More percent problems

Let's say I go to a store and I have $50 in my pocket. $50 in my wallet. And at the store that day they say it is a 25% off marked price sale. So 25% off marked price means that if the marked price is $100 the price I'm going to pay is going to be 25% less than $100. So my question to you is if I have $50, what is the highest marked price I can afford? Because I need to know that before I go finding something that I might like. So let's do a little bit of algebra. So let x be the highest marked price that I can afford. So if the sale is 25% off of x, we could say that the new price, the sale price will be x minus 25% of x is equal to the sale price. And I'm assuming that I'm in a state without sales tax. Whatever the sale price is, is what I have to pay in cash. So x minus 25% x is equal to the sale price. The discount is going to be 25% of x. But we know that this is the same thing as x minus 0.25x. And we know that that's the same thing as-- well, because we know this is 1x, x is the same thing is 1x. 1x minus 0.25x. Well, that means that 0.75x is equal to the sale price, right? All I did is I rewrote x minus 25% of x as 1x minus 0.25x. And that's the same thing as 0.75x. Because 1 minus 0.25 is 0.75. So 0.75x is going to be the sale price. Well, what's the sale price that I can afford? Well, the sale price I can afford is $50. So 0.75x is going to equal $50. If x is any larger number than the number I'm solving for, then the sale price is going to be more than $50 and I won't be able to afford it. So that's how we set the-- the highest I can pay is $50 and that's the sale price. So going back to how we did these problems before. We just divide both sides by 0.75. And we say that the highest marked price that I can afford is $50 divided by 0.75. And let's figure out what that is. 0.75 goes into 50-- let's add some 0's in the back. If I take this decimal 2 to the right. Take this decimal, move it 2 to the right, goes right there. So 0.75 goes into 50 the same number of times that 75 goes into 5,000. So let's do this. 75 goes into 50 zero times. 75 goes into 500-- so let me think about that. I think it goes into it six times. Because seven times is going to be too much. So it goes into it six times. 6 times 5 is 30. 6 times 7 is 42. Plus 3 is 45. So the remainder is 50. I see a pattern. Bring down the 0. Well, same thing again. 75 goes into 500 six times. 6 times 75 is going to be 450 again. We're going to keep having that same pattern over and over and over again. It's actually 66.666-- I hope you don't think I'm an evil person because of this number that happened to show up. But anyway, so the highest sale price that I can afford or the highest marked price I can afford is $66 dollars. And if I were to around up, and $0.67 if I were to round to the nearest penny. If I were to write this kind of as a repeating decimal, I could write this as 66.66 repeating. Or I also know that 0.6666 going on forever is the same thing as 2/3. So it's 66 and 2/3. But since we're working with money and we're working with dollars, we should just round to the nearest penny. So the highest marked price that I can afford is $66.67. So if I go and I see a nice pair of shoes for $55, I can afford it. If I see a nice tie for $70, I can't afford it with the $50 in my pocket. So hopefully not only will this teach you a little bit of math, but it'll help you do a little bit of shopping. So let me ask you another problem, a very interesting problem. Let's say I start with an arbitrary-- let's put a fixed number on it. Let's say I start with $100. And after one year it grows by 25%. And then the next year, let's call that year two, it shrinks by 25%. So this could have happened in the stock market. The first year I have a good year, my portfolio grows by 25%. The second year I have a bad year and my portfolio shrinks by 25%. So my question is how much money do I have at the end of the two years? Well a lot of people might say, oh, this is easy, Sal. If I grow by 25% and then I shrink by 25% I'll end up with the same amount of money. But I'll show you it's actually not that simple because the 25% in either case or in both cases is actually a different amount of money. So let's figure this out. If I start with $100 and I grow it by 25%-- 25% of $100 is $25. So I grew it by $25. So I go to $125. So after one year of growing by 25% I end up with $125. And now this $125 is going to shrink by 25$. So if something shrinks by 25%, that means it's just going to be 0.75 or 75% of what it was before, right? 1 minus 25%. 0.75 times $125. So let's work that out here. $125 times 0.75. And just in case you're confused, I don't want to repeat it too much, but if something shrinks by 25% it is now 75% of its original value. So if $125 shrinks by 25% it's now 75% of $125 or 0.75. Let's do the math. 5 times 5 is 25. 2 times 5 is 10 plus 2 is 12. 1 times 5-- 7. 7 times 5 is 35. 7 times 2 is 14. Plus 3 is 17. Sorry. 7 times 1 is 7. Plus 1 is 8. So it's 5, 7, and then this is 7 actually. 14. 9. 94.75, right? Two decimal points. 94.75. So it's interesting. If I start with $100 and it grows by 25%, and then it shrinks by 25% I end up with less than I started with. And I want you to think about why that happens. Because 25% on $100 is the amount that I'm gaining. That's a smaller number than the amount that I'm losing. I'm losing 25% on $125. That's pretty interesting, don't you think? That's actually very interesting when a lot of people compare-- well, actually I won't go into stock returns and things. But I think that should be a pretty interesting thing. You should try that out with other examples. Another interesting thing is for any percentage gain, you should think about how much you would have to lose-- what percentage you would have to lose to end up where you started. That's another interesting project. Maybe I'll do that in a future presentation. Anyway, I think you're now ready to do some of those percent madness problems. Hope you have fun. Bye.