Same length in different units

We're asked, how many inches are in 4 and 1/2 yards? And we'll do it a couple of ways. One, we could just say how many inches are in 4 yards and how many inches are in a 1/2 of a yard? And this really 4 plus 1/2 yards. Or we could convert this into an improper fraction first and then convert. But before I even do that, let's just think about how many inches there are in a yard. So if I have 1 yard, we know that there are 3 feet for every 1 yard, Right? And when you say, why am I multiplying by 3 instead of saying there's 1 yard for every 3 feet? And the easiest way to think about it is you're going to have a larger value over here, and you're also going to want to have these units right over here cancel out. So yard is canceling out with yard. So you have 1 yard is equal to 3 feet, which is kind of what we already knew. I'm just showing you how the dimensions cancel out. And how many inches are there per foot? Well, we know that there are 12 inches for every 1 foot. And same logic over here, inches is a smaller unit of measurement so it makes sense that we're multiplying by 12. 3 feet is going to be more inches, so we're multiplying by 12. And also these units cancel out-- foot in the numerator, foot in the denominator. 3 times 12 divided by 1 is equal to 36 inches. So you might have already known it. But this is nice to have the dimensions cancel out like this. We know that 1 yard is equal to 36 inches. Or there are 36 inches for every 1 yard. And so we can now either break this down, or we can turn this into an improper fraction. First I'll just break it down into 4 yards plus 1/2 yards. So we could say that this is 4 yards. So 4 yards is going to be equal to-- well, let's just multiply it times 36 inches, 36 inches for every 1 yard. The yards cancel out. 4 times 36 is 120, plus 24, so that's 144. So this is equal to 144 inches. That's just the 4 yards. And then if we do the 1/2 yards, so 1/2 of a-- I guess I say 1/2 yard, once again, times 36 inches per yard, the yards cancel out. 1/2 times 36 is going to be equal to 18 inches. So 4 and 1/2 yards is the same thing as 4 yards plus 1/2 yards, which is the same thing as 144 inches plus 18 inches, which is going to give us-- let's just add it up over here on the right, 144 plus 18. 4 plus 8 is 12. 4 plus 1 is 5. You have this 1 up here, so it's 6, and then we have a 1. So when you add them all together, you get 162 inches. The other way to do this would have been to convert this into an improper fraction and then multiply by the unit. So let's do it that way. If I have 4 and 1/2 of anything, really-- so let me write 4 and 1/2. I'm trying to find a suitable color. So if I have 4 and 1/2, this is the same thing-- 4 is the same thing as 8/2. This is the same thing-- so let me write it this way. 4 and 1/2 is the same thing as 4 plus 1/2, which is the same thing as-- if we want to have the same denominator as this 2 over here or as this 1/2 over here, this is the same thing as 8/2. Or you could say 4/1 is the same thing as 8/2, if we want to have a common denominator, so 8/2 plus 1/2. Actually, let me write it that way, just so you really understand what we're doing. 4 is the same thing as 4/1. So it's 4/1 plus 1/2. If we want to find a common denominator, it's 2. So 4/1 is the same thing as 8/2 plus 1/2, which is equal to 9/2. Now, I did it this way, which takes longer, just so you really understand how we converted it, why it makes-- hopefully conceptually why it just makes intuitive sense, why 4 and 1/2 is the same thing as 9/2. But if you want a simple process for it, you could just say, look, 4 times 2 is 8. 8 plus 1 is 9. And that gives you that 9 right over there, so 9/2. So we have 9/2 yards that we want to convert to inches. Same process-- times 36 inches per yard. Yard in the numerator, yard in the denominator. We are left with 9/2 times 36. We could say times 36/1 if we like, 36 Inches for every 1 yard. 36-- or the number 36 really is the same as 36/1. And then we're left with just inches in our units. We're just left with inches. And over here there's several ways that we can simplify it. Probably the easiest way to simplify it is we can divide both our numerator and our denominator by 2. So let me write it this way. I don't want to skip steps. So we have 9 times 36 over 2 times 1, or over 2 inches. And we can divide the numerator and the denominator by 2 to simplify it. They're both divisible by 2. 36 divided by 2 is 18. 2 divided by 2 is 1. So we're really just left with 9 times 18 inches. We can just multiply 9 times 18. Let me do it over here. 18 times 9. 8 times 9 is 72. 1 times 9 is 9, plus 7 is 16, so we get 162 inches. So all of this simplifies to 162 inches, and we are done.