 Specifically, this section will contain segments on ratios, proportion, and percent. You hear a set of related topics that are all key basic understanding elements of lab math. They're mathematical concepts and procedures that are used frequently in the laboratory. So, let's start first in our discussion of ratios. A ratio is a mathematical relationship to numbers. And usually we'll say if the numbers are say A and B, we'll express the ratio as A to B. And let's say we had a bowl of apples and oranges and we have three apples and five. Okay, we could express this as a ratio of apples to oranges as three to five. We can also write that with a colon. Or in terms of oranges to apples, we could say it's five to three. Of course, we have to state that this is apples to oranges and this is oranges to apples. And that's a relationship of oranges to apples in this case. So, that's a ratio. Additionally, the ratio of apples, say to the total amount, would have a ratio of three to A. Where this is the apples to total fruit, would be a three to eight ratio or three of A. So, three apples out of a total. So, three apples, eight total fruit. Okay, so in this example, if we just had apples and oranges in there, so we had just the eight total fruit, right? Then in this ratio of three apples to total fruit, we have established a proportion, right? So, the proportion, right, is a comparison of a specific quantity to the whole, right? So, in this case, our apples, a specific quantity of three to the whole of A. And we often express these as percentages, right? So, three eight, right? Or three divided by eight is equal to 0.375, right? Or if we do that times 100%, it's 37.5%. So, our proportion of apples in the bowl is 37.5%. Now, in the examples we've been working with, you know, our units in this case are all fruit. Apples are fruit, oranges are fruit. So, we've been working with the same type of unit. In the lab, we often kind of use this same notion of a proportion to items of different types of units. For instance, if we may say if we're making a cake, we may have two cups of flour in the whole of a cake. Now, in this case, this proportion, we really can't do a percent because flour and cake, you know, are not the same units. We have the other components of the cake, the sugar and eggs and milk, you know, an assortment of things that are not of the same material. So, it's a little bit different situation than the pure proportion we're talking about. But it's useful in the lab, for instance, if this is one cake, what if we're wanting to make three cakes, right? Well, it's pretty easy to see from this that for three cakes, it's going to take, what, six cups of... Because here, in this proportion or ratio of two cups of flour per one cake, right, we're keeping the same relationship when we have six cups of flour to three cakes, so we're saying that these proportions or ratios are equivalent. And this is a useful technique if you're making cakes, if you want to make more cakes and you have your basic components, you see that you just proportionately increase them in this case times three. If you did this for the sugar and the eggs, likewise, you'd be multiplying them by three. Okay, so that example was straightforward enough. Hopefully, we can just see that without calculating it. But let's do another example here. And let's say, for instance, for our cake, I guess we're going to make an orange cake. So, we want to put eight ounces of orange juice in our cake. Our recipe calls for eight ounces of orange juice, but we're going to a big gathering and we want to make five cakes. And we want to know how much orange juice do we want to use for five cakes. Now, it's very necessary to keep track of units. In this case, our units are ounces of orange juice and our number of cakes. And in a proportional equation like this, the denominators have to be the same unit. In this case, the cakes. You couldn't write this as five cakes over some amount of orange juice. It has to be the same units in the denominator. So, the way we're going to solve for this is to cross-multiply. So, we have eight ounces of orange juice for five cakes over one cake are unknown. So, here we have an denominator like cakes like. So, that will cancel. So, we wind up then with five cakes at eight ounces per cake. So, we didn't need 40 ounces of orange juice cakes. You see, these were over here, these were unlike units orange juice and cakes, but we had the like unit in the denominator. So, we have to have the like unit in the denominators of these proportional relationships so that when we cross-multiply, we can solve for that denominator. So, we can express then this is eight ounces of OJ per cake as per, eight ounces per cake. Let's look at a kind of laboratory example here. And let's say that we have 10 to the third bacteria per 10 mils of bacteriological media, right? And so, our question here is at this ratio or proportion of bacteria to media, if we had one liter, right, we had one liter of this suspension of bacteria, then how many bacteria would we have? So, we can say 10 to the third bacteria per 10 mil is equal to how many bacteria per one liter. And so, then if we cross-multiply, we've got 10 to the third bacteria times one liter over milliliters times our own bacteria. Ah, now, what do we say we needed the same units in the denominators? Well, 10 milliliters is not the same units as liters, but they are related. So, from our discussion on metric volume relationship, we know that there are 1,000 milliliters in a liter. So, we can say that this is 1,000 milliliters, right? So now, we have white units that we can divide so that we know 10 to the third times then 10 into 1,000 would give us 100, right? Which is going to give us then 10 to the fifth bacteria, right? How many bacteria 10 to the fifth bacteria? Okay, so let's call that strategy one. Let's call this strategy two. Now, strategy one is perfectly fine. I'm just going to show this alternative because some folks, when they solve these types of proportional ratios, will like to take it to a smaller unit and then go up. So, that person would take the same problem, 10 to the third bacteria for 10 milliliters, and they would say how many in 1 ml, right? So, that their equation 10 to the third times 1 ml divided by 10 ml times question. So, 1 divided by 10 is 0.1, so we would have 10 to the square bacteria per 1 ml. Then you have to say, okay, in the liter, we have 1,000 ml. So, we have to take this 10 to the square bacteria, 1,000 ml is going to give 10 to the fifth bacteria, right? Same answer. There's two ways of getting there. Directly in the first one, we're using the 10 ml in the liter. There again, we have to know as here that there are 1,000 ml in the liter. In strategy two, this strategy just takes the ratio down to the smaller unit, 1 ml unit, and then it in has to deal once again with the 1,000. Two strategies, either one is fine depending on how your mind works. So, this brings us around then to our kind of third topic of percent. Now, if we go all the way back up here, right, we're talking about our apples and oranges, and we had that we had three apples out of eight total fruit, and we said that was 37.5%. So, we've already talked about percent. We saw, so a percent, which has this symbol, right, mean of every hundred, of every hundred, right? Program into your brain that when you see percentages and you see that symbol, that relationship is of every hundred. Let's go back to our apples. So, we're just talking about apples. So, we bought some apples, we bought 200 apples, and when we went through them, we found that 18 of them were right. And so, we got 18 rotten apples out of 200 apples, right? So, to convert this information to a percent, it's important to remember, right, that this is out of every hundred. So, if we had 18 out of 200, 18 out of 200, how many out of 100? So, right, if we could then cross multiply 18 times 100, right, divided by 200, 1800 divided by 200 is equal to 9. So, we can say 9 out of 100 are rotten. So, that's 9%. So, we can generalize this by saying the number with a characteristic out of a total number times 100%. So, in our case, back over here, if we had 18 out of 200, right, times 100%, that would be 0.09 times 100 is equal to 9%, clean 9%. Now, let us notice once again that the units here, right, in both the numerator and denominator are the same. They're apples to apples, right? So, they have to be comparable units. So, in the laboratory, one really big use of the percent are in percent solutions. You're often going to see instructions, SOPs, procedures, recipes to make up solutions that are percent. So, let's say for instance that our instruction tells us that we want a 15% solution of ethanol. And we'll say that that's in water. And let's say that we want 500 ml of that solution. So, how do we determine how much ethanol that we're going to have in 15 ml? So, when we say percentage, take us back to 100. So, it's 15%, and that's 15 ml of ethanol per 100 ml of solution. And we're wanting 500 ml, and we want to know how much ethanol. So, 15 ml 500 ml over 100 ml is us. There's 100 into 5 is 5. So, we have 15 ml times 5 is 75 ml of ethanol, right, of 500 ml of solution. So, we're going to take 75 ml of ethanol and bring to volume of 500 ml. So, we're going to have to add 425 ml of water, right, to give us 500 ml of a 15% ethanol solution. Okay, so that was strategy one, which is our cross multiplication method. Taking it to 100. Nothing wrong with that. And alternative is a strategy two. Okay, in this case, if we know that 15% is equivalent to the decimal form of 0.15, we can just take that 0.15, multiply it by 500 ml. That's telling us that 15% of 500 ml is equal to what? Again, 75 ml of ethanol, bring to volume 500 ml to give us 500 ml of a 15% ethanol solution. So, that concludes this session on ratio proportions and percentages. We'll have some follow-up sessions on some specific rules for manipulating percentages and some specific practice application.