 So we're now going to talk about ratios and proportions, starting first with ratios and a definition. So a ratio is just a comparison between two quantities, an example that you see on a daily basis, 50 miles per hour. So I can actually rewrite 50 miles per hour as 50 miles over or per one hour. So I'm now comparing miles to hours. So let's look at a couple of examples. A laboratory centrifuge is spinning at 50,000 RPM. First off, what does RPM stand for? Revolutions per minute. So I can take the 50,000 RPM and change that into a fraction. So 50,000 revolutions per one minute. Can always add a one in the denominator that doesn't change anything because 50,000 divided by one is still 50,000. So you're not changing anything by doing that. And so that's an example of a ratio. Another example, 43% of the class is male. Comparing what percent means, percent means out of 100, I can change the 43% to a fraction easily by writing it as 43 over 100. But what do those two numbers really represent? 43 is going to represent the number of males. So for every 43 males in the class, the 100 represents the total number of students. There'll be 100 total students. So we're comparing the males in the class to the total number of students in the class. One more example, a solution requires 50 milligrams of sodium chloride for each 0.5 liters of solution. So again, we can easily set up a ratio. We're comparing sodium chloride to the solution. So 50 milligrams of sodium chloride for every 0.5 liters of solution. So again, a ratio is nothing more than comparing two different quantities. And that will lead us right into a proportion, which we'll be using a lot. Proportion. So nothing more than when two ratios are set equal to each other. For example, I have two ratios here. It took me three eggs to bake one cake. And then I want us to see if we can figure out this missing piece. How many eggs would it take us to bake three cakes? Well obviously it's going to take you nine. So again, we have two ratios, three eggs for one cake, and then over here, nine eggs for three cakes. I want you to notice something really important about this proportion that I've set up. If you were to do what's called cross-multiplying, which means you're going to multiply the top of one fraction by the denominator at the bottom of the other, three times three would give you nine, and then if you look at the other side, one times nine would also give you nine. So something that's always true about proportions is that when you cross-multiply they should be equal to each other. So let's try just a simple example with that. How to solve a proportion, and then we'll look at some actual application problems. So to solve this proportion, first off I have a ratio on the left, a ratio on the right, and they're set equal to each other. So that's the definition of a proportion. So what you're going to do is simply cross-multiply. So four times three, and you're going to set it equal to nine times x. Now remember when you're solving, you always want to simplify each side of the equation before you try to actually solve for the variable. So on the left side, four times three of course would be twelve, and on the right side, nine times x can be rewritten as just simply nine x, means the same thing. Now to solve this, of course I'm trying to get the x by itself. Since the nine and the x are being multiplied, I'm going to do the opposite, which is division. So I'm going to divide both sides by nine. So I end up with twelve over nine equals x, twelve over nine will reduce, three will go into both of them. So three will go into twelve, four times, and three will go into nine, three times. So my answer actually ends up being four-thirds. So again to solve a proportion you just cross-multiply. So let's look at some examples dealing with proportions. If one bag of chips contains three ounces, how many ounces are contained in 4.5 bags of chips? So I would start with the first piece of information given that comes before the comma to start with. So I have one bag of chips, so one bag, and it contains three ounces. So for every one bag I'd have three ounces, a ratio. And then we're going to set it equal so that we can create a proportion. And then let's go after the comma. How many ounces, so I'm looking for the number of ounces. Notice the way that I have this set up, the bags have to go in the numerator, the ounces have to go in the denominator. You have to be consistent, the units have to match up. So I'm looking for the number of ounces, I don't know the ounces so I'm going to put x in the denominator, which would stand for x ounces. And then it says they're contained in 4.5 bags, the 4.5 bags of course would have to go in the top because that's where bags are located. So again with proportions it's extremely important that the units match up. I could have written this three ounces over one bag, if I had done that then ounces would have to be on top here and bags would have to have been in the denominator here. So again make sure that your units are matched up. And then this becomes a simple problem. To solve a proportion you simply cross multiply. And it doesn't matter if I start with the one or the three first. So one bag times x and then bring your equal sign down. And then three ounces times 4.5 bags. So in this particular case since we have lots of units involved what I'm going to do is go ahead and solve for x and then we'll work with the units. So on the left side we have x and one bag is being multiplied by x. Of course to get rid of that one bag I'm going to divide by sides by one bag and let's see what happens. First off the one bag on top here cancels with the one bag on the bottom and we're left with x equals bring that down with you. And then lots of things are going to happen on this other side. First off the bag unit cancels out and then we see there's some simplifying that we can do. So remember when you're dealing with multiplication and fractions you do the top and of course in the bottom. So let's start with the top. Three times 4.5 would give us 13.5 and in the numerator we still have a unit of ounces. And in the denominator we still have one. There's still something else you can do and that's just to remember that anything divided by one stays the same. So 13.5 divided by one is still 13.5 so this is really 13.5 ounces. So what have we found? We have found that in 4.5 bags of chips you will have 13.5 ounces. Our second example if there are about one times 10 to the second blood cells in a one times 10 to the negative two milliliter sample about how many blood cells would be in one milliliters of this blood. So starting with what comes before the common, let's see if we can come up with the two things that we're comparing. Obviously we're comparing blood cells to milliliters. So I'm going to go ahead and start my first ratio with blood cells in the numerator and milliliters in the denominator. So how many blood cells did we have? Well it said one times 10 to, well one times 10 to the two is the same thing as 10 to the second power. I mean the same thing because one times anything of course is going to be itself. With milliliters we know that's one times 10 to the negative two. Again this is the same thing as 10 to the negative two. Because one times anything of course is itself. So we have 10 to the second blood cells over 10 to the negative two milliliters. We're setting up a proportion so we're going to set it equal to another ratio. So looking at what comes after the comma about how many blood cells. So we're looking for blood cells so I'm going to let that be x. Of course that has to be in the numerator because in the numerator of the other fraction we had blood cells would be in one milliliter of this blood. So since milliliters are in the bottom I'm going to put one milliliter in the bottom. So we have our proportion, we have our setup, and we're ready to cross multiply. And this time I'm going to start in the denominator to do the multiplication. Doesn't matter if I start with the top or the bottom. But 10 to the negative two milliliters times x. And we're going to set that equal and then do our other cross multiply in here. 10 to the second blood cells times one milliliter. And then of course we're going to solve for x. And remember again to solve for x here since 10 to the negative two is being multiplied I'm going to divide. So divide by 10 to the negative two milliliters on the left and the right. And then let's see the things that will cancel out. On the left side of course 10 to the negative two milliliters will cancel. And I'm simply left with x. On the right side milliliters will cancel. So on the right side we're left with 10 to the second blood cells times one. Which is still going to be 10 to the second blood cells. And in the denominator we're left with 10 to the negative two. And this is a really good example to help you review exponent rules. So we can still simplify this a step further. Focusing on just the 10 to the second and 10 to the negative two. If you remember when you are dividing like basis and these are like basis because we have 10 for both of the bases. You keep the base of 10. And then remember you're going to subtract the exponents top minus bottom. So two minus a negative two. And the units are still blood cells. So keeping our base of 10 then we're going to simplify those exponents. Remember a negative and a negative course here. I'm going to make a positive. So two plus two becomes four. So my answer becomes 10 to the fourth blood cells. So if I have one milliliter of this blood I'll end up with 10 to the fourth blood cells. So this sums up proportions.