Structure and Calculation Notes

Q: How do we use the four operations with any number?

Use formal written methods and place value for integers, decimals and fractions, including negatives. Add, subtract, multiply and divide integers, decimals and fractions (proper, improper and mixed), including negatives.

Q: What are inverse operations and the order of operations?

Inverse operations undo each other, and a calculation is done in the order: brackets, powers, multiplication and division, addition and subtraction. Inverse operations undo each other: addition and subtraction, multiplication and division, squaring and square rooting. Use them to check answers.

Q: How do we list outcomes systematically?

Use ordered lists, tables or diagrams, and the product rule to count (Higher). Systematic listing makes sure you count or list every outcome once, using ordered lists, tables or diagrams.

Q: What financial words do I need?

Know the terms for buying and selling, accounts, tax and interest. Household finance questions use specific words. Profit is selling price minus cost price when positive; loss is the same difference when negative.

AQA GCSE Maths: Revision notes

Key facts

Use $=$ , $\neq$ , $<$ , $>$ , $\leq$ , $\geq$ to compare numbers; a number line helps order them.
Apply the four operations to integers, decimals and fractions, including negatives.
Order of operations: brackets, powers/roots, multiplication and division, addition and subtraction.
Inverse operations undo each other; the reciprocal of $x$ is $\frac{1}{x}$ .
Product rule: $m$ ways and $n$ ways give $m \times n$ ways (Higher).
Learn the finance vocabulary: profit, loss, cost price, selling price, debit, credit, balance, income tax, VAT.

1
Brackets
first
2
Powers and roots
next
3
× and ÷
left to right
4
and −
left to right

Order of operations

Ordering and comparing

Convert to a common form, then use the symbols and a number line.

Numbers can be integers, decimals or fractions. To compare, convert to the same form first, for example a common denominator or the same number of decimal places.

Symbols: $=$ equal, $\neq$ not equal, $<$ less than, $>$ greater than, $\leq$ less than or equal, $\geq$ greater than or equal.

Reading the number line left to right gives ascending order: $-3 < -1.5 < 0.25 < 2$ .

Which is the smallest?

The four operations

Use formal written methods and place value for integers, decimals and fractions, including negatives.

Add, subtract, multiply and divide integers, decimals and fractions (proper, improper and mixed), including negatives. Use formal written methods such as column addition and long division for calculations that cannot be done mentally.

Place value matters: line up decimal points, and remember that shifting a digit one place left multiplies it by 10.

Integers

Column methods
Take care with negative signs

Decimals

Line up the decimal points
Not the right-hand digits

Fractions

Common denominator to add or subtract
Proper, improper and mixed

Worked example

Work out $3.45 + 12.8$ using a column method.

1
Line up the decimal points and add a zero: $3.45$ and $12.80$ .
2
Hundredths: $5 + 0 = 5$ .
3
Tenths: $4 + 8 = 12$ , write $2$ , carry $1$ .
4
Units: $3 + 2 + 1 = 6$ ; tens: $1$ .

What is $3.5 + 0.25$ ?

Operations and order

Inverse operations undo each other, and a calculation is done in the order: brackets, powers, multiplication and division, addition and subtraction.

Inverse operations undo each other: addition and subtraction, multiplication and division, squaring and square rooting. Use them to check answers.

The reciprocal of $x$ is $\frac{1}{x}$ . A number times its reciprocal is always 1.

Work left to right within multiplication and division, and within addition and subtraction.

1
Brackets
first
2
Powers, roots, reciprocals
second
3
Multiply and divide
left to right
4
Add and subtract
left to right

Order of operations

Worked example

Work out $12 - 2 \times (3 + 1)^2 \div 8$ .

1
Brackets: $3 + 1 = 4$ .
2
Powers: $4^2 = 16$ .
3
Multiply and divide: $2 \times 16 = 32$ , then $32 \div 8 = 4$ .
4
Subtract: $12 - 4 = 8$ .

What is $3 + 4 \times 2$ ?

Systematic listing

Use ordered lists, tables or diagrams, and the product rule to count (Higher).

Systematic listing makes sure you count or list every outcome once, using ordered lists, tables or diagrams.

The product rule for counting: $m$ ways of doing one thing and $n$ ways of doing another give $m \times n$ ways of doing both.

A starter from 3 options and a main from 4 options gives $3 \times 4 = 12$ meals.

The product rule: 3 starters and 4 mains give 3 × 4 = 12 different meals.

$m$ ways and $n$ ways $m \times n$ combinations

Worked example

A shop sells 5 shirts and 3 pairs of trousers. How many outfits (one shirt and one pair of trousers) are possible?

1
$m = 5$ shirts.
2
$n = 3$ trousers.
3
Product rule: $5 \times 3$ .

A café has 4 drinks and 6 cakes. How many drink and cake combinations?

Financial vocabulary

Know the terms for buying and selling, accounts, tax and interest.

Household finance questions use specific words. Profit is selling price minus cost price when positive; loss is the same difference when negative.

A debit takes money out of an account and a credit pays money in. The balance is what remains.

Illustrative example: bought for £40 (cost price) and sold for £55 (selling price) gives a profit of £15.

Buying and selling

Cost price: paid to buy or make
Selling price: sold for
Profit or loss

Accounts

Debit: money out
Credit: money in
Balance: amount left

Tax and interest

Income tax: on earnings
VAT: added to prices
Interest rate

Money paid into an account is a

Try an exam question

Work out $12 - 2 \times (3 + 1)^2 \div 8$ , showing each step.

[4 marks]

That's the notes covered.

Carry on to the next subtopic.