Rectangle 𝑄𝑅𝑆𝑇 is similar to rectangle 𝐽𝐾𝐿𝑀, with their sides having a ratio of eight to nine. If the dimensions of each rectangle are doubled, find the ratio of the areas of the larger rectangles.
So let’s, first of all, consider our ratio. So we have the ratio eight to nine. And this is the ratio of the sides of 𝑄𝑅𝑆𝑇 to 𝐽𝐾𝐿𝑀. Well then, what it tells us is that the dimensions of each rectangle are doubled. So we look at our ratio and we think, “Oh, okay, so let’s double both parts.” So that gives us 16 to 18. Well, in fact, we don’t need to do this. And that’s because when we’re talking about a ratio, we’re not talking about the actual lengths of the sides. What we’re talking about is a ratio, so the proportion between the two things that we’re looking at.
So therefore, as we can see, we’d be given the ratio 16 to 18 if we multiply both sides by two. But this would just cancel down to eight to nine because in fact as long as you multiply each of the rectangles or each of the sides in the rectangles by the same amount, the ratio will in fact stay the same. I’m gonna show you a little example which will highlight why that is.
Well, if we took a look at two rectangles, so the two rectangles here, and I’ve put the dimensions on. So we’ve got the side lengths two and three and side lengths three and 4.5. Well, if we looked at the ratios of the two longer sides, then what we’d have is a ratio of three to 4.5. However, if we doubled both of these sides, so we wanted to make both of our rectangles bigger, then what we’d have is six and nine. So the ratio between these sides would be six to nine.
Well, if we consider the shorter sides and these of the original rectangles, well, in theory, they should have exactly the same ratio if what we said earlier is true. Well, let’s see what we’ve got. We’ve got six to nine when we looked at the larger sides that have been doubled. Well, in fact, we know that six to nine have a common factor, and that is three. So if we divide six and nine both by three, what we’re gonna get is two and three. So therefore, we’re gonna get the ratio two to three, which is what we’d get if we looked at the ratio between the shorter sides.
Okay, so we’ve shown that doubling the sides on the rectangle does not affect the ratio. So now let’s solve the problem, because what we want to do is find the ratio of the areas of the larger rectangles because in fact the ratio of the areas of the larger rectangles is gonna be the same as the ratio of the areas of the smaller rectangles.
Well, in fact, all you need to do if you’re looking to find the ratio of the areas is square both parts of the ratio of the sides. And that’s because if we think about it, if we have a length, then that’s just a single value. However, if we’re looking at area, then it’s gonna be a squared value. So therefore, what we do is square each part of our ratio. So we’re gonna have eight squared to nine squared. And this will give the ratio 64 to 81.
Okay, so that was nice and straightforward. But let’s show how this would actually work by using the example we had earlier. Well, when we looked at our rectangles that we had in our example, we could see that the ratio between the side lengths was two to three. And that was the same whether it was the same rectangles we have here or if we double their sizes or if we triple their sizes. It’d still give us the same ratio.
Now, what we want to do is have a look at the ratio of the areas. Well, if we were going to find the area of the first rectangle, this would be found by multiplying the two side lengths, so two multiplied by three, which would be six. And then if we wanted to find the area of the second rectangle, it’d be three multiplied by 4.5, which would be 13.5. So then, if we wanted to find the ratio of the areas, this is gonna be six to 13.5.
Okay, but it probably doesn’t quite look like what we want at the moment. So what are we gonna do with this? Well, if we double both sides of our ratio, because obviously we want to try and get rid of the decimal, we’re gonna have 12 to 27. And then if we’ve got 12 and 27, we can see that three is in fact a factor to both of these. So if we divide both 12 and 27 by three, we’re gonna get four to nine as our ratio. So this is our ratio of our areas.
Well, if we go back to our original ratio of sides, we can see that it was two to three. Well, two squared is four, and three squared is nine. So yes, this identifies and shows what we mentioned earlier. And that is if we’re looking to find the ratio of the areas, what we do is square the ratio of the sides.