Solving percent problems

We're asked to identify the percent, amount, and base in this problem. And they ask us, 150 is 25% of what number? They don't ask us to solve it, but it's too tempting. So what I want to do is first answer this question that they're not even asking us to solve. But first, I want to answer this question. And then we can think about what the percent, the amount, and the base is, because those are just words. Those are just definitions. The important thing is to be able to solve a problem like this. So they're saying 150 is 25% of what number? Or another way to view this, 150 is 25% of some number. So let's let x, x is equal to the number that 150 is 25% of, right? That's what we need to figure out. 150 is 25% of what number? That number right here we're seeing is x. So that tells us that if we start with x, and if we were to take 25% of x, you could imagine, that's the same thing as multiplying it by 25%, which is the same thing as multiplying it, if you view it as a decimal, times 0.25 times x. These two statements are identical. So if you start with that number, you take 25% of it, or you multiply it by 0.25, that is going to be equal to 150. 150 is 25% of this number. And then you can solve for x. So let's just start with this one over here. Let me just write it separately, so you understand what I'm doing. 0.25 times some number is equal to 150. Now there's two ways we can do this. We can divide both sides of this equation by 0.25, or if you recognize that four quarters make a dollar, you could say, let's multiply both sides of this equation by 4. You could do either one. I'll do the first, because that's how we normally do algebra problems like this. So let's just multiply both by 0.25. That will just be an x. And then the right-hand side will be 150 divided by 0.25. And the reason why I wanted to is really it's just good practice dividing by a decimal. So let's do that. So we want to figure out what 150 divided by 0.25 is. And we've done this before. When you divide by a decimal, what you can do is you can make the number that you're dividing into the other number, you can turn this into a whole number by essentially shifting the decimal two to the right. But if you do that for the number in the denominator, you also have to do that to the numerator. So right now you can view this as 150.00. If you multiply 0.25 times 100, you're shifting the decimal two to the right. Then you'd also have to do that with 150, so then it becomes 15,000. Shift it two to the right. So our decimal place becomes like this. So 150 divided by 0.25 is the same thing as 15,000 divided by 25. And let's just work it out really fast. So 25 doesn't go into 1, doesn't go into 15, it goes into 150, what is that? Six times, right? If it goes into 100 four times, then it goes into 150 six times. 6 times 0.25 is-- or actually, this is now a 25. We've shifted the decimal. This decimal is sitting right over there. So 6 times 25 is 150. You subtract. You get no remainder. Bring down this 0 right here. 25 goes into 0 zero times. 0 times 25 is 0. Subtract. No remainder. Bring down this last 0. 25 goes into 0 zero times. 0 times 25 is 0. Subtract. No remainder. So 150 divided by 0.25 is equal to 600. And you might have been able to do that in your head, because when we were at this point in our equation, 0.25x is equal to 150, you could have just multiplied both sides of this equation times 4. 4 times 0.25 is the same thing as 4 times 1/4, which is a whole. And 4 times 150 is 600. So you would have gotten it either way. And this makes total sense. If 150 is 25% of some number, that means 150 should be 1/4 of that number. It should be a lot smaller than that number, and it is. 150 is 1/4 of 600. Now let's answer their actual question. Identify the percent. Well, that looks like 25%, that's the percent. The amount and the base in this problem. And based on how they're wording it, I assume amount means when you take the 25% of the base, so they're saying that the amount-- as my best sense of it-- is that the amount is equal to the percent times the base. Let me do the base in green. So the base is the number you're taking the percent of. The amount is the quantity that that percentage represents. So here we already saw the percent is 25%. That's the percent. The number that we're taking 25% of, or the base, is x. The value of it is 600. We figured it out. And the amount is 150. This right here is the amount. The amount is 150. 150 is 25% of the base, of 600. The important thing is how you solve this problem. The words themselves, you know, those are all really just definitions.