### 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.