The length of a rectangle is eight metres and its width is five metres. What is the perimeter of a similar rectangle with length 19 metres?
Okay, great! To solve a problem like this, I will start with a couple of diagrams to really help me visualize what’s going on. Okay, these are couple of sketches that I need to be to scale. And now what I’m going to do is add the values that I know. Okay, I’ve added the values that we know. I’ve called the width that we don’t know of rectangle 𝑏 𝑥 metres. And I’ve also said the perimeter we don’t know cause that’s what we want to find to solve the problem.
Great! So we’ve got our sketches. Right, now we’re gonna start solve the problem. And the key thing about this problem is one word in particular. And that word is “similar.” Because it says that our rectangles are similar, this means that one of the rectangles is going to be enlargement of the other, which means that there’s going to be a consistent scale factor of enlargement between the dimensions on our rectangles. So bearing that in mind, we’re gonna try and find out what the scale factor is.
Okay, I’ve written out the formula on the right-hand side in green that says the scale factor is equal to new lengths, so any length on a enlarged rectangle divided by the original length. So let’s apply this to our problem. So with our problem, we actually know the length of rectangle 𝑎 is eight and the length of rectangle 𝑏 is 19. So our scale factor is gonna be equal to 19 divided by eight. Now, this is really handy cause it’s actually says to us that any dimension on rectangle 𝑏 is 19 divided by eight times bigger that on rectangle 𝑎.
Great! So now we can use this scale factor to help us find the missing width. Okay, I will find that by multiplying the original width by our scale factor. So in this case, it’ll be 𝑥 is gonna be equal to five multiplied by 19 divided by eight. You can do that by using a calculator or if you just want to do it by hand, you’d multiply the five by the numerator. So it’ll be five times 19, which would give us 95 over eight.
And now, we just turn this into a mixed number. Okay and that gives us this 𝑥 is equal to 11 and seven-eighths. So we’ve got that because how many eighths are going to 95? So we know that there’re 11 eighths in 95 to give us 88. That gives us a remainder of seven. So it’s gonna be 11 and seven over eight. Fantastic! So we’ve now found the missing width.
Now that we found the missing width, we can actually solve the problem because we want to find the perimeter of the new rectangle, so the perimeter of rectangle 𝑏. So let’s solve that now. Okay, to find the perimeter, well, perimeter is the distance all the way around the outside of our rectangle. So the formula for that is that the perimeter is equal to two 𝑙 plus two 𝑤. So two times the length plus two times the width. So we can do that now. So our perimeter is gonna be equal to two times 19 plus two times 11 and seven-eighths.
Again, we can actually solve this using a calculator. But to just give you an idea of how you do it by hand, again if you had two times 11 is seven-eighths, one way of doing is to split it into two components. So you have two times 11 plus two times seven-eighths. And do them and then add them together. This will give you 22 plus 14 over eight, which will give us 22 plus one and six-eighths. Again use the same method as before to convert it into a mixed number and then finally add those together. And that’s going to give us 23 and six-eighths. Again, we can simplify that fraction to leave us with 23 and three-quarters.
Great! Okay, let’s get back to find the perimeter. So our perimeter is gonna be equal to 38 plus 23 and three-quarters, which gives us a final total of 61 and three-quarters metres. Okay, great! So we’ve now managed to solve the problem and we’ve worked through what we’re doing. So first of all, because it was similar rectangles, we found the scale factor. We found the scale factor by dividing a new length by an original length. So that gives a scale factor of 19 over eight. We then used that scale factor to find the missing length, which is the width. And once we had the missing width, we then found the perimeter using the formula 𝑏 is equal to two 𝑙 plus two 𝑤.
However, how can we check this? So let’s show you a quick way of checking it. Okay, we can find the perimeter of 𝑎 using the same formula. So perimeter of 𝑎 would be two times eight plus two times five, which is equal to 26. That tells us the perimeter of 𝑎. And now what we can actually do is a little check or you can do this actually with your method if you prefer this method to work out the perimeter. As we can now multiply this perimeter by our scale factor cause we’ve already said because this similar rectangles is an enlargement of the other. So we can multiply. So let’s try that.
So that’s gonna give us perimeter 𝑏. It’s gonna be equal to 26 multiplied by 19 over eight. And then either using the methods I’ve shown to you before, doing it by hand or by calculator, we get a final answer of 494 divided by eight. If you want to then change this into a mixed number, it will give us 61 and three-quarter metres. Fantastic! We’ve checked and we’ve actually got the same answer as we got with our first methods. So we know yes, we are right.