Question Video: Solving a Real-World Problem Involving Proportions Mathematics

It cost 3,799 pounds to fit wooden flooring in a class with dimensions 28 m and 10 m. How much would it cost to fit wooden flooring in a similar room with dimensions 84 m and 30 m?

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Video Transcript

It cost 3,799 pounds to fit wooden flooring in a class with dimensions 28 meters and 10 meters. How much would it cost to fit wooden flooring in a similar room with dimensions 84 meters and 30 meters?

So, what we have here are two rooms that we’re looking at flooring for. Now, this problem is actually a problem that can be solved in two ways. And the first way would be to find the area of each of our rectangular rooms. And that’s remembering that area is equal to length multiplied by width. Well, for our first rectangular room, the area will be equal to 10 multiplied by 28 which would be 280 meters squared. Then, for the larger room, we’d have 30 multiplied by 84 which would be 2,520 meters squared.

And then what we would do is work out the cost per meter squared. But how would we do this? Well, what we’d do is take the cost for fitting wood in the smaller room, which is 3,799 pounds, then divide it by 280. So then from this, what we could do is work out the cost for the larger room. And we’d do that by multiplying our cost per meter squared — so 3,799 over 280, and we’ll keep it like this to maintain accuracy — by 2,520. And what this would give us would be a cost for fitting flooring in the larger room of 34,191 pounds, and this is rounded to the nearest pound.

But like we said, there are a couple of ways we could solve this problem, so what we’re gonna do is check this answer using another method. And to use the second method, what we’re gonna do is base our answer around the fact that both of the rooms are similar. What this means is that one is an enlargement of the other. Well, if they’re an enlargement of each other, then the first thing we want to do is work out the scale factor. And to work out the scale factor, what we have is the formula the scale factor is equal to the new length over the original length. Well, therefore, if we picked two corresponding lengths, we’ve got 30 and 10. So therefore, we can say the scale factor is equal to 30 divided by 10, which is three.

Well, it’s at this point we’d actually see a common mistake. And that common mistake would be — right, okay, so we know the scale factor is three. So therefore, all we need to do is multiply the cost, which is 3,799 pounds, by three. And that will give us the answer we’re looking for. But this is incorrect. We’ll have a look why. Well, the reason why is because the scale factor we’ve worked out is the scale factor of our dimensions. It’s not the scale factor of our area. And if we’re gonna fit flooring to a class, then what we’re looking at is the area of the flooring needed. So therefore, if we’re looking for the scale factor of our areas, then this is gonna be equal to our original scale factor, so the scale factor of our dimensions, squared. And this is gonna give us a result of nine.

And if we think about how we got here, well it’s because if we’re trying to find the area of something, we’re gonna multiply two dimensions together and our units are in fact gonna be something squared. So in this case, it’d be meters squared. So that’s why we square the scale factor for our dimension which gives us the scale factor for our areas of nine. So therefore, we can say that the cost for the larger room is gonna be equal to 3,799 multiplied by nine which, once again, gives us the answer 34,191 pounds.

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