Ratios (also called scales) have a format X:Y for example 1:500 or 2:3. The number of items involved may also be more than two, for example 2:3:5. Some examples of maths problems are:

1. The model is in the scale 3:500. The model is 12cm, how big is the real thing?

2. For every 2 red beans there are 3 green beans and 5 blue beans. There are 10 red beans, how many beans are there in total?

The first problem is a ‘scale’ problem and the second is a ‘ratio’ problem. The ‘scale’ problems are simpler than the ‘ratio’ problems, as all you are expected to do is work out the size of one thing in terms of the other thing. The ratio problems are a little more tricky as they may ask you to work out the total amount of something and often involve more than two ‘things’.

Scale and ratio problems basically require a good understanding of how fractional arithmetic works and in particular, to understand the following:

If you divide A by A you get 1 (which can be written as A / A = 1)
If you multiply anything by 1 you get the same thing (which can be written as A x 1 = A)
If you divide anything by 1 you get the same thing (which can be written as A / 1 = A)
If you have a multiplication followed by a division, then you can swap them around e.g.

A x B / C = A / C x B (which is known as transitivity)

You can cancel out fractions (make them smaller) if the top and bottom divide into each other using a common multiple e.g. 6 divided by 8 is the same as 3 divided by 4

The problems your child will face may be simpler than the ones on this website and can usually be tackled intuitively.

Intuitive Solution

For a simple scale problem, we might say the real object is 5 times bigger than the model, so the ratio is 1:5. Intuitively, a child will know, in this case, that if the model is 3cm, for example, then the real thing is going to be 15cm. They can probably also work out that if the real thing is 15cm then the model is 3cm. The reason they can intuitively see this is because the small ratio has the value ‘one’. This means the solution is found using either one multiplication or one division. However if you present them with the ratio 2:5, then this immediately involves two actions (a division and a multiplication) and this is more difficult to understand.

The way of thinking about this more complex problem is to visualise that you want to bring one side of the ratio down to ‘one’ and then the other side can be worked out using a single multiplication or division. However, the ‘complexity’ is to also comprehend that you do not want to actually work out the division immediately as it is often easier to wait to do this after cancelling out the factions. The following sections illustrate the technique.

Three types of question

Find the size of one thing, given the size of another thing

Example, the model is in the scale of 3:500. Find the size of the real thing given the model is 12cm?

The key is to bring ratio of the given number down to 1 and then multiply both sides by the given number.

So in this problem, the given number is for the model.

First bring the model ratio down to 1 by dividing both sides by 3, which can be visualised as follows:

3/3 : 500/3

Do not actually do the division yet! Leave that for later as most problems are laid out to make an easy division of the given number.

Now multiply both sides by the given number (500 in this case). Write down,

12x3/3 : 12x500/3

Working out both sides using basic fractional arithmetic (see the rules above) gives,

12:2000

The answer is 2000cm

If you had been given the real value was 2000cm, you can work out the model value using the same technique as above.

3/500 : 500/500 (bring the ratio for the given number down to 1)

2000x3/500 : 2000x500/500 (multiply both sides by the given number)

12:2000 (work out the fractional arithmetic)

The answer is 12cm

Find the total, given the size of one thing

In this type of question you are given the size of one thing and must work out the total. Note this type of question only applies to general ratio questions and not scale questions.

Example, for every 2 red beans there are 3 green beans and 5 blue beans. There are 12 red beans, how many beans are there in total?

Work out the sum of all the ratios 2 + 3 + 5 to give 10. You can now create a dummy scale problem and proceed as before. The dummy scale problem is

2:10

2/2 : 10/2 (bring the ratio of the given number down to 1)

12x2/2 : 12x10/2 (multiply both sides by the given number)

12:60 (work out the fractional arithmetic)

The answer is 60 beans

Find the size of one thing, given the total

This is the exact opposite of the above:

Example, for every 2 red beans there are 3 green beans and 5 blue beans. There are 60 beans in total, how many green beans are there?

Work out the sum of all the ratios 2 + 3 + 5 to give 10. You can now create a pretend scale problem in the form

3:10

3/10 : 10/10 (bring the ratio of the given number down to 1)

60x3/10 : 60x10/10 (multiply both sides by the given number)

18:60 (work out the fractional arithmetic)

The answer is 18 green beans.

How can the graph paper help?

In general, if you are confident, then the above rules can be applied easily in your head. However, it may help you to use the graph paper to visualise the necessary multiplication and division as follows.

Suppose you get the following question:

We only need to write down the information about the right had side (the real item) as we know the left hand side will end up as the given value (14).

Firstly, enter the fraction you are looking for (note this requires you to keep moving the cursor back once space).

Now enter the given value

Hopefully this is enough to visualise the solution – 14 cancels out with 2 to give 3 x 7, which gives 21. The answer is 21cm.