Solving a scale drawing word problem

Sally is an architect who creates a blueprint of a rectangular dining room. The area of the actual dining room is 1,600 times larger than the area of the dining room on the blueprint. The length of the dining room on the blueprint is 3 inches. What is the length of the actual dining room in feet? So there's a couple of really interesting things going on here. They give us the dimensions of the blueprint in inches. We want the actual length in feet. And then they tell us that the area of the actual dining room is 1,600 times larger. So they're not saying that the scale of the blueprint is at 1/1600. It's going to be something less than that, and let's think about what that scale is going to be. Let's just think about some different scales. Let's say that this is my blueprint, and this is the actual reality of the dining room that we're thinking about. And my blueprint is let's just say 1 by 1, just for the sake of argument. Now, if this was a 1 by 1 square and we increased the dimensions by a factor of 2, so it's a 2 by 2 square, what's the area going to be? Well, this area is going to be 4. This area is 1, this area is 4. So you notice that if we increase by a factor of 2, it increase our area by a factor of 4. Or another way of saying, if we increase each of our dimensions by a factor of 2, we're going to increase our area by a factor of 4. If instead we increased each of our dimensions by a factor of 3, this would be a 3 by 3 square, and we would increase our area by a factor of 9. So notice, whatever factor we're increasing the area by, it's going to be the factor that we're increasing the dimensions by squared. So let's just think about it that way. So they're telling us that we're increasing the area by 1,600 times. Actually, let me just clean this thing up a little bit. So one way we could imagine it, if our drawing did have an area of 1, which we can't assume, but we could for the sake of just figuring out what the scale of the drawing is. Let me clear all of this here. So the area of the actual dining room is 1,600 times larger, and so if the drawing had an area of 1, then the area of the actual dining room would be 1,600 So what would I have to multiply each of the dimensions by to get an area factor of 1,600? Well, if I multiply this dimension by 40 and this dimension by 40, we see 40 times 40 is 1,600. You might say, hey, Sal, how did you figure out 40? Well, the 16 is a big clue. We know that 4 times 4 is equal to 16, and so if you gave a 0 to each of these 4's, if you made it 40 times 40, then that is going to be 1,600. So this information right over here tells us that the scale factor of the lengths is 40. That would result in an scale factor for the area of 1,600. So that's a good starting point. Now let's go to the actual dining room on the blueprint. So the actual dining room on the blueprint doesn't have these dimensions. We just used that to figure out the scaling factor. The actual dining room on the blueprint has a length of 3 inches. So maybe it looks something like this. They don't give us any of the other dimensions, so we can even imagine a 3 inch by 2 inch, 1 inch, whatever we want. We could even imagine a 3 inch by 3 inch square. They only care about the length. Now let's multiply both of these by a factor of 40. And we only care about the length here. They actually say what's the length of the actual dining room. So let's multiply it, and obviously, this is not drawn to scale. Let's multiply this times a factor of 40. So 3 times 40 is 120, and this, of course, is what we're referring to as the length. Now, you might be tempted to say OK, we're done. This will be 120. But remember, this is 120 inches. So what is 120 inches in terms of feet? Well, 1 foot is equal to 12 inches. If we were to multiply both of these times 10, we know that 10 feet is equal to 120 inches. Or another way you could have thought about it, you have 120 inches divided by 12 inches per foot is going to give you 10. So 120 divided by-- 120 inches-- let me write it this way. 120 inches divided by 12 inches per foot is going to give you 10 feet. So that's the actual length of the dining room in feet.