Maths

Strengthening your numeracy skills may be a requirement for your university study.

Here we introduce the basics of working with numbers through the topics of number types, operations, fractions, decimals and percentages, and ratios and proportions. Across each section, we provide explanations, worked examples and knowledge checks to enable learning at your own pace.

Types of numbers

It is useful to know that numbers come in different forms, and you may encounter one or more of these forms. Knowing the different types of numbers helps you understand how they behave, how they can be written, and how they are used in calculations.

Have a look at the example below for further explanation.

Once you understand the different families of numbers, the next step is to see how these numbers are positioned and compared using the number line.

The number line

The number line arranges real numbers from left to right. Smaller numbers are further left; larger numbers are further right. This rule is especially useful for negative numbers: -1 is greater than -5 because -1 is closer to zero and lies to the right of -5.

Watch the video below explaining the number line.

After using the number line to compare numbers, we can also use it to understand how far a number is from zero. This is called the absolute value of a number. It is written using vertical bars, such as |x|, and it gives the distance from zero regardless of direction. For example, |7| = 7 and |-7| = 7.

It is not uncommon to find that you need the knowledge of number types, the number line and or absolute values in your course. For example, you may find them:

• when reading graphs,
• when comparing positive and negative values,
• interpreting measurements,
• working with data,
• understanding and interpreting coordinates, or
• checking how far a value is from a target or starting point.

Operations and order of operations

In many calculations, there is more than one operation to do. For example, a calculation may include the following operators all in the same line: adding, multiplying, brackets and powers. The order matters because doing the steps in the wrong order can give a completely different answer.

BODMAS

To avoid confusion over the order of calculation, we adopt the acronym BODMAS (See below)

Why is BODMAS useful?

It helps us follow the order in which we need to read and solve calculations involving theses operators. For example, if a lab formula, or a finance calculation includes brackets and multiplication, the answer depends on doing the steps in the correct order. Without this order of operation, two people may approach the same question or try to solve same problem but arrive at a different answer/ outcome.

Note: When operations are of the same rank, we work from left to right. For example, addition and subtraction have equal priority, so if subtraction appears before addition in an expression, we do the subtraction first. Similarly, multiplication and division have equal priority, so we also work from left to right.

Now that we know the order to follow in a calculation, the next thing to watch is the sign of each number. This is especially important when a calculation includes positive and negative numbers, because the signs can change the final answer. To have a grasp of how this is done, we briefly show how common operators work below.

Adding, subtracting, multiplying and dividing integers

As we have seen under the section where we discussed the different types of numbers, Integers include positive numbers, negative numbers and zero. You will often meet them in calculations involving scores, money, temperatures, data changes, measurements, coordinates or values above and below a starting point.

If you tried to think through the examples in above table based on the operations, you would have seen that for addition and subtraction operations, it is helpful to use the number line. However, division and multiplication operations are not exactly straightforward with the number line. There is another very handy tool that helps.

Sign rules for multiplication

Watch the video below for further explanation and an example of the table above.

Fractions, decimals and percentages

Fractions, decimals and percentages are three ways of expressing parts of a whole. In academic, professional and everyday contexts, fluency between these representations is essential.

Before we proceed, it is important to define some of the terms that are critical to understanding the concept fractions, decimals, and percentages. Peruse the table below to familiarise yourself with these terms.

Click through the slides below for very basic examples where we define the operation and the method employed in fractions, decimals and percentages.

To learn more about converting decimals in to fractions, watch the video below.

Fractions, decimals and percentages are different ways of showing parts of a whole, so it is useful to move confidently between them. When working with fractions, pay close attention to denominators and reciprocals, and when working with percentages, use multipliers to make increases, decreases and reverse percentage problems quicker and ensure correctness. As you practise, notice which form feels easiest for you, then use that strength to improve the others by converting from your strongest form into the one you find most difficult.

Ratios and proportions

Ratios and proportions are both about comparing quantities. You use them when you want to know how much of one thing relates to another, how to scale something up or down, or how quickly something changes.

You may already use these ideas without calling them “ratios” or “proportions”. For example, you might compare prices in a shop, increase a recipe for more people, work out speed, share money fairly, read a map scale, or calculate how long a task will take if more people help. Let’s walk through a few examples to see how to handle maths relating to ratios and proportions.

A ratio compares amounts of the same kind. For example, if a drink is made using squash and water, the ratio tells us how much squash is used compared with how much water is used. Ratios are often written using a colon, such as 1:4, but they can also be written in words as 1 to 4. In some situations, they may also be written as a fraction, but the colon form is usually clearer when comparing parts.

Ratios can also be simplified, just like fractions. Simplifying a ratio does not change the relationship between the quantities; it only makes the comparison easier to read. This is useful because large numbers can sometimes hide a simple relationship.

Sometimes, a ratio tells us how to share something. This could be money, ingredients, your module assessment marks between assessment types, materials or any other quantity. To share an amount in a ratio, add the ratio parts, divide the total amount by the number of parts, multiply one part by each number in the ratio, and then check that the shares add back to the original total.

Reasoning with ratio: If a drink is mixed using 2 parts squash and 7 parts water, the ratio of squash to water is 2:7. This means that for every 2 parts of squash, there are 7 parts of water.

Proportional reasoning

Proportional reasoning is about noticing how quantities change together. A useful question to ask is: if one amount changes, what happens to the other amount? This helps you decide whether a problem is about direct proportion or inverse proportion.

In direct proportion, both quantities move in the same direction. If one quantity increases, the other also increases. If one quantity decreases, the other also decreases. For example, if each snack from the University Cafeteria costs the same amount, buying more snacks would simply mean paying more money. Whereas, in inverse proportion, one quantity increases while the other decreases. This often happens when the same task, distance or amount is being shared differently.

For example, the more workers you hire, the faster a job gets completed. If you quadruple the number of workers, the time taken takes is divided by four. Thus, if one worker would complete the job in 12 hours, a team of 4 should finish it in 3 hours.

Practical ways to learn and use quantitative thinking

Mathematics and quantitative reasoning are skills that can be developed with practice. They are not fixed talents that some students have and others do not. The most effective way to improve is to be active: try questions yourself, write out your method, check your answers, and look carefully at any mistakes.

When you make an error, do not treat it as failure. Use it as information and learn from it.

It is also better to practise in short, regular sessions than to leave everything until the last minute. Returning to a topic several times helps it become more familiar. Mixing different types of questions is especially useful because it trains you to recognise which method is needed, instead of only repeating the same procedure.

You can also strengthen your understanding by explaining your method to someone else, comparing approaches with a peer, booking and attending maths support sessions, and using feedback from tutors. If something feels unclear, ask for help early. Small gaps are much easier to fix before they affect later topics.

Key takeaways

The maths and numeracy skills you have learned should come in handy on your course and everyday life. This page has covered some essential parts of this great subject and has hopefully boosted your confidence in types of numbers, operations, fractions, decimals and percentages, and working with ratios and proportions.

Credit

Written by Ade Olowolayemo and Bojan Koltaj

Last updated 2026