Direct and Inverse Proportion

One of the old favourites – Proportion is often something that many students struggle with. It sits in the ‘sweet spot’ of level 4 / 5, and so in order to get those all-important marks at that level, it is one of the topics that I suggest is well worth spending some time practising and getting to grips with. Also, it is perhaps one of the more useful skills to have in your toolbox for ‘real-life’ as well, somewhere that maths doesn’t always seem to live!

To start with, you will need to be confident in two other skills, namely basic rearranging of formulae and solving basic linear equations.

In its most basic form, it is how one amount changes as another changes. So, for example, (and it is a classic example!) if you want to make twice as many cakes, you need twice as many ingredients. On that level, it seems to be obvious. However, there are many ways in which the difficulty can be ramped up in GCSE maths.

Direct Proportion

Here is an example of a question at the first level of difficulty:

A recipe requires 3 eggs to make 12 cupcakes. How many eggs do I need to make 36 cupcakes?

Here we are looking at how many times more cupcakes we are looking at making :

So, 3612 = 3

This means we need 3 times more eggs, 3 x 3 = 9

OR to put it another way :

12 Cupcakes 36 Cupcakes
Needs Needs
3 Eggs 9 Eggs

This is an example of DIRECT PROPORTION – as one increases, so does the other at the same rate, or in the same proportion – think how many times more?

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Pancakes!

Next up, we get the less obvious increases, again, in direct proportion, but with slightly more difficult numbers involved, but only because the number of times each is increasing isn’t so obvious. It is easier to go through this using a worked example:

A recipe states that you need 200g of flour to make 8 pancakes. How much flour do you need to make 30 pancakes?

This is about adopting a slightly different strategy, based on working out how much flour one pancake needs in order to then multiply it back up by 30. Ask yourself, “How much do I need for one?”

From the first part of the question, we can see that 200 ÷ 8 = 25g of flour per pancake

So, it follows that 25 x 30 is how much we need to make 30 pancakes = 750g.

Decreases in Direct Proportion

Now let’s have a look at an example of a decrease, remaining in direct proportion. In this case, we are looking at as one decreases, so does the other at the same rate, or in the same proportion – think how many times less?

Using the first question we looked at, let’s see how this might work:

A recipe requires 3 eggs to make 12 cupcakes. How many cupcakes can I make if I only have 1 egg?

We have 3 times less in terms of the number of eggs, so we can make three times fewer cupcakes

3 Eggs 1 Egg
Makes Makes
12 Cupcakes 4 Cupcakes

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How Much For One?

Once again, let’s do the same with some less obvious numbers, using the “How much for one?” approach.

A recipe for making 16 pancakes uses 800g of flour. How much flour do we need to make 10 pancakes?

So for this – let’s look at the “How much for one?” approach :

If 16 pancakes need 800g of flour, it stands that 1 pancake will need (800÷16)g = 50g

We now know we need 50g per pancake, so 10 pancakes will need 10 x 50 = 500g.

Once you have got to grips with the “how many times more / less?” and “how much for one?” ideas then we can finish with a look at the most common way these questions are framed in a GCSE assessment – the ‘Best Buy” question.

Proportion in Daily Life

Go to any supermarket and you will see all kinds of offers and similar products from different brands. Best Buy questions are all about value for money and being able to compare two very similar products or offers.

There are two main types of question at level 4 / 5 and are as follows:

What we need to ask ourselves here is “how much for one?” – but that leads to “one what?”

We need to identify a common unit of volume in this question that can be applied to both buying options – given the amounts in question, I would suggest finding out how much one-litre costs in each case:

Multipack 6 x 330ml = 1980ml = 1.98 litres
Cost of this is £1.70
So cost per litre = £1.70 ÷ 1.98 L = £0.85858585……, so rounded to 86p per litre.

For the bottle, we have 1.5 litres
The cost of this is £1.50
So cost per litre = £1.50 ÷ 1.50 L = £1.00 per litre.

From the two calculations, we can now compare the costs and can see that the multipack offers better value for money as the cost per litre is less.

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Special Offers

Another classic example is the “special offer” scenario and involves the use of percentages and other skills:

Laundry liquid normally costs £6 per bottle.

Two different supermarkets have competing offers as follows:

Tesdi supermarket – 30% off all laundry liquid.

Snozbury’s supermarket – buy 3 get one free on laundry liquid.

Which offers the best value?

For Tesdi, we now have a new price of £6 – 30% = £4.20 per bottle.

For Snozbury’s we are now buying 4 bottles, but only paying for 3.
Total cost = 3 x £6 = £18
This is now spread over the 4 bottles, so cost per bottle is £18 ÷ 4 = £4.50 per bottle

So from this, we can see the 30% off offer from Tesdi is the better value.

For these types of question, the two questions that we need to consider are: “how many times more/less?” and/or “how much for one?” in order to do a direct comparison and are all linked directly to proportion.

A bit about the author of this article, David M:

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