 Life often presents us with choices. Now, sometimes the choices, although are hard, it's not such a big deal when we choose. I mean like Pokemon sword versus Pokemon shield. Which one do I get? You know, in the end, it's probably not gonna make much of a difference. But in some situations, particularly when money is involved, the different choices, they're sort of a clear-cut best choice, right? You know, if I do this, I have these two options, right? They'll give me the thing I want. But it's one cheaper than the other, right? We can often use some type of quantitative analysis to make decisions about what's the best thing to do with respect to money and other quantitative variables of some kind. So what I want to do right here is in this example, I want to present you a hypothetical situation where an individual has to make a purchasing choice and clearly the individual wants to make the best choice so that he can save the most money. So he has two options to do and which one's gonna be the best one for him. So this is the situation. You need to buy chicken for dinner tonight, right? And so let's say we're in a situation where it's like, oh, you started making the food, you threw the marinades and the seasonings or the sauces, whatever together. It's like, oh, we start the meal then turns out we don't have enough chicken to feed everyone. Therefore, we really can't stop making it because then all the ingredients we use so far would go to waste. So we'll just make a quick run to the store and purchase some chicken. Well, you remember noticing an ad for chicken on sale at the grocery store that's across town. They're selling their chicken for two dollars and ninety-nine cents per pound. While the grocery store that's really close to you and you usually shop at, you remember, they sell chicken usually about three dollars and fifty-nine cents a pound. So the other store has cheaper chicken, but it is farther away. So there's sort of like these competing forces going on, cheaper chicken, but the farther you travel, the more fuel costs you'd have to spend and there's also usage of time right there. And so what is the best option? What's the best option for for you or for this hypothetical person? Well, we have to actually collect some data first before we can really answer that question because if we just look at the surface value, it's like, well, the other store is cheaper, so maybe we go there. But let's actually stop and think about it. Some questions we might ask ourselves to help us decide whether this is the right choice or not. Which choice is the right choice? Well, first of all, how much chicken will we actually be buying, right? And so with the amount of chicken we already have and the amount of people we have to feed and looking at what the recipe demands, we realize that we still need five pounds of chicken. So we're going to go to the store and buy five pounds of chicken. Well, how far are the two stores away? Well, the local store that you usually buy, I mean, they're both local inside the town here, but the store you usually go to is 2.2 miles away. And that usually takes about nine minutes of driving where it's 2.2 miles one way. But total, it's about nine minutes of driving there and come back, right? The other store, it is 8.2 miles away. And I do love, I love on this problem how the store is across town, but then it's only 8.2 miles away. You know, it tells you how big the town is when you say things like this. That's fine. So it's 8.2 miles away and around trip would take about 26 minutes to drive there and back with the speed limits and things like that. So we have that information. OK, well, does distance cost me anything? Well, not directly, right? When you drive your car, let's just assume the individual is driving a car right here. There's fuel cost, right? Your typical combustion engine will consume gasoline based upon how far it travels. And then there is a difference between driving in the city and driving on the freeway and things like that. So we're just doing city driving here. So what kind of mileage does your car get? It gets 23 miles per gallon when you drive through the city. And so again, that's not exactly the cost, right? There's going to the cost is going to be coming from the gasoline. So we have to somehow convert from miles to gasoline. But then gasoline has the cost. We know our car can hold about 13 gallons when it's full. And the current rate of gas for this town would be $3 and 81 cents per gallon. So this kind of is some data we've collected that's going to be pertinent to the problem at hand. What is the cheaper option? The close store or the far store when it comes to buying this chicken? So we actually want to sit down and think about as we as we look at this problem, what are the actual costs? And so what we're doing right now is essentially a modeling problem. We're trying to model using functions, whether we realize it or not. If if you were actually thinking about this problem, whether you actually come up with here's my linear function, or you just came up with some numbers and crunch them together, we are modeling using functions that are cognizance of that doesn't actually matter here. So some things to consider is there's going to be the cost of the chicken, right? So the cost of chicken. How are we supposed to determine the cost of the chicken? Well, the cost of the chicken is going to be determined by which store we go to, right? So we can think of in the following way, we're going to buy five pounds of chicken that is going to be fixed. And then what is somewhat variable here is our variable X, which will say that X is going to equal the price per pounds of chicken here. Now, one store, we see that the X value, you know, so for the for the close store, we see that X is going to equal $3.49, so you're going to write this way. And so this is of course, if we go to our usual store. On the other hand, if we go to the farther store with the store across town, in that situation, our X value is going to equal just $2.99. So this variable depends on which of the two options we're going to take. Great. That's part of the function. But another important part of the function is going to be the cost of travel, traveling to the store, right? So as we travel, what what's going to be affecting this? Well, some things we can see here, that'll be the same as that the distance we travel is going to make a difference, right? So one store is one distance, one store is the other distance. And we can analyze that real quick. So the locals, the closer store, the usual store, we see that it's 2.2 miles away. And so in that situation, that would be our D value. The fact that it's nine minutes driving doesn't really matter too much because we're I mean, time means something. But is there are we actually going to attach some monetary value to the time? Some people like to say the thing like, oh, pick your favorite rich person. Like Bill Gates is so rich that if he saw a hundred dollar bill on the floor, he wouldn't even stop to pick it up because he would lose money doing that, which, of course, is a ridiculous statement to say. Like you think if he ever takes a moment to like cough or go to the bathroom, he's no longer making money. No, that's not how it works. So yes, he can afford to stop for what, five seconds at most to pick up a hundred dollar bill. Time doesn't exactly work that way. Maybe if you're on the clock with a wage position, you know, taking a break or something might mean you're not making money. But we're talking about making dinner, which most likely we're not on the clock right now with our job. The time is an opportunity cost. Yes, but it's like, oh, you know, the nine minutes here because the other one would take about 17 minutes longer to do. What are you going to do with that 17 minutes play Minecraft or whatever? It turns out, I would say here that the time is really sort of this extra information that's not going to have really any bearing on it. Again, it's about a difference of 20 minutes in time, but that's somewhat inconsequential, I think, in the consideration here. I mean, maybe maybe it'll be relevant, but we'll come back to it later on. The distance is really going to be the big deal here. The distance to travel to your store is 2.2 miles. And then the distance to travel to the other store is going to be 8.2 miles. But that's not exactly true because when you go the distance, you actually do have to come back, right? And so because it's a round trip, it's 2.2 miles one way or 8.2 miles one way. So we really need to take that distance in times by two. So 2D, make sure we're getting the correct value. But that's just the distance traveled. That's not exactly the cost, right? So this is the distance that you're going to have to drive your car. And it's useful to keep track of measurements here. This will be this will be measured in miles right here. Well, what else is going to affect the cost then? Well, we saw earlier, like we mentioned this here, that our car that we're driving to get there, it'll average 23 miles per gallon. So that's going to affect things, right? I can use this to convert from distance to gallons, which has a cost, right? It costs three point eight one dollars per gallon. Well, here's a question you have to often ask yourself, am I going to multiply by this rate or am I going to divide by this rate and how do I even know it's a rate? Well, you'll see that the units right here, miles per gallon. When you see the word per right here, this is in some little kitty cat. This is division, right? Miles per gallon, same thing going on down here. We talk about dollars per gallon, which is very explicit with the division right here when you see that fraction bar. So these these rates have to do some type of vision. And if you're ever uncertain about, should I be dividing by the rate or multiplying by the right, use the units to help you clarify things. If I take something that's measured in miles and I multiply it by miles per gallon, that would give me miles squared per gallon, which is like, I don't think I need that. I would like to have gallons. So if you take miles and you divide it by miles per gallon, you're going to take miles and times it by gallons over miles for which the miles then cancel out, giving you gallons. So if we take the miles we have to travel and we divide it by. So let's actually write this as division here. And if we divide it by the rate of miles per gallon, so we take the 23 miles, whoops, miles per gallon like so. If we take the 2D and divide it by miles per gallon, this then will come together and this will give us something in gallons. Like I said, miles times miles per gallon gives you gallons. But again, gallons isn't the cost. The cost is going to come from dollars per gallon. So we have, we know the cost is going to be three dollars and 81 cents per gallon. So what should I multiply by that? Or should I should I divide by that? Same thing, use the units, right? Currently, what I have built is something measured in gallons. If I times gallons by dollars per gallon, I would get dollars, which is what I want. If I took gallons and divided it by dollars per gallon, that would be given gallons squared per dollar, which is not what I want. You can often use the units in play to help you determine whether you should be multiplying or dividing by the rate. So in this situation, we're going to multiply by the dollars, three dollars and 81 cents per gallon, knowing that the gallons would cancel out. And then the unit here is going to be in dollars. It's money, which is how you measure costs. And that's the same thing we saw over here, right? Because the X value right here is being measured in dollars per pound. And so the pounds will cancel out, giving you just a measurement in dollars. So let's put this together and see what we now have. So our cost function, when we put things together, we get the following cost is going to equal, of course, the cost. Let me make this a little bit bigger. When we put these things all together, it's going to be the cost of chicken. Plus the cost of travel. So those are the things we have to put together when we work with this cost function. And so therefore, when we put these things together, we see the following. I'm going to try to simplify things a little bit. So we're going to have this five X, which came from the cost of the chicken. And then we're going to add this to what I'm going to actually kind of simplify this thing a little bit because we have this variable D. But if you take two times the three dollars and 81 cents and you divide that by twenty three, I'm just putting this in my calculator real quick, you're going to get a rate of zero point three, three, one. And you don't need to have every decimal here. But if you just kind of write the, you know, I'm going to put way more than I need that way. When I'm done, I can round and be perfectly fine. So if you take that number two times and I'll summarize what we did right here. So this number, I just took all the constants here, two times three point 81 divide that by twenty three. That's where this came from two point three point eight one over twenty three. That's this value right here. And so this now gives me a linear function for my cost. It is a linear function with two variables, but we know what the information is in this situation. So there's of course the there's the close store. I think we did that one in green, actually. There was the close store, which remember our X value was three dollars and forty nine cents. And our D value was eight point two. And I'll just double check to make sure that's our data. Oh, no, I'm sorry. The close or was close right for a reason, right? So it's two point two, two point two. And before I forget, then, the far store, it had an X value of two dollars and ninety nine cents. And then the distance was that was eight point two. And let me just double check the numbers one more time to make sure. And I should mention that we we also talked about how the car it can hold 13 gallons, although that is a quantitative measurement that doesn't really have any effect on the decision right here. So we're going to ignore that. So now we want to plug these into our function and see what happens. So the cost of going to the close store, this is going to be five times three point four nine. And then we're going to add that to point three three one times two point two. And so I'm going to put that in my calculator real quick times that by, you know, putting the two point two. And you're going to add also the cost of the chicken. Sorry, I can't really show you this. I mean, this is something you want to use a calculator for. This, when you when you do these numbers, you're going to get. Eighteen dollars and 18 cents. I'm just going to round to the nearest penny because with US dollars, we don't have a denomination smaller than penny. So all these extra decimal places, we can just kind of round. We don't necessarily want to round to the 100th place too early. We want to have at least a thousandth place, maybe a 10,000th place would be good enough so that when we round here, we're not off and I know a penny is not much, but we it's not too much extra to hold on to those extra decimal places. So just do that. If we do the cost of going to the farther away store, we're going to get five times two point nine. That's where the savings will be made. But then we have to do point three three one times eight point two. That one's a farther away store. So the cost might be higher to travel there. So let's take a look at that. If we plug in the numbers, you'll excuse me for one second. As I do that, we get the following again, rounding this to two decimal places. You're going to get seventeen dollars and sixty seven cents. So we can now see that when you compare these things side by side, that the farther away store is the is the cheaper option, right? The savings that you make on the chicken outweigh the the cost of traveling there. It does take another about 20 minutes extra time. But also when you take the difference of these things, right? If you take the difference of the two, how much savings are you actually going to make? You're going to get a savings of, well, point five one. That is fifty one cents, so half a buck. And so that's sort of like the cost benefit analysis that we can make based upon this decision. Is the 17 minutes of extra travel worth saving fifty one cents? Fifty one cents, of course, is not a large amount of money. And therefore, many of us might actually choose the convenience over the price, right? But, you know, if you really want to save money, take the extra time and save yourself the fifty one cents. That we have to make these decisions based upon that. Now, of course, on different types of purchasing, like if you purchase a car or furniture or a home, these situations are just much more complicated. But also the savings can easily come to hundreds if not thousands of dollars. And so this type of analysis is very useful. And I want to mention here that the type of function that was in play here was just a linear function. Well, we had basically this direct variation between the cost of chicken and the amount of chicken we're going to buy or, in this case, the store we go to. And then we also had this direct variation of the cost between traveling, the cost of traveling and then how much we actually travel. And so, yeah, we might have an equation with multiple variables, but we're able to find those variables and we set up a linear function for each of the variables individually and then we added them all together. This type of quantitative analysis, these problem solving skills, are really essential to be a successful citizen in life. And it doesn't often take much more than just a little bit of use of linear models. And so I hope this can kind of prep you for our quiz, which you have to you'll have to deal with a similar problem, helping Alejandro make a decision about a big purchase coming up in his life.