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