Structure and CalculationAQA GCSE Maths: Revision notes

Section 1

How Do We Order and Compare Numbers?

Numbers can be integers, decimals or fractions, and you must be able to order them and compare them using the correct symbols.

$=$ means equal to
$\neq$ means not equal to
$<$ means less than, $>$ means greater than
$\leq$ means less than or equal to, $\geq$ means greater than or equal to
A number line can be used to order positive and negative integers, decimals and fractions visually

When comparing fractions or decimals, convert them to the same form (e.g. a common denominator, or the same number of decimal places) before comparing.

Key termsintegerinequality symbol

Exam tip

When ordering negative numbers, remember that $-5$ is smaller than $-2$ — the further left on the number line, the smaller the value.

Section 2

How Do We Apply the Four Operations to Any Number?

The four operations — addition, subtraction, multiplication and division — must be applied confidently to integers, decimals and fractions (proper, improper and mixed numbers), including when negative numbers are involved.

Formal written methods (e.g. column addition/subtraction, long multiplication, long division) should be used for calculations that cannot easily be done mentally.

Place value is essential when calculating with decimals or very large/small numbers — for example, correctly lining up decimal points, or knowing that shifting a digit one column left multiplies it by 10.

Key termsplace valueformal written method

Common mistake

A common error is misaligning decimal points when adding or subtracting decimals — always line up the decimal points, not the right-hand digits.

Section 3

What Are Inverse Operations and the Order of Operations?

Inverse operations undo each other: addition undoes subtraction, multiplication undoes division, squaring undoes square rooting. Recognising inverse operations lets you check answers or simplify calculations (e.g. cancelling a multiplication with a division).

When a calculation has more than one operation, follow the conventional order of operations:

Brackets
Powers, roots and reciprocals
Multiplication and division (left to right)
Addition and subtraction (left to right)

A reciprocal of a number $x$ is $\frac{1}{x}$ ; multiplying a number by its reciprocal always gives 1.

Key termsinverse operationreciprocal

Example

$3 + 4 \times 2 = 3 + 8 = 11$ , not $14$ , because multiplication is done before addition.

Section 4

How Do We List Outcomes Systematically? (Higher)

Systematic listing strategies help you count or list every possible outcome without missing any or repeating any — using ordered lists, tables or diagrams.

For larger counting problems, the product rule for counting states that if there are $m$ ways of doing one thing and $n$ ways of doing another, there are $m \times n$ ways of doing both.

Key termsproduct rule for counting

Example

Choosing a starter from 3 options and a main from 4 options gives $3 \times 4 = 12$ possible meal combinations.

Section 5

What Financial Vocabulary Do I Need?

Household finance calculations use specific vocabulary that you must understand and apply correctly:

Term	Meaning
Profit	Selling price minus cost price, when positive
Loss	Selling price minus cost price, when negative
Cost price	The price paid to buy or produce an item
Selling price	The price an item is sold for
Debit	Money taken out of an account
Credit	Money paid into an account
Balance	The amount remaining in an account
Income tax	Tax paid on earnings
VAT	Value Added Tax, a percentage added to the price of goods/services
Interest rate	The percentage charged or earned on money borrowed or saved

Key termsprofitVAT

Must Know

Use $=$ , $\neq$ , $<$ , $>$ , $\leq$ , $\geq$ correctly to order and compare integers, decimals and fractions
Apply the four operations with formal written methods to positive and negative integers, decimals and fractions
Follow the correct order of operations: brackets, powers/roots/reciprocals, multiplication/division, addition/subtraction
Recognise and use inverse operations to check or simplify calculations
Use systematic listing and the product rule ( $m \times n$ ) to count outcomes without missing or repeating any
Know the household finance terms: profit, loss, cost price, selling price, debit, credit, balance, income tax, VAT, interest rate

That's the notes covered.

Carry on to the next subtopic.