Old school equations with Sal
Let's get started with some problems. Let's see. First problem: what is 15% of 40? The way I do percent problems is I just convert the percentage to a decimal and then I multiply it times the number that I'm trying to get the percentage of. So 15% as a decimal is 0.15. You learned that from the percent to decimal conversion video, hopefully. And we just multiply this times 40. So let's say 40 times 0.15. 5 times 0 is 0. 5 times 4 is 20. Put a 0 there. And then 1 times 0 is 0. 1 times 4 is 4. And you get 6 0 0. Then you count the decimal spots. 1, 2. No decimals up there, so you go 1, 2 and you put the decimal there. So 15% of 40 is equal to 0.15 times 40, which equals 6.00. Well, that's just the same thing as 6. Let's do another problem. Hopefully, that didn't confuse you too much. And I'm going to try to confuse you this time just in case you weren't properly confused the last time. What is 0.2% of-- let me think of a number-- of 7. So a lot of people's inclinations would just say, oh, 0.2%, that's the same thing as 0.2. And if that was your inclination you would be wrong. Because remember, this isn't 0.2. This is 0.2%. So there's two ways of thinking about this. You could say that this is 0.2/100, which is, if you multiply the numerator and denominator by 10, is the same thing as 2/1,000. Or you can just do the technique where you move the decimal space over 2 to the left. In which case, if you're starting with 0.2 and you move the decimal space 2 to the left, you go bam. Whoops! Bam, bam. That's where the decimal goes. So it's 0.002. This is key. 0.2% is the same thing as 0.002. This can always trip you up and I've made this careless mistake all the time, so don't feel bad if you ever do it. But just always pay careful attention if you see a decimal and a percentage at the same time. So now that we've figured out how to write this percentage as a decimal we just have to multiply it times the number that we want to take the percentage of. So we say 0.002 times 7. Well, this is pretty straightforward. 7 times 2 is 14. And how many total numbers do we have or how many total digits do we have behind the decimal point? Let's see. It's 1, 2, 3. So we need 1, 2, 3 digits behind the decimal point. So 0.2% of 7 is equal to 0.014. And you're probably thinking, boy, that's a really, really small number. And it makes sense because 0.2%, if you want to think about it, that's smaller than even 1%. So that's even smaller than 1/100. And actually, if you think about it, 0.2% is 1/500. And if you do the math, 1/500 of 7 will turn out to be this number. And that's an important thing to do. It's always good to do a reality check because when you're doing these decimal and these percent problems, it's very easy to kind of lose a factor of 10 here or there. Or gain a factor of 10. So always do a reality check to see if your answer makes sense. So now I'm going to confuse you even further. What if I were to ask you 4 is 20% of what number? So a lot of people's reflex might just be, oh, let me take 20%. It becomes 0.20. And multiply it times 4. And in that case, again, you may be wrong. Because think about it. I'm not saying what is 20% of 4? I'm saying that 20% of some number is 4. So now we're going to be doing a little bit of algebra. I bet you didn't expect that in the percent module. So let x equal the number. And this problem says that 20% of x is equal to 4. I think now it's in a form that you might recognize. So how do we write 20% as a decimal? Well, that's just 0.20 or 0.2. And we just multiply it by x to get 4. So 20%, that's the same thing as 0.2. It's the same thing as 0.20, but that last trailing 0 doesn't mean much. 0.2 times x is equal to 4. And now we have a level one linear equation. I bet you didn't expect to see that. So what do we do? Well there's two ways to view it. You can just divide both sides of this equation by the coefficient on x. So if you divide 0.2 here and you divide by 0.2 here. So you get x is equal to 4 divided by 0.2. So let's figure out what 4 divided by 0.2 is. I hope I have enough space. 0.2 goes into 4-- I'm going to put a decimal point here. And the way we do these problems, we move the decimal point here one over to the right. So we just get a 2 and then we can move the decimal point here one over to the right. So this 0.2 goes into 4 the same number of times that 2 goes into 40. And this is easy. 2 goes into 40 how many times? Well, 2 goes into 4 two times and then 2 goes into 0, zero times. You could've done that in your head. 2 into 40 is twenty times. So 4 divided by 0.2 is 20. So the answer is 4 is 20% of 20. And does that make sense? Well, there's a couple of ways to think about it. 20% is exactly 1/5. And 4 times 5 is 20. That makes sense. If you're still not sure we can check the problem. Let's take 20% of 20. So 20% of 20 is equal to 0.2 times 20. And if you do the math that also will equal 4. So you made sure you got the right answer. Let's do another one like that. I'm picking numbers randomly. Let's say 3 is 9% of what? Once again, let's let x equal the number that 3 is 9% of. You didn't have to write all that. Well, in that case we know that 0.09x-- 0.09, that's the same thing as 9% of x-- is equal to 3. Or that x is equal to 3 divided by 0.09. Well, if we do the decimal division, 0.09 goes into 3. Let's put a decimal point here. I don't know how many 0's I'm going to need. So if I move this decimal over to the right twice, then I'll move this decimal over to the right twice. So 0.09 goes into 3 the same number of times that 9 goes into 300. So 9 goes into 30 three times. 3 times nine is 27. I think I see a pattern here already. 30, 3, 3 times 9 is 27. You're going to keep getting 33-- the 3's are just going to go on forever. So it turns out that 3 is 9% of-- you can either write it as 33.3 repeating or we all know that 0.3 forever is the same thing as 1/3. So 3 is 9% of 33 and 1/3. Either one of those would be an acceptable answer. And a lot of times when you're doing percentages you're actually just trying to get a ballpark. The precision might not always be the most important thing, but in this case we will be precise. And obviously, on tests and things you need to be precise as well. Hopefully, I didn't go too fast and you have a good sense of percentage. The important thing for these type of problems is pay attention to how the problem is written. If it says find 10% of 100. That's easy. You just convert 10% to a decimal and multiply it by 100. But if I were to ask you 100 is 10% of what? You have to remember that that's a different problem. In which case, 100 is 10% of-- and if you did the math, it would be 1,000. I think I spoke very quickly on this problem on this module, so I hope you didn't get too confused. But I will record more.