 Good day. This is Mr. Beard. I've put together a couple videos to support the basic math techniques. You see here I'm calling these practice problems for basic math techniques. These practice problems are from your text, the basic laboratory method for biotechnology, second edition by Sideman and Moore from chapter 13. And the first problems, the example problems we're going to look at are proportions. So on page 232 we have this problem. If there are about 100 paramecia in a 20 milliliter water sample, then about how many paramecia would be found in 10 to the third mil of this water? And we're going to look at a couple ways to, or strategies to solve this problem. First of all, let's look at the 10 to the third milliliters, right? You know, 10 to the 3 milliliters, how many is that? Remember when we talked about exponents, 10 to the 3 is equivalent to 10 times 10 times 10, which is equal to 1,000. So that's the question is if there are 120 mills, about how many are going to be in 1,000 milliliters? So strategy one is the proportional method where we can say 100 paramecia in 20 mills is proportional to how many paramecia in 1,000 milliliters. So we have this proportional setup. Remember to solve it, we can cross multiply, right? So we can say 100 paramecia times 1,000 mL is equal to 20 mL times how many paramecia. So go down and rewrite this to solve for the question mark. We're going to divide both sides by 20 mL. So we're going to rewrite this to find our question. Now, since these units cancel out, we can do our multiplication and say 100 times 1,000, right? We're just going to give us 100,000 paramecia divided by 20 is going to equal 5,000 paramecia. That was strategy one, right? So strategy two is just going to look at it as saying how many first, how many in 1 mL and then basically multiply it by 500, right? So if we have 100 paramecia in 20 mLs, right? So if we just divide by 20, right? Then we get what? 5 paramecia per mL. I'll put one for 1 mL divided by 1 mL. So that equals 5 per 1 mL. So if there is 1,000 mLs, remember from 10 to the third mLs, 0 to 1,000 mLs, right? So 5 paramecia, 1,000 mL, paramecia per mL, keep track of our unit, gives us 5,000 paramecia. So two ways to get there to sort out how many paramecia in 1,000 mLs if there are 120 mLs. Okay, let's go on and here we have a second problem. Before I proceed to the second problem, let me correct an error I made when I started that these are actually from chapter 14 of Sideman instead of originally said 13. So this is actually, this problem, this example problem comes from page 230. And it's actually two problems, first part and then the part here. So we're going to look at the first part first and problem states, a transgenic animal is one that produces a protein or trait from another species as a result of incorporating a foreign gene into its genome. In 1993, transgenic sheep were born that secrete a human protein, AAT, into their milk. AAT is valuable in the treatment of emphysema. A sheep can produce 400 liters of milk each year. If a transgenic sheep secrete 15 grams of AAT in each liter of milk she produces, how many grams of AAT can this sheep produce in a year? Once again, we can look at this from two strategies. And the first way is looking at if there are 15 grams per liter times 400 liters, that's how many she makes in a year. So that would be 400 liters we'll cancel out. So it's 400 times 15 grams or 6,000 grams. Now the other way we can do it is just in the proportions that we've looked at. So we say 15 grams per one liter is equal to how many in 400 liters. And we cross-multiply, so we get 15 grams times 400 liters is equal to one liter, right, times, unknown grams. And then once again, we put 15 grams times 400 liters over the one liter is equal to our question mark. So liters are going to cancel out and basically it's one divided into 15 grams times 400 is what's the same answer, 6,000 grams. So this is 6,000 grams per sheep per year. So we go back to the second part of that. If AAT is worth $110 per gram, what is the value of a year's production of AAT? So if it's $110 per gram and the year supply is 6,000 grams, then once again there's two strategies, right? We can just say $110 per gram times the 6,000 grams is going to give us $660,000, right? In a year, remember the grams are going to cancel out so we run it with $6,660,000. Sorry about that. And the other way is our, once again, our proportional cross-multiplication method. So we have our $110 per one gram is equal to how many dollars for our 6,000 grams. So once again we can cross-multiply, right? So we get $110,6,000 is equal to one gram times some number of dollars. Okay, let me erase that, let me rewrite that. $110,6,000 grams is equal to one question mark. So once again we're going to divide each side by one gram. We have $110 times 6,000 grams by one gram is equal to some question mark. So that once again boils down to the same $110 times 6,000 is equal to $660,000. So both these questions have been dealing with proportions. And I say most of these have the two strategies depending on how you want to look at them. I've got a couple kind of little problems here that I'll just call common. I mean, these are not biotechnology problems per se related to proportions. But sometimes students get, you know, when we start talking about transgenic animals and grams and liters, they get thrown off by the units and, you know, it lets them obscure the process. So let me do a couple problems here in kind of your more common everyday lingo. And maybe you'll feel better that, oh, well, I already know how this works. Let's say your car gets 30 miles per gallon, right? And you need to go to the gas station and get some gas because you're going to go to the beach and that is 200 miles away. And so your question is how much gas, how many gallons are you going to buy? Right? So if you get 30 miles or one gallon, how much gas is it going to take for 200 miles? Right? So once again, we cross-multiply. So one gallon times 200 miles is equal to so many gallons times 30 miles. So we're going to divide both sides by 30 miles, right? Solving for the question mark. So the miles are going to cancel out. So we get 200 times 1 or 200, right? Divided by 30, right? Which is going to equal 6.67 gallons. So you have to go to the gas station and get about 7 gallons of gas. Now carrying on that logic, let's say that gas costs you $3.45 per gallon. And I'm not going to say the .9. I'm just going to say it's $3.45 per gallon. And let's say your vehicle has a 20 gallon capacity or 20 gallon tank. And say you're dead empty. How much is it going to cost you to go buy that 20 gallons of gas? So I mean, you've done this in your head without doing proportions. You just say that's $3.45 per gallon times 20 gallons is $69. Now you can set it up as a proportion and say $3.45 per gallon is equal to some amount for 20 gallons, right? Cross multiply, right? So you get 3.45 times 20. Question mark, that's 3.45 times 20 gallons divided by one gallon is equal to your question mark. The gallon unit is going to cancel out. So you get 3.45 times 20 is equal to $69. So you do proportions anytime you think about what you're buying at the service station. Just in the biotechnology lab, we have milligrams per liter and those type of things. So the units are a little bit foreign, but the math really isn't that far. So when you're doing one of these problems, see what the problem gives you, what values you're given, like the 3.45 a gallon and 20 gallons, and then sort out what the problem is asking you for and how you need to solve it. So the best way of getting comfortable with these, as I said before, is just working through the problem. So it would work through as many of the sample problems in this section of the chapter as you can. The text in the back of the text gives you the answer. But work the problem, see if you got them right. Don't work backward from the answer and think that you understand the process when you see how they got the answer. Make sure you feel comfortable about going through the process and getting that answer.