Let’s solve a few verbal problems using exponential increase and decrease. In the first task we have the decomposition of a radioactive substance at a rate of 3.5% per hour. What percentage of the substance will be left after 6 hours? Let me make a little table here, to imagine what is happening. And then we will try to derive a general formula how much is left after n hours. Let’s say hours have passed and other percentages, What percentage is left after 0 hours? Well, it hasn’t fallen apart yet, so we have 100% left. What happened after 1 hour? The substance decomposes at a rate of 3.5% per hour. So 3.5% are gone. In other words, this is 0.965. Remember that if you take 1 minus 3.5%, or if you take 100% minus 3.5% – this shows how much we lose every hour – and equals 96.5%. So every hour we will have 96.5% of the quantity in the previous hour. In hour 1 we have 96.5% of hour 0 or 0.965 per 100, by the amount per hour 0. What happens in hour 2? Time 2. We have 96.5% of the amount in the previous hour. We will lose 3.5%, which means we have 96.5% of the previous hour. It will be 0.965 on that, multiplied by 0.965 by 100. I think you see what happens in general. In the first hour we have 0.965 to degree 1, multiplied by 100. At zero hour we have 0.965 to the power of zero. We don’t see it, but there is a unit multiplied by 100. In the second hour we have 0.965 to the 2nd power, multiplied by 100. And in general in the nth hour – let me write it in nice bright color – in the nth hour we will have 0,965 to the nth power multiplied by 100, which is left of our radioactive substance. Very often you will find it written this way. You have the initial value multiplied by the quotient of 0.965 to the nth power. That’s all that’s left in n hours. Now we can answer the question: How much is left after 6 hours? We will have the remaining 100 multiplied by 0.965 to the sixth power. We can use a calculator to calculate how much this is. Let’s use our true calculator. We have 100 at 0.965 to the sixth power, which is equal to 80.75. It’s all in percentages. So there are 80.75% of our original substance left. 80.75%. Let’s make another similar example. It is given that Nadia owns a fast food chain, which in 1999 had 200 sites. If the growth rate is … Oh, there’s a typo. It should be 8% here – the growth rate is 8% per year how many stores will the chain have in 2007? Let’s apply the same logic. Here we have years after 1999. And here we have how many restaurants the restaurant will have Nadia, or hers fast food chain. 1999 itself is 0 years after 1999. and it has 200 stores. Then in 2000, which is 1 year after 1999, how many restaurants will there be? Well, it is growing at a rate of 8% per year. So there will be all the restaurants she had before, plus 8% of the restaurants it had before. 1.08 multiplied by the number of restaurants it had before. And as you will see, the quotient here is 1.08. If it grows by 8%, this is equivalent to multiplication by 1.08. Let’s make this clear. 200 plus 0.08 multiplied by 200. Well, that’s only 1 over 200 plus 0.08 multiplied by 200. That’s 1.08 over 200. Then what happened in 2001? It’s been 2 years since 1999, and there’s been an increase by 8% of that number. You’re going to multiply 1.08 by that number, multiplied by 1.08, multiplied by 200. I think he got the gist. n years after 1999 this will be 1.08 – let me write it this way: will be 200 to 1.08 to the nth power. After 2 years it is 1.08 per square. After 1 year it is 1.08 in the first degree. 0 years, that’s the same as 1 in 200, which is 1.08 to zero. The condition asks how many restaurants work will there be in 2007? 2007 is 8 years after 1999. So here n is equal to 8. Let us replace n with 8. The answer to our question is 200 to 1.08 to the eighth degree. Let’s take out our calculator and calculate it. We want to calculate 200 by 1.08 to the eighth degree. It will have 370 restaurants and will be in the process of opening a few more. So if we round it down, it will have 370 restaurants. 8% growth may not seem like something that is so fast or so exciting. But in less than a decade, in just 8 years, it would have increased its chain of restaurants from 200 to 370 restaurants. You see that in 8 years an increase of 8% in fact, it is very impressive. .