Ratio and Scale
Introduction

How do architects visualize big ideas? How do people find their way to new places without getting lost? How do sportsmen plan set moves? The answers to all these questions involve scaling.

Here we will be looking at how we use scaling (reduction and enlargement) to represent simplified versions of objects and plans.

Scaling up and down
Architects make models (of bridges, buildings, and so on), people use maps so as not to get lost, and sportsmen look at set formations on a scaled-down version of their playing field.
Figure 1.   A model of Edinburgh.
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Figure 2.   A 3-5-2 soccer formation.
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These are cases where rough models or sketches are not adequate. An effective model is one which gives as much detail as is required for the situation. So if a building is to have a square base and a height 4.3 times its width, only a properly scaled model will show what this will look like.

When important measurements are kept in proportion like this, we say that the model is a scale version of the original. So if a football pitch's width is half of its length, then a scaled-down version should also have these proportions.

A tennis court is 7 m wide and 21 m long. A scale plan of it is drawn with a width of 3.5 cm. What is the length of the plan?
  • cm
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Figure 3.   A plan of a tennis court.
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Ratio and models
When we make a scale copy of an object, the original and the copy have the same proportions. So, in Fig.4, you see that the dinosaur's tail length is about half the length of the whole skeleton for both the real skeleton and the model. You can see this more clearly by using the animation to zoom in on the model.

Figure 4.   A real dinosaur skeleton and a model skeleton.
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To describe how much an object has actually been scaled down (or up), we often use ratios. So the dinosaur model above has a scale of 1:12. This means that for every 1 cm that a particular part of the model measures, the corresponding part of the real skeleton measures 12 cm. This also means that the skeleton is 12 times the length of the model dinosaur, as you can see using the animation.

The head on the dinosaur model is 8 cm in length. How long is the head of the real dinosaur?
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Let's say you make a scale model of a car and all its measurements are one tenth of the actual measurements.

What is the scale of the model?

  • 1:
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Below are details of two measurements of the real car.

Figure 5.   Measurements of a car.
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What will be the length of your model of the car?
  • m (or cm)
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What will the height of your model be?
  • m (or cm)
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Ratio and maps
Whereas models are three-dimensional (3-D) representations of objects, maps are flat (2-D) representations of a plan view of an area. However, we still use
ratio
A ratio compares two quantities. The ratio of a to b is often written a:b. For example, if the ratio of the width to the length of a swimming pool is 1:3, the length is three times the width.
ratio
to relate maps to real-life dimensions. Maps differ in the amount of detail they contain, but the one feature common to most maps is the fact that they are to scale, whether they are used to direct motorists, or to show weather patterns or the heights of mountain ranges.

Figure 6.   A map of Europe.
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The map above has been made at a scale of 1:10,000,000.

What distance on the ground does 1 cm on the map in Fig.6 represent?
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A road represented by a line 8.7 cm long on a map is 104.4 km long in real life.

a) How many centimetres are there in 1 kilometre?

b) What is the scale of the map?

  • a) cm = 1 km

    b) 1:
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Scale factors
It is more usual in mathematics to express
enlargement
An enlargement is a type of transformation in which lengths are multiplied whilst directions and angles are preserved. The transformation is specified by a scale factor of enlargement and a centre of enlargement. For every point in the original shape, the transformation multiplies the distance between the point and the centre of enlargement by the scale factor.
enlargement
or reduction in terms of scale factors rather than ratios. The
scale factor
The scale factor is the ratio of distances between equivalent points on two geometrically similar shapes.
scale factor
is the number you multiply the original lengths by to get the lengths on the model. An exact full-size copy of something, therefore, has a scale factor of 1. In the model car example we looked at earlier, the scale factor is 0.1, since each of the measurements of the model car is one tenth of the corresponding measurement in the original.


Click on the figure below to interact with the model.

 Figure 7.  A car body and a model.



Fig.7 above shows a car body and a corresponding scale mode. Vary the size of the model using the handle and see how the scale factor changes.

Complete the sentences below.
  • A scale factor of less than 1 is a(n)...
    A scale factor of more than 1 is a(n)...
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In the model we can see that:





– where the original and model measurements are the lengths of equivalent parts (for instance, the height of the car's body). Rearranging this, we see that we can find the scale factor in the following way:





So, as long as we are given measurements of the original object and the scaled copy, we can work out what the scale factor is. Look at the following example.

A design for an invitation has been scaled down using a photocopier. The original and scaled copy measurements are shown in Fig.8 below.

Figure 8.   Scaling a design.
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Follow the steps to work out the scale factor of the enlargement.

We start by looking at equivalent measurements from the original and the scaled copy.
Let's look at the height measurements.
If this is the correct result, then it will also be the multiplier for any other measurement pairs – for example, the original and scaled widths.
     We can check the answer by multiplying the original width, 32 cm, by the scale factor, 0.25. This gives us 8 cm, which is indeed the correct width of the photocopy. Our answer must therefore be correct.



Now look at the model in Fig.9 below.


Click on the figure below to interact with the model.

 Figure 9.  Right-angled triangles.

What is the length of the
hypotenuse
The hypotenuse of a right-angled triangle is the side opposite the right angle.
hypotenuse
of
triangle
A triangle is a three-sided polygon.
triangle
1 on the left? (Place the mouse cursor over the relevant line and read off the length.)
  • cm
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What is the length of the hypotenuse of triangle 2 on the right?
  • cm
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Triangle 2 is an enlargement of triangle 1. What is the scale factor of this enlargement?
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If triangle 1 were to be resized using a scale factor of 0.4, what would be the length of the hypotenuse of the new triangle?
  • cm
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Summary

A scale model or drawing has exactly the same shape as the original object, and all the same proportions, but a different overall size.

The scale of a model can be represented as a ratio, which indicates the relative size of the model and the real object.

Ratios can be used to express the scale of a map. You can use the scale of a map to calculate the real distance between two towns.

Scale factors express the degree of enlargement or reduction.

The scale factor can be calculated using measurements of corresponding lengths on the original and the model, using the equation:




A scale factor of less than 1 reduces the size of an object. A scale factor greater than 1 increases its size.
Exercises
1. The scale of a map used by walkers is 1:25,000. What distance on the map represents 1 km (1000 m) on the ground?
  • m ( cm)
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2. An archaeologist discovers a scale drawing of a lost ancient Egyptian city. On it is a road that measures 2 cm long. A written account of the city describes this road as being 1000 cubits long. (Cubits are an ancient Egyptian unit of measurement.) On the map, the entire city measures 10 cm long by 7 cm wide. What are the dimensions of the real city in cubits? (You do not need to know the size of a cubit to calculate the answer.)

  • The lost city was cubits long and cubits wide.
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Figure 10.   The plan for the ground floor of a house. The scale is 1:250.
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3. Using the plan of a house in Fig.10 above select the correct answer for each of the measurements below. The dining room and the entrance room are both square. Length always refers to the longest dimension of a room and width to the shortest dimension.
  • The length of the lounge
    The width of the lounge
    The length and width of the dining room
    The length of the garden
    The width of the garden
    The length and width of the entrance room
    The length of the kitchen
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4. A photograph measures 6 cm by 4 cm. It is sent through a photocopier with a scale factor of 0.75. What are the dimensions of the copy of the photograph?

  • The copy measures cm by cm.
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