 Hi, I'm Zor. Welcome to a new Zor education. We continue talking about certain, not exactly standard problems in mathematics. Some of them are a little bit more difficult. Some of them are simple, just maybe to repeat certain important topics of math. So today we will just solve a couple of very, very simple arithmetic problems. This is arithmetic 03 lecture. To get to this lecture, it's presented on unisor.com. You have to choose the course called math plus and problems. And there is a topic arithmetic in there, which will lead you to one of the arithmetic lectures, including this one, which is 03. Now this website contains prerequisite course called math for teens. There is also physics for teens and some other courses over there. So you're welcome to take them. All these courses are totally free. There are no advertisements. Signing is optional. I mean, if it's just self study, you don't really have to sign in. If it's supervised study, you have to do it because you have to establish the connection between the supervisor and yourself. Regardless, this is totally free, as I said, no strings attached. So, well, basically enjoy. Now let's get back to our topic. So today we will solve a couple of problems related to percentages. Now, they do have certain applications in the real life, primarily in financial sectors, for example, and examples which I will use, they are from the financial industry. Basically, it's related to increasing or decreasing the price of something, like price of a stock, for instance. You bought a stock at a certain point, at a certain price. Now, what do we mean when we are saying that the stock price has increased by a certain percentage? Well, let me just start from this definition so we don't have any kind of misunderstanding. So if price was, let's say, let's call it P, and we are saying that the price has increased by, let's say, 5%. What does it mean? It means that, first of all, it increases, which means we have to add something. And if we are saying that it increases by 5%, it means 5% of the previous price. So we have to 5% times price P, basically. Now, what is 5%? This is 5 over 100, or 0.05. So when we are saying that the price has increased by 5%, it means that the new price, new price, is equal to old price plus old price times 0.05. Or, which is exactly the same thing, 0.5 times old price. Right? If we will take old price out of the parenthesis, we will have 1 plus 0.05, and that's what will be the result. So increasing the price by 5% means basically either this or multiplying by 1.05. Now, decreasing means basically the opposite result. This is minus, this is minus, and in this case, now this is for plus, and for minus, it will be 0.95 times old price. So if this is minus, it will be 1 minus 0.5 times P old, which is 0.95. So that's basically the definition. And based on this definition, we will solve a couple of problems. So the problem number one is, let's say you bought something for $100. Then, within a month, let's say, the price has increased by 10%. Then, during the next month, price decreased by 10%. So my question is, what happened with the price after this increase and decrease? Well, they seem to be balancing each other, right? Now, will the price be the same or greater or smaller? Okay. So that's where you have to pause the video and do it yourself. And then you can just continue with doing whatever I'm doing. So let's just calculate. If you had $100 and you add 10% to this price, it will be plus 0.1. 10% is $100, which is 0.1 times $100. And that will be what? This is $10, so it's $110. That's the first increase. Now, then the decrease, what do we have? We have $110 minus 10%, which is 0.1 times $110. And what will be the result? Well, this is $110 of $110, which is $11. So $110 minus $11.99. So as you see, after increasing by 10% and decreasing by 10%, we actually lost $1 relative to original price. Why? Because the same percentage is applied after increase. It's applied to a bigger amount. So this 10% was applied against $100. This 10% was applied after we have increased the price to $110. The same percentage multiplied by a bigger amount will give you a bigger result. So we are subtracting a bigger amount than we are adding. We are adding $10 and subtracting $11. Okay, what if it's the other way around? First, you had minus, first loss, 10%. And then you gain. What happens here? Well, here we will have minus here, which will give me $100 minus $10.90. And here we have $90 plus $1.10, 10% of $90, which is $9. And the result is the same $99. And the reason is exactly the same, because we are increasing by 10% after the original amount was decreased, which means again, we are going up against a smaller amount. Then we are going up per smaller amount, then we are going down. Same thing. So the point is, whenever market goes up and down, up and down by the same percentage, you lose money, which means that either the amount it goes up should, percentage actually, it goes up should exceed the percentage it goes down to at least get even, or maybe even gain some money, or it should be more frequent, not just increase, decrease, increase, decrease, maybe one decrease and two increases or whatever. It doesn't really matter. It goes, it should go up more than it goes down, percentage wise or frequency wise or both, to earn something, to gain money. Okay, so my next problem is, what exactly is the percentage, let's say it goes down by 10%. What is the percentage it should go up to get even? Well, again, let's just calculate. First, it goes down by 10%. So from 100, we subtract 10% of 100, which is one tenth, which is $10 and get 90. Now 90 should add certain amount, certain percentage multiplied by 90 to get 100. I would like to get even. What is this factor? Well, that's a simple arithmetic. 90 goes to here, that would be 10. This is 90 times. So question mark is equal to one ninth. Now one ninth is what? It's 0.111111 up to infinity, right? So let's just cut it somewhere. So approximately it's this, in percentage, it's 11.11%. So after it goes down by 10%, it should go up by 11.11. Well, actually a little bit more. But we're rounding to basically two decimal points. So it should go to 11.11 up to only to get even. And if you want to earn a little bit more, even if you would like to gain some money, it should be greater than this amount of this. Okay, on the way back. So this is one ninth in exact numbers. Now this is 11.11% of this approximation, which is good enough, I guess. Now let's say we go up first and then go down. So one ninth is greater than one tenth. So 0.10, which is 10%. And we had 0.111111, which is greater. So up, it's supposed to be greater than down. Now, how about the reverse? What if first we go up? That means that we will get 100 plus 0.1 times 100, that's 110. Now we have to 110 minus something times 110 and it should go to 100. So question is by how much we can allow it to go down in order to get even. Because if it's 10%, it will be 99, as we know. So it should be less than 10%. Well, again, that's kind of easy. So we'll have 10 here equals the question mark times 110 there. So the question mark is equal to 10 divided by 110 or 1 over 11, which is less than 110. So it goes up by 110, but down it should go to 1 over 11, which is smaller percentage. Now in percentages, 111 is equal to 0.09090909, et cetera. So approximately it's 9.10% more or less. So if it goes up by 10%, it should not go up down by 10%. Because if it goes down by 10%, we will lose money. It should go by less than 10. By factors 111 in percentages, it's 9.10. Okay, so these are basically manipulations which everybody should really think about when investing money and checking what exactly is the result of price movement, up or down, et cetera. So that's where all the financial calculations basically are done. On a bigger scale, obviously, and it's not just one particular movement of the price up and down. If you're talking about real stock market, the price goes up and down every fraction of a second. So somehow this movement of the price, people who are involved in this, they are kind of trying to predict the future movement up or down. Because the markets usually are fluctuating around certain values. So there are some macro movements from day to day, but there are some micro movement from a second to a second. In contemporary computers, which are actually doing this particular trading, they are obviously doing it on every fraction of a second. Now, so I have introduced these problems, not obviously because they have some financial implication, just to get a little bit more information about percentages. I think it's important that people who are thinking about percentages are relatively comfortable using them in moving up or moving down. Plus the definition, what does it mean that the price has increased by 10% or whatever number of percentages? So we wanted to clarify these items because, unfortunately, it's not really high school stuff. It's probably, it belongs to much, much earlier stages. Unfortunately, I mean, my experience shows that people even in high school don't really understand properly how it's all manipulated. Some people. Okay, now one more problem, which is also related to percentages, and it's not about finance, it's about chemistry. Now, in chemistry, we are dealing with multiple substances mixed together. So we will talk about salt solution in water and again, percentages because from the vocabulary standpoint, there is some kind of a statement. Okay, this solution is, let's say, 1% salt. What does it mean? Or alcohol, by the way, that's the same thing. Let's say 40% by value. What does it mean? Well, it means basically the following. So if you have a statement saying that something is like 1% solution of salt in water, what does it mean? It means that if you have a 100 gram solution, it has 1% of 100, which is 1 gram of salt and 99 grams of water. So that's what it means. So let's just clarify this. From the very beginning, what does it mean that we're talking about 1% solution, or 2% or 7% or whatever. So that basically is amount, mass. Let's talk about physical kind of words, mass or weight. You can say weight or mass in this particular case, since everything is on this planet. So of the main ingredient, in this case, we are talking about salt. So 1% solution of salt means 1% is salt from what? From the all together, from the solution, which includes both water and salt, either by weight or by mass or by whatever. So that's what basically is the definition. Now the problem. Let's say you have one solution and you have 100 gram solution. Of salty water, which means 1% is 1 gram of 100 and 99 grams of water. Now, this is in a glass, let's say, and the glass is stinging on the table and the water evaporates. Now after a certain amount of time, whatever remains in the glass is 2% solution. Question is how much water has evaporated. Now the first answer which, how much water is evaporated, how much salt and how much water is in this final solution. So let's just cover all the topics. So what exactly is the result? Okay, now how can we calculate it? Well, first of all, we have to, now again, you have to pause the video and do it yourself and then you have continued. All right, so what does it mean that it's 100%, that it's 100 gram of 1% solution? It means we have 1 gram of salt and the rest, which is 99 gram of water. That's what we have in the beginning. All right, now when the water evaporates, salt remains. So we still have 1 gram of salt. Now, what is the weight of the solution? Well, if the weight of the solution is something, then 1 gram is supposed to be 2% of this. So 2% of weight of the solution, weight of solution equals exactly the same 1 gram because the salt is the same. Well, 2% is actually 0.02, right? It's 200. So 200 times something gives me 1 gram. So what is something? Well, it's 1 divided by 200, which is 50. So weight of the solution is 50. It used to be 100. Now the total weight is 50. It's half of this. And that's why the same amount of salt becomes twice as more, twice concentrated than it was before. Everything is logical. Now, if the total weight of the solution is 50, we still have 1 gram of salt, which means we have only 49 gram of water. So this is the result. This is what it was before. So 50 gram of water evaporated. So from 99, we went down to 49. But this is the composition of the solution after it was evaporated. Again, it's very simple things. It's really simple arithmetic, and it's all resolved in one operation division in this particular case. Same as before, the previous problems about the price going up and down. So these are very simple problems. However, again, experience shows, but for whatever reason, certain people do not really feel comfortable about all these manipulations with percentages. And that's why I decided to present them as a couple of problems, very simple problems. But nevertheless, I think it's useful. So I suggested to read the notes for this lecture where whatever I'm saying is basically in front of your eyes, and you can read it. Try to do your calculations first yourself, and then read the calculations which I put in my notes. And again, to get to these notes, you have to go to the website unisor.com, choose math plus and problems course, choose arithmetic as part of this course, and this will be on the menu as lecture number three. That's it. Thanks very much and good luck.