Ratio and Scaled Figures

What is a ratio?

Let's actually see a ratio in action, shall we:

Hmm! Yes, very impressive, but what does it actually mean? Well, let's suppose that the ratio represents a comparison of the number of boys and the number of girls in a class of children. We say that the ratio of boys to girls is 3:4, pronounced "3 to 4".

This means that there are more girls than boys, and that for every 3 boys there are 4 girls. It doesn't mean that the class contains only 3 boys and 4 girls, but simply lets us compare the number of boys to the number of girls.

In fact, let's suppose the class contains 28 children. We can imagine those children being split up into groups, so that each group contains 3 boys and 4 girls, as shown in the diagram below:

In this case, you can see that there are 4 groups all together, each with 3 boys and 4 girls. Clearly, there must be 4 x 3 = 12 boys, and 4 x 4 = 16 girls. This does mean that there are 28 children all together (12 + 16 = 28, a useful double check!) and that for every 3 boys there are 4 girls.

We could equally say that the ratio of boys to girls is 12 : 16 (matching the numbers of boys and girls exactly), but it is easier to cancel this ratio in a similar manner to fractions, to give 3 : 4.

We cancelled the ratio by dividing both the first and the second number by 2, not once but twice. This should remind you of cancelling the numerator and denominator of fractions.

Here's another example. An alloy of metal is formed by melting copper, zinc and tin in the ratio 1:3:2 and mixing them together. What does this tell us? Well, you can see just by that ratio that there is going to be more tin in the alloy than copper (2 parts as opposed to 1 part) and more zinc in the alloy than tin (3 parts as opposed to 2 parts). You can also see that we need three figures in the ratio now instead of two as we are comparing measurements of three metals instead of two types of children. copper : zinc : tin = 1 : 3 : 2

How much of each metal would be contained in 600g of the alloy? Well, we can split 600 up in the same ratio as before by dividing it by 6. Where did 6 come from? Well, we found it by adding 1 + 3 + 2 = 6. This gives us 100, which is the size of 1 "part" of the alloy.

This term "part" is a rather woolly one. It is equivalent to the 7 children (3 + 4) in each of the groups in the first example. In this case, the copper accounts for 1 part of the alloy, the zinc accounts for 3 parts (as there is 3 times as much zinc as copper in the alloy) and the tin accounts for 2 parts (as there is 2 times as much tin as copper in the alloy). This makes 1 part + 3 parts + 2 parts = 6 parts in total, each of 100g.

There must be 100g of copper, 300g of zinc and 200g of tin in every 600g of the alloy. The ratio between the metals is 100:300:200 which you can see at a glance does indeed cancel to give 1:3:2.

How much of the alloy would you need in order to be sure of having exactly 130g of tin? Well, we know that tin represents two "parts", so in this question, a part must represent 130 / 2 = 65g. Now we can work out the quantities of the other two metals. Copper is 1 part, i.e. 65g. Zinc is 3 parts, i.e. 3 x 65 = 195g. The total weight of the alloy must be 65 + 195 + 130 = 390g.

What you have learned is how to split a figure in a given ratio. The steps are as follows:

1. Add the numbers that form the ratio to get the total number of equal parts into which figure must be split.
2. Divide the figure by the total number of parts.
3. Examine the ratio figure by figure to see how many parts each figure corresponds to.

For another worked example of this, please click here.

For a question on this, please click here.

Comparing using Ratios

Peter's height and the height of his brother are in the ratio 5:7. If Peter is 120cm tall, how tall is his brother? In this case, we don't have to split the 120 into the ratio 5:7, as it does not represent the total height of Peter and his brother together. Instead, the 120 is Peter's height, which represents 5 "parts" on some scale. Clearly, 1 "part" must be 120 / 5 = 24 cm.

We know that Peter is 5 parts tall (5 x 24 = 120cm, yes, we know that!) and the brother is 7 parts tall on the same scale. This means that the brother is 7 x 24 = 168cm. The ratio between the two heights is 120:168, which does indeed cancel down to give 5:7 (Try the maths and see for yourself!)

In this type of question, we have been given the size of one side of the ratio and we are asked to calculate the size of the other side. We can again talk in terms of "parts", where we can find out the size of one part by looking at the figure in the ratio corresponding to the quantity that we have been given (Here the 5 in the ratio corresponds to 120cm). Once we have the size of one part, we can then look at the ratio to see how many parts each of the figures corresponds to.

If you would like another worked example of this, please click here.

If you would like a question on this, please click here.

Similar Figures

Similar figures are two objects, diagrams or shapes which are exactly the same shape but different size. They can be simple mathematical figures such as triangles or rectangles, or complex images such as a building and a photograph (or blueprint) of the building. The important thing to remember is that all the measurements of distance are in the same ratio when comparing the small object to the large object.

Here is an example: "A building is 120m long. An exact plan view of the building is constructed so that the ratio of distances on the plan to distances in the real building is 1 : 200. Calculate the length of the building in the plan view."

This is the same as Peter and his brother, except that we are going from large to small rather than the other way round. We know that the building is 200 units long on some scale and that the building on the plan is 1 unit long on the same scale. This means that the real building is 200 times as long as the plan. The building on the plan is 120 / 200 metres = 0.6 m or 60 cm.

Because the plan is a similar figure, we can compare other dimensions. Suppose the width of the building on the plan is 45 cm. How wide is the real building? Using the same logic, the real building is 200 times wider than that on the plan, so the real building is 200 x 45 cm = 9000cm or 90m wide.

The door of the main building has a width of 3.5 metres. The door on the plan will have a width of 3.5 / 200 = 0.0175 metres, which is 1.75 cm. We can compare any measurement of distance on the plan with the corresponding measurement on the real building and work out one from the other.

Please note: Complications arise when you compare derived measurements, such as the area of similar figures, or their volumes (for three-dimensional similar figures). However, this tutorial is about ratios rather than similar figures as such, so I won't go into that here.

Here's another example involving mathematical shapes:

The diagram shows two similar parallelograms - they are the same shape, but different in size. The side lengths of the two parallelograms are in the same ratio - in this case 2 : 1, so that each side on the larger parallelogram is twice the size of the corresponding side on the smaller parallelogram.

I have conveniently drawn both parallelograms in the same orientation (i.e. both the same way up), so we can easily compare measurements. If you look at the base of the parallelograms, you will see that they are 20cm and 10cm, which is a ratio of 2 : 1. Similarly (no pun intended!) the shorter sides of the parallelograms are 12cm and 6cm, which are also in the ratio 2 : 1.

Let's try a question involving mathematical shapes. The following diagram shows a vertical flag-pole which casts a shadow 26.5m long on flat, horizontal ground. A vertical stick, 1.5m long, casts a shadow 1.25m long, also on flat, horizontal ground. How can we use this information to calculate the height of the flag-pole? Because the rays of the sun come from many millions of miles away, they are effectively parallel, and so the flag-pole, the stick and their two shadows form two similar triangles. We don't know all the measurements, but we can calculate the ratio between the two triangles.

The large shadow is 26.5m and the small one is 1.25m. This gives a ratio of 26.5 : 1.25 (funnily enough!) which cancels to give 21.2 : 1 (we find this by dividing both original numbers by 1.25). This tells us that every measurement on the flag-pole is 21.2 times as large as the corresponding measurement on the stick. Since the stick is 1.5m long, the flag-pole must be 1.5 x 21.2 = 31.8m tall.

Map scales

A scale of 1 : 25 000, for instance, means that any measurement in real life is 25 000 times bigger than the equivalent measurement on the map. For instance, a distance of 1 cm on the map represents a distance of 25 000 cm (i.e. 250 m) in real life. Similarly, a distance of 1 inch on the map represents a distance of 25 000 inches in real life (which would then probably be converted to yards and feet). The map scale does not specify a unit of measurement (i.e. the scale is not given as 1m : 25 000m or as 1 inch : 25 000 inches) as it works for any measurement scale you choose to use.

Let's try an actual question. An electricity transmission line (pylon line) is 9.2cm long on a map whose scale is 1 : 15 000. Calculate the length of the transmission line in real life.