Lesson Video: Scale Drawings and Models Mathematics

Video Transcript

So what we’re going to look at in this lesson is scale drawings and maps. And in fact, a map is an example of a scale drawing because it’s a scale drawing of a location and the terrain within it. We could also have a scale drawing which would be the plans, the plans to a house or the plans to a building. And when we look at scale models, we’re looking at scale models that are used by town planners or a scale model of a car. But what we’re not looking at is the type of scale we can see here on the right because that’s used to weigh things, and that’s not what we’re gonna be using in this lesson today.

Now, before we actually look at some examples of some questions, what we’re gonna have a look at is some scales and how they work and also remind ourselves of a few conversion factors. So let’s start off with an example of a scale. So let’s think about a wall map of the world. Well, with a wall map of the world, a typical scale could be one to 30 million. But what does this mean? Well, the first thing that I noticed when I read it out was that I said one to 30 million. So what we have here is our colon. This is read as “to.” So we got one to 30 million.

Okay, so what does this mean? Well, what it means is that on the map, if we have a measurement of one centimeter, for instance, then what this will be equal to is 30 million centimeters in real life. Well, that wouldn’t be much use because it’s hard to really visualize what 30 million centimeters would be like. However, if we convert this to kilometers, then what we can see is that one centimeter on the wall map would be worth 300 kilometers in real life. And we get that because if we remember our conversion factors, well one meter is equal to 100 centimeters, one kilometer is equal to 1000 meters. So therefore, one kilometer is gonna be equal to 100,000 centimeters. And 100,000 multiplied by 300 would give us 30,000 [30,000,000].

Okay, great. So that’s an example of a wall map of the world. So if we look at something else here, so we got the road map for motorists, we can see that typically the scale of this would be one to 250,000. So what we can imagine is that actually this is more zoomed in because you’ll need to see more detail if you’re traveling as a motorist or driving. So once again, we can see that one centimeter on the map would be equal to 250,000 centimeters in real life. But again, that’s quite hard for us to kind of picture what that would be. So once again, we can convert. And we see that one centimeter on the map is gonna be worth 2.5 kilometers in real life.

Okay, great. So that’s another example of map that we’ve looked at, one which is more zoomed in and also closer up more detailed than the previous one. We’ll have a look at one more example of a scale and a scale drawing. Well, if we think about architects plans, so the plans for making an extension to a building or a new building, well, the scale of these are typically one to 100. So here what we can use is our top conversion, because we can see that one to 100 is gonna be the same as one centimeter to one meter, so we know that every centimeter on the plan is gonna be one meter in real life.

And if you think about the scenario that’s involved in, this can be much more sort of drilled down or much more zoomed in because when you’re actually building a building, there’s gonna be a lot more accuracy needed to construct each one of the parts. So what we’ve looked at there are some scales, how they’re used, and also how we can convert. But it is worth remembering that although each of the scale drawings that we’ve looked at in this example have been where the drawing itself is smaller than the real-life scenario, that in fact a scale drawing or model can be used to represent a smaller or larger object, shape, or image. So it can work both ways. Okay, so now let’s have a look at some examples and solve some problems.

Given that the distance between two cities on a map is 4.4 centimeters and the actual distance between them is 26.4 kilometers, what is the drawing scale of the map?

So if we think about the information we’ve been given, we know that we’ve got a map, but we don’t know the scale of the map. But what we do know is that the distance between two cities is 4.4 centimeters on the map. And what this represents in real life is 26.4 kilometers. Well, the first thing we’ll want to do if we want to work out the scale of the map is get each of our measurements in the same units. So let’s think of some conversions. First of all, we know that one meter is equal to 100 centimeters and one kilometer is equal to 1,000 meters. So therefore, we can say that one kilometer is gonna be equal to 100,000 centimeters cause we’re gonna get that by putting those together.

Okay, great. So this is now gonna help us cause what we can do is convert 26.4 kilometers into centimeters. So therefore, what we can say is that our scale is gonna be 4.4 centimeters is equal to 2,640,000 centimeters. And we get that because what we do is multiply 26.4 by 100,000. And when we multiply by 100,000, then what we do is move each of our digits five place values to the left. So we can see that our two goes from being a 10 to going to be a million.

Well, if we think about what we get on the map, typically, we’ll have a scale of one to something. So let’s see if we can do that with the scale that we’ve got here. But what we’re gonna do then is divide each side by 4.4. And when we do that, what we’re gonna get is a scale of one to 600,000. Well, if you think about what that’s actually going to mean, it means that one centimeter on our map is worth six kilometers in real life.

Okay, great. So we’ve had a look at one example. Let’s take another look at a different scenario. And what this scenario is going to do is actually develop the skills we’ve had here because we’re gonna use the distance between two cities on a map and then their actual distance to help us calculate what the actual distance is of two different cities on the same map.

The distance between two cities on a map is 17 centimeters. The actual distance between them is 68 kilometers. If the distance between two other cities on the same map is 15 centimeters, determine the real distance between them.

So the first thing we need to do to solve this problem is work out what the scale of our map is going to be. And to do that, what we’re gonna do is use the two distances that correlate. And that is that we know that 17 centimeters on the map is worth 68 kilometers in real life. Well, the first thing we want to do here is convert it so that both of our measures are in the same units. Well, if we remember that one meter is equal to 100 centimeters and one kilometer is equal to 1,000 meters, then therefore one kilometer is gonna be equal to 100,000 centimeters.

Okay, great. Well, now, if we convert, we can convert 68 kilometers to 6,800,000 centimeters by multiplying it by 100,000. And if we wanted to do that without a calculator, then all we do is we shift each of the digits from our 68 five spaces, so five places to the left. Well, when we’re looking at a scale of a map, we tend to see it as one to something. So let’s try and work out what ours is going to be if it’s one to something else. And to do that, what we’re gonna do is divide both sides of our scale by 17.

Well, when we do that, what we’re gonna get is a scale of one to 400,000. And we get that because we divided 6,800,000 by 17. Well, we can do this on a calculator. But if you want to use a mental method, then what we could do is do 68 divided by 17, which is four. Then we could see that we’d have one, two, three, four, five zeros that we’d then have to put on. So that would be 400,000. Okay, great. So we now have the scale of the map. So now what we want to do is use our map scale to work out what 15 centimeters on the map would be if we’re looking at the real distance.

Well, to work this out, what we’re gonna do is multiply each side of our scale by 15. So we do; we’re gonna have 15 centimeters on the map is worth six million centimeters in real life. Well, now what we need to do is convert this into something that’s meaningful because six million centimeters isn’t very useful. So what we’re gonna do is use our conversion factor where we know that one kilometer equals 100,000 centimeters. So we’re gonna divide six million by 100,000. So that’s gonna give us 60 kilometers. So therefore, we can say that the real distance between the two cities on the map that have a distance on the map of 15 centimeters is 60 kilometers.

So with both examples we’re given here and also with the introduction, we’ve looked at maps and how the scales are used within a map. What I’m gonna look at next is actually an example where we’re looking at an engineer’s model.

An engineer makes a model of a 600-meter-tall tower to a scale of one to 750. What is the model’s height in centimeters?

So if we think about the model the engineer’s made, well, we know that the scale is one to 750. And we know that the real height is 600 meters. However, we don’t know the height of our model. But what we do know from the scale is that one meter on our model is worth 750 meters in real life. Well, if we think about the scale factor — well, in this case, the scale factor is 750 — then what we know is that the scale factor is equal to the real length over the model length. So therefore, if we want to find the model length, this is gonna be equal to the real length divided by the scale factor.

So therefore, in our example, it’s a model height that we’re looking at. So the model height is gonna be equal to the real height, which is 600 meters, divided by the scale factor, which is 750, which is gonna be equal to 0.8 meters. Well, we know that one meter is equal to 100 centimeters, so therefore we could write this as 80 centimeters. So we know that the model’s height in centimeters is 80. And we’ve given it in centimeters because that was what was asked for in the question.

So here we’ve looked at some different examples involving maps and models. But what we’re gonna move on to now is something that we mentioned in the introduction. And that is the fact that when we’re looking at scales or looking at enlargements, it can involve either a magnification or a reduction. And we’re gonna work out whether what we’re looking at is in fact a magnification or a reduction.

The table below shows some information about a scale drawing. Complete the missing details, including whether the drawing is a magnification or reduction from real life.

And what we’re gonna do first of all is take a look at the scale. And what the scale tells us is that it’s 22 to one. And what we can see is that the 22 refers to the drawing distance, so it’s 22 parts, and the one refers to the real distance. So therefore, what we can see is that the drawing distance is in fact going to be bigger than the real distance. So therefore, we can surmise that this is actually going to be a reduction. However, what we can do is check this out by putting in our values.

So what we know is that the drawing distance is 66 centimeters. And we need to see how we get from 22 to 66. Well, what we do is we multiply by three. And if we multiply one by three, we’re gonna get three. So we know that the drawing distance is 66 centimeters. And therefore, the real distance is gonna be three centimeters. However, this is not quite the answer we want because if we check the table, the table says the real distance in millimeters.

So we need to remind ourselves of one of our conversion factors. And that is that one centimeter is equal to 10 millimeters. So therefore, the real distance is gonna be 30 millimeters because it’s three multiplied by 10, which gives is 30. And we know and can confirm that in fact it’s going to be a reduction from real life because we can see that the real-life distance is in fact smaller than the drawing distance.

So, as we said, we had a question here where we looked at whether something was a magnification or a reduction. So now what we’re gonna do is compare distances on two different maps with two different scales.

A map of Egypt was drawn at a scale of one to 340,000. A second map was drawn at a scale of one to 160,000. The distance between two cities on the first map is 16 centimeters. What is the distance between the same two cities on the second map?

But what we know is our first map has a scale of one to 340,000. And we know that the distance on the first map is 16 centimeters between the two cities. What we can see is that if we multiply one by 16, that gives us our 16. And that’s because our scale means that one centimeter on the map is worth 340,000 centimeters in real life. And we could use a calculator to work this out. So we get 340,000 multiplied by 16, which would give us 5,440,000, and that’s centimeters.

We could’ve also worked it out with a written method. That’s by multiplying 16 and 34, which would’ve given us 544. Then we’d see that we’d have one, two, three, four zeros out on the end, which would give us our 5,440,000 centimeters. Okay, so what would we do now? Well, in usual questions like this, what we would do is convert this into something that makes more sense, like kilometers. However, as we’re gonna transfer onto the second map, I’m gonna keep things in centimeters. We won’t change here what the units are in.

Well, with the second map, we know the scale is one to 160,000. So we know one centimeter on the map is worth 160,000 centimeters in real life. Well, we know the distance between the two cities in real life is 5,440,000 centimeters. So we’ve got to see how would we get there from 160,000. Well, in fact, we’d multiply by 34. If we take a look at the first map, we can see that because we’ve actually got numbers 34, 16, and 544 all involved with the calculations over there.

However, if we couldn’t work that out from the first map, what we could’ve done is use the bus stop method for short division. And we can see that 16 goes into 544 34 times. So therefore, 160,000 goes into 5,440,000 34 times. So therefore, if we do this for the first part of our scale multiplied by 34, we’re gonna get 34 centimeters. So therefore, we can say that the distance between two cities on the second map is 34 centimeters.

Now, finally, what we’re gonna do is we’re gonna look at a question that involves area and a scale. So let’s have a look at this example now.

A square-shaped yard with a side length of 91 meters was mapped with a drawing scale of one to 700. Determine its area in the scaled drawing in square centimeters.

Okay, so in this problem, the first thing we want to do is work out what the length would be on our drawing. So we have a scale which is one to 700 which means that on our drawing, one centimeter equals 700 centimeters in real life, or one meter equals 700 meters in real life. Well, because the question wants us to look at square centimeters, what we’re gonna do is use one of our conversion factors to convert the real distance to centimeters. We know that one meter is equal to 100 centimeters. So then we know that the real-life distance is gonna be equal to 9,100 centimeters cause 91 multiplied by 100 is 9,100.

Well, we can see that to get from 700 to 9,100, you multiply by 13. And we can do that again by dividing 9,100 by 700 using the calculator. Or if we use the bus stop method or the short division, we can see that 91 divided by seven is 13. So the same as 91, 100 or 9,100 divided by 700 is gonna give us 13. So then if we multiply one by 13, we get 13. So therefore, we know that each side of the yard, so our square yard, is gonna be 13 centimeters on our diagram.

Well, now we’ve got the length of a side of the square. Then what we need to do is work out the area cause the question wants us to work out the area. Well, the area of a square is equal to 𝐿 squared, so the length squared. So the area is gonna be equal to 13 squared, which will give an area in the scale drawing of 169 centimeters squared.

So, great. What we’ve looked at is a number of different examples and shown how we can use scales in both drawings and models. But now what we’re gonna do is have a look at a quick summary of the lesson.

Our first key point is that a scale drawing or model can be a magnification or a reduction. So it can in fact be bigger or smaller than the original. And with the second key point, we can see that the scale is usually written as a number to another number. And that’s written as a number colon another number. And what this would mean that if we had a scale drawing that one centimeter on the drawing would be worth 1,000 centimeters in real life.

What we’ve also seen is we’ve got a scale, for instance, like 22 to one, where we’ve got the large number on the left-hand side. This is going to be a reduction because what we say is that the value on the drawing or model is in fact bigger than that in real life. And finally, what we’ve done is we’ve worked out real distances using the distance on our map or scale model, or in fact vice versa. And we’ve done that using a number of our conversion factors. For instance, one meter is equal to 100 centimeters, one kilometer is equal to 1,000 meters, and one kilometer is equal to 100,000 centimeters.