 Today we're going to talk about ratios and proportions. Ratios are the mathematical relationship between two numbers. It's usually expressed as A to B, or written A colon B. And so for example, if we had eight bears and five elephants, the ratio of bears to elephants would be eight to five, and the ratio of elephants to bears would be five to eight. So a proportion is related to a ratio, but not quite the same. A proportion compares a specific quantity to the whole. So for example, if we use those same bears and elephants, and we compare the number of bears to the number of animals, we have a proportion. So there are eight bears, and there are 13 animals. This is our proportion of bears to animals. If we have a number of elephants to a number of animals, we have five to 13. So you can see how you can set up our proportion if you have a ratio. So because of what we just saw in terms of proportion, it might have looked familiar to you. It might have looked like a fraction. So if we have our bear proportion, eight to 13, we can express that as a percent. And this is just a reminder of ways to work with percentages and work with fractions and decimals. We have eight out of 13. We're going to convert that to a decimal, and then we're going to convert that to a percent. And so we do that by moving the decimal point. And what we have is that 62% of the animals are bears. And so you can use this original proportion, which you got from your original ratios, to determine the percentage of animals that are bears. So when you're cross-multiplying to solve proportions, it can really help you to pay attention to the units. And it's always very important to pay attention to units so that you get the correct answer, but it can also help you move through the problem. So we're going to work to solve this question. We're going to work it in two different ways, and they're both correct. You'll see that we get the same answer using both. So whatever you're more comfortable with, whatever makes the most sense to you, is the way you want to solve these proportions. But I will suggest that you continue to write the units as you set up the proportions and work through the problem, because it'll help you keep track. So our question here is if there are 10 to the third bacteria in 10 mLs, how many bacteria are in one liter? If we're going to directly solve it, we just set up our proportion, 10 to the third bacteria in 10 mLs is equal to how many bacteria in one liter? So you're going to cross-multiply 10 to the third bacteria times one liter. And then I'm going to put in a conversion factor, because we have mLs on the other side, and we want to make sure that we know how to get between the two. So there's 1,000 mLs in one liter. And then you've got X bacteria in 10 mLs. And so if you multiply this out, you'll get 10 to the sixth, the liters cancel, bacteria mLs equals 10X bacteria mLs. And you'll divide by 10. So you'll get 10 to the fifth, you'll divide by the 10 mLs, so the mLs will cancel. And you'll get 10 to the fifth bacteria equals... So that answers our question, how many bacteria are in one liter? 10 to the fifth. So here's another way of solving that same proportion. We can set up our proportion of 10 to the third bacteria in 10 mLs gives us how many bacteria in one mL so that we're going to determine how many bacteria are in one milliliter, and then we'll use that to get us to one liter by multiplying by 1,000. So again, 10 to the third bacteria mLs is equal to 10X bacteria mLs. We divide by the 10 mLs, and so that'll get rid of that. And you just... I'll show you over here that I am doing this so that we are left with X bacteria. 10 to the second bacteria is equal to X. So then we're going to have to use that, and I'll add a page here. 10 to the second bacteria in one mL times 1,000 mLs. 10 stuff to work in here. Now let's start over. Times 1,000 mLs for one liter gives us 10 to the fifth bacteria in one liter. So you see we got the same answer. It's just a different way. If you don't want to use your unit conversions right in your proportions, it's a different way of thinking about it and a different way of solving. But because you write the units out, you can keep track of where you are and whether you've done the right steps to answer your problem.