence? Now the hard way of doing this is to note the following. Tom's rate is one fence in three hours, or one third. Ben's rate is one fence in five hours, or one fifth. So the combined rate will be one third plus one fifth, which is eight fifteenths. There's also an easy way. So notice that there are times of three hours and five hours. Now suppose they paint fences separately for three times five, fifteen hours. Since it takes Tom three hours to paint a fence, he'd finish five fences. We just solve that problem. Since it takes Ben five hours to paint a fence, then his rate is one fence per five hours, and in fifteen hours he'd paint, which we can solve, and find that he would finish three fences. So altogether, Tom and Ben will paint five plus three eight fences, and so their combined rate is eight fences in fifteen hours. To find the time to finish one fence, we can solve eight fences per fifteen hours. That's equal to one fence in however many hours it takes, and so we want to solve the equation eight fifteenths equals one divided by x. We can eliminate the denominator x by multiplying both sides by x. We can eliminate the denominator fifteen by multiplying both sides by fifteen. And this gives us, which we can solve, and so it takes fifteen eights hours for Tom and Ben to paint the fence. It's worth keeping in mind that we can express the ratio either way, either as eight fences per fifteen hours, or as, keeping the units with the numbers, fifteen hours per eight fences. Generally, one form will make the problem easier to solve. The important thing to remember here is that once you've chosen quantities for the numerator and denominator, be consistent. As a general rule, it's a little easier if we make our unknown denominator, and so this allows us to set up our problem as x hours for one fence. And we want to choose the form that has hours in the numerator, so this should equal fifteen hours per eight fences. So that's x divided by one equals fifteen eights, and we solve, well that was actually pretty easy, and we get the same answer. Nothing important changes if we have three or more rates. So for example, a pipe can fill a container in eight hours, a second pipe can fill the same container in five hours, and a third pipe can fill the container in 20 hours. How long will it take all three pipes to fill seven containers? So again, our times are eight, five, and twenty, so it's convenient if we suppose our pipes fill different containers for eight by five by twenty eight hundred hours. The first pipe can fill a container in eight hours, so it will fill 800 divided by eight or 100 containers. The second pipe can fill a container in five hours, so it will fill 800 divided by five or 160 containers, and the third pipe will fill containers in 20 hours, so it will fill 800 divided by 20 or 40 containers. So the three pipes together fill 100 plus 160 plus 40 or 300 containers, and so we can express the combined rate as 300 containers per 800 hours, or also 800 hours per 300 containers. We'll want to use one of these, but we're not sure which one, so we might as well put down both. Now the unknown time is for seven containers, and again it's easier if our numerator is the unknown, so that's some number of hours per seven containers, and so we'll want to use the form that has hours in the numerator and containers in the denominator. So our equation will be, and we can solve this, and get approximately 18.67 hours.