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In this example we grow a whole number by a percentage of itself. Growing by percentage is a common skill often used when figuring how much is owed or earned with interest. Created by Sal Khan.
Video transcript
Let's do some more percentage problems. Let's say that I start this year in my stock portfolio with $95.00. And I say that my portfolio grows by, let's say, fifteen percent. How much do I have now? OK. I think you might be able to figure this out on your own, but of course we'll do some example problems, just in case it's a little confusing. So I'm starting with $95.00, and I'll get rid of the dollar sign. We know we're working with dollars. ninety-five dollars, right? And I'm going to earn, or I'm going to grow just because I was an excellent stock investor, that ninety-five dollars is going to grow by fifteen percent. So to that ninety-five dollars, I'm going to add another fifteen percent of ninety-five. So we know we write 15% as a decimal, as 0.15, so 95 plus 0.15 of 95, so this is times 95-- that dot is just a times sign. It's not a decimal, it's a times, it's a little higher than a decimal-- So 95 plus 0.15 times 95 is what we have now, right? Because we started with ninety-five dollars, and then we made another fifteen percent times what we started with. Hopefully that make sense. Another way to say it, the ninety-five dollars has grown by fifteen percent. So let's just work this out. This is the same thing as 95 plus-- what's 0.15 times 95? Let's see. So let me do this, hopefully I'll have enough space here. 95 times 0.15-- I don't want to run out of space. Actually, let me do it up here, I think I'm about to run out of space-- 95 times 0.15. five times five is twenty-five, nine times five is forty-five plus two is forty-seven, one times ninety-five is ninety-five, bring down the five, twelve, carry the one, fifteen. And how many decimals do we have? one, two. 15.25. Actually, is that right? I think I made a mistake here. See five times five is twenty-five. five times nine is forty-five, plus two is forty-seven. And we bring the zero here, it's ninety-five, one times five, one times nine, then we add five plus zero is five, seven plus five is twelve-- oh. See? I made a mistake. It's 14.25, not 15.25. So I'll ask you an interesting question? How did I know that 15.25 was a mistake? Well, I did a reality check. I said, well, I know in my head that fifteen percent of one hundred is fifteen, so if fifteen percent of one hundred is fifteen, how can fifteen percent of ninety-five be more than fifteen? I think that might have made sense. The bottom line is ninety-five is less than one hundred. So fifteen percent of ninety-five had to be less than fifteen, so I knew my answer of 15.25 was wrong. And so it turns out that I actually made an addition error, and the answer is 14.25. So the answer is going to be ninety-five plus fifteen percent of ninety-five, which is the same thing as 95 plus 14.25, well, that equals what? 109.25. Notice how easy I made this for you to read, especially this two here. 109.25. So if I start off with $95.00 and my portfolio grows-- or the amount of money I have-- grows by fifteen percent, I'll end up with $109.25. Let's do another problem. Let's say I start off with some amount of money, and after a year, let's says my portfolio grows twenty-five percent, and after growing twenty-five percent, I now have $one hundred. How much did I originally have? Notice I'm not saying that the $one hundred is growing by twenty-five percent. I'm saying that I start with some amount of money, it grows by twenty-five percent, and I end up with $one hundred after it grew by twenty-five percent. To solve this one, we might have to break out a little bit of algebra. So let x equal what I start with. So just like the last problem, I start with x and it grows by twenty-five percent, so x plus twenty-five percent of x is equal to one hundred, and we know this 25% of x we can just rewrite as x plus 0.25 of x is equal to one hundred, and now actually we have a level-- actually this might be level three system, level three linear equation-- but the bottom line, we can just add the coefficients on the x. x is the same thing as onex, right? So 1x plus 0.25x, well that's just the same thing as 1 plus 0.25, plus x-- we're just doing the distributive property in reverse-- equals one hundred. And what's 1 plus 0.25? That's easy, it's 1.25. So we say 1.25x is equal to 100. Not too hard. And after you do a lot of these problems, you're going to intuitively say, oh, if some number grows by twenty-five percent, and it becomes 100, that means that 1.25 times that number is equal to one hundred. And if this doesn't make sense, sit and think about it a little bit, maybe rewatch the video, and hopefully it'll, over time, start to make a lot of sense to you. This type of math is very very useful. I actually work at a hedge fund, and I'm doing this type of math in my head day and night. So 1.25 times x is equal to 100, so x would equal 100 divided by 1.25. I just realized you probably don't know what a hedge fund is. I invest in stocks for a living. Anyway, back to the math. So x is equal to 100 divided by 1.25. So let me make some space here, just because I used up too much space. Let me get rid of my little let x statement. Actually I think we know what x is and we know how we got to there. If you forgot how we got there, you can I guess rewatch the video. Let's see. Let me make the pen thin again, and go back to the orange color, OK. X equals 100 divided by 1.25, so we say 1.25 goes into 100.00-- I'm going to add a couple of 0's, I don't know how many I'm going to need, probably added too many-- if I move this decimal over two to the right, I need to move this one over two to the right. And I say how many times does one hundred go into one hundred-- how many times does one hundred and twenty-five go into one hundred? None. How many times does it go into one thousand? It goes into it eight times. I happen to know that in my head, but you could do trial and error and think about it. eight times-- if you want to think about it, eight times one hundred is eight hundred, and then eight times twenty-five is two hundred, so it becomes one thousand. You could work out if you like, but I think I'm running out of time, so I'm going to do this fast. eight times one hundred and twenty-five is one thousand. Remember this thing isn't here. one thousand, so one thousand minus one thousand is zero, so you can bring down the zero. one hundred and twenty-five goes into zero zero times, and we just keep getting zero's. This is just a decimal division problem. So it turns out that if your portfolio grew by twenty-five percent and you ended up with $100.00 you started with $80.00. And that makes sense, because twenty-five percent is roughly one / four, right? So if I started with $80.00 and I grow by 1/4, that means I grew by $twenty, because twenty-five percent of eighty is twenty. So if I start with eighty and I grow by twenty, that gets me to one hundred. Makes sense. So remember, all you have to say is, well, some number times 1.25-- because I'm growing it by 25%-- is equal to 100. Don't worry, if you're still confused, I'm going to add at least one more presentation on a couple of more examples like this.