Ratios, rates, and proportions — Harder example

- [Instructor] Erika plans to purchase seed to plant grass in a large field. She has a map of the region, and calculates the area of the region on paper to be 180 square centimeters. The scale on the map shows that one centimeter is equal to 20 feet. So they're giving us linear centimeters to linear feet, not square centimeters to square feet, so I think we're gonna have to figure that out. If Erika plans to cover every 400 square feet with one pound of seed, approximately how many pounds of seed will she need to cover the entire field? So what we wanna do is we wanna figure out, well, how many square feet is the field? How many square feet is the field? And we know on her map, her map has, well, a map of the field, and she's, on the map it's 180 square centimeters. And they tell them or they tell us that one centimeter is equal to 20 feet. But if this is square centimeters, then we wanna convert this to square feet. So to convert, if one centimeter's equal to 20 feet, to convert it from, if we wanna figure out the conversion from square centimeters to square feet, we could just square both sides of this. We could just square both sides of this. And what will that be? One squared is still one, but now we have one centimeter squared. And I could write it like this just to be clear that I took both of them to the second power, but one squared is just one. And this is going to be equal, 20 squared is going to be 400 square feet. So one square centimeter on this map is going to be equal to 400 square feet. So how many square feet does 180 square centimeters represent? So I could write 180 centimeters squared. And we know that every centimeter squared represents 400 feet, or we could say 400 feet squared per centimeter squared. And we want the units to cancel out, so this makes sense. You have centimeters squared over centimeters squared. So the actual field is going to be 180. 180 times 400. Times 400 square feet. And that makes sense. The region on paper was 180 square centimeters. Each square centimeter is equal to 400 square feet in reality. So for each of those square centimeters, it's going to be 400 square feet. So it's 180 times 400. So this is the area in square footage of the actual field, and then they tell us we're gonna use one pound of seed. One pound of seed for every 400 square feet. So let's see, the area of the entire field, let me give myself a little bit more space, is, and I'm not gonna even multiply it out because this 400 is showing up, so this seems useful. So the area is 400 feet squared. I just wrote that. And we're going to use one pound, one pound of seed, pound of seed, for every 400 square feet. For every 400 square feet. And if we look at it, the units cancel out. Square feet divided by square feet. Then we have 400 divided by 400. And we're left with 180 times one pound of seed, which tells us that we're going to use 180 pounds, 180 pounds of, 180 pounds of seed, which is that choice right over there. Now, if you're under time pressure on the SAT, although I would be careful, because this one is a little bit, it could be a little confusing. But you might've said, okay, there's 180 square centimeters. Each of those square centimeters is 400 feet. This is actually the key thing that you have to realize, that each square centimeter is not just 20 feet. It's 400. It's 400 feet. So you're gonna have 180 times 400 square feet for the entire field. But then each of those 400 feet, each of those 180 400 square feet, you're gonna use one pound of seed. You're gonna use 180 pounds of seed. So you could've done it without all of this writing, but it is helpful to write it so that you don't confuse things.