How to calculate ratio part IV: scale drawings and maps

Another in a series of hubs designed to help readers tackle problems in everyday maths

Ratio and proportion is a subject that many people find tricky, but the trickiest part of all (for quite a few of us) is working with scale. As with one of my earlier hubs (How to calculate ratio part II), I will try and demystify the subject by giving a few worked examples, followed by some questions for you to try… answers to follow in another hub!

Scale drawings are used whenever the actual size of an item is too large to go down on paper. Maps of countries are always drawn to scale, for obvious reasons! Scale drawings are also used in design work, e.g. architecture. On a more everyday level, it can also be extremely helpful to use scale drawings to help plan out a new kitchen or rearrange the furniture in a room.

You can express scale in several ways, either as a ratio without any units (e.g. "1 : 50") or with the units written down.  So a map might well have a scale of 5 cm : 2 km, for example.


Worked example 1

The scale of the map of Kent shown here is  5 cm : 25 km.  How would you write this as a ratio?

Your aim in doing this is to lose the units and end up with an answer like "1 : 10" or "3 : 20".

In order to be able to get rid of the units, you have to make sure that both units are the same to start with (once they are the same, you can just delete them). In other words you need to convert the 25 km to centimetres (cm)! This is quite a daunting task so a lot of people prefer to do it in stages, by first converting to metres and then to cm.

1 km = 1,000 m. Therefore you can write the scale as

5 cm : 25,000 m (since 25 km = 25 x 1,000 m)

We're still not there yet, because the 25,000 m has to be converted to cm!

1 m = 100 cm

So the scale can be written as

5 cm : 2,500,000 cm (since 25,000 m = 25,000 x 100 cm)

Deleting the units gives

5 : 2,500,000

We're still not quite there yet, because you can do a bit of cancelling - divide both numbers by 5 and you get:

1 : 500,000

So the scale of the map of Kent can be written as 1 : 500,000

Worked example 2

Rachel is redecorating her living room, which is rectangular and measures 6.5 metres by 3 metres. She decides to do a scale drawing of the room and its furniture so that she can see how well certain items fit, and whether there would be room for more. The scale drawing measures 13 cm by 9 cm. What scale has she used in the drawing?

When solving this type of question, you can take either the length or the width of the real room to work out the scale used in the drawing - it doesn't matter which.

We'll use the width here, because the number is slightly simpler.

Expressed using the relevant units, the scale is:

9 cm : 3 m

As with worked example 1, you need to change the units so that they're the same throughout.

3 m = 300 cm, so the scale becomes:

9 cm : 300 cm

You can now get rid of the units altogether to give:

9 : 300

which cancels down to give:

3 : 100

Now try the following questions yourself.

Question 1

Rearrange the following map scales in order, from smallest to largest:

(1) 1 cm : 5 km
(2) 2 : 5,000
(3) 1 cm : 100 m
(4) 5 cm : 2 km
(5) 3 : 40,000

Hint: the larger the scale, the closer the "zoom" - for example, a map of London on a single A4 page has a larger scale than a map of the entire British Isles on the same sized page. So a scale of 1 : 10 is larger than 1 : 100, which is larger than 1 : 1,000 and so on.

Question 2

You are designing a theatre set. The actual size of the stage is 12 metres by 8 metres, and you want to create a scale drawing using a scale of 1 : 200. What are the dimensions of your scale drawing in cm?


Question 3

You decide that the scale drawing you created in Question 2 isn't big enough, so you decide to create another one on a scale of 3 : 400. Again, what are the dimensions of your scale drawing in cm?


Question 4

A model boat enthusiast has built a scale model of his boat, which is 500 mm long. The actual boat is 25 metres long. What is the scale of the model written as a ratio? (10 mm = 1 cm)


Question 5

The UK Ordinance Survey's "Explorer^TM" map has a scale of 1 : 25,000. How would you express this in cm and km (e.g. "3 cm : 5 km")?

Answers are in this hub: How to calculate ratio part V.

© Empress Felicity August 2010