N2, N3NumberFoundationGrades 1–4

The Four Operations

Paper 1 is a non-calculator paper, so reliable written methods for adding, subtracting, multiplying and dividing are worth marks on almost every question. This lesson makes those methods secure, then adds negative numbers and the order of operations — the two places where quick mental maths most often goes wrong.

40 min Video by Maths Genie AQA GCSE Maths

The big picture

Written methods are place value put to work. Column addition works because you only ever add ones to ones and tens to tens; long multiplication works because $34 \times 26$ is $34 \times 20 + 34 \times 6$ .

The order of operations — brackets, indices, division and multiplication, addition and subtraction (BIDMAS) — is an agreement that means every calculation has exactly one answer. Calculators follow it too, which is why typing a calculation in the wrong way gives a wrong answer.

Multiplying Whole Numbers and DecimalsCurated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

What you'll learn

Use written methods to add, subtract, multiply and divide, including decimals
Add, subtract, multiply and divide negative numbers
Apply the order of operations (BIDMAS)
Use inverse operations to check answers

Written methods

Column addition and subtraction: line up the digits by place value (line up decimal points), start from the right, and carry or exchange as needed.

Long multiplication: multiply by each digit of the second number in turn, remembering to shift one place left (put a zero) for the tens digit, then add the rows. The grid method gives the same result.

Short division ("bus stop"): divide digit by digit from the left, carrying remainders to the next digit. For long division, write out each subtraction.

Example 1Work out

347 \times 26

347 \times 6 = 2082

347 \times 20 = 6940

3Add:

2082 + 6940 = 9022

Answer

9022

Your turnSame method, new numbers — try it before moving on.

Work out $268 \times 34$ .

Your answer

Show hint

Find $268 \times 4$ and $268 \times 30$ , then add.

Show worked solution

Solution

1 $268 \times 4 = 1072$ .
2 $268 \times 30 = 8040$ .
3 $1072 + 8040 = 9112$ .

Answer: $9112$

Example 2Work out

1584 \div 6

15 \div 6 = 2

remainder 3.

38 \div 6 = 6

remainder 2.

24 \div 6 = 4

Answer

264

Your turnSame method, new numbers — try it before moving on.

Work out $2961 \div 7$ .

Your answer

Show worked solution

Solution

1 $29 \div 7 = 4$ r 1.
2 $16 \div 7 = 2$ r 2.
3 $21 \div 7 = 3$ . Answer $423$ .

Answer: $423$

Example 3Work out

4.7 \times 3.6

1Ignore the points:

47 \times 36 = 1692

2There are 2 decimal places in the question, so 2 in the answer.

Answer

16.92

Your turnSame method, new numbers — try it before moving on.

Work out $2.8 \times 0.45$ .

Your answer

Show worked solution

Solution

1 $28 \times 45 = 1260$ .
23 decimal places in the question: $1.260 = 1.26$ .

Answer: $1.26$

Tip — Estimate first: $4.7 \times 3.6 \approx 5 \times 4 = 20$ . If your answer is 169.2 or 1.692, a decimal point has gone astray.

Negative numbers

Adding a negative is the same as subtracting: $5 + (-3) = 2$ . Subtracting a negative is the same as adding: $5 - (-3) = 8$ .

For multiplication and division: same signs give a positive answer; different signs give a negative answer.

Example 1Work out (a)

-6 - (-9)

(b)

-4 \times -7

(c)

36 \div -9

1(a) Subtracting a negative:

-6 + 9 = 3

2(b) Same signs:

+28

3(c) Different signs:

-4

Answer

(a)

3

(b)

28

(c)

-4

Your turnSame method, new numbers — try it before moving on.

Work out $-8 - (-3) \times 4$ .

Your answer

Show hint

Multiplication comes first: $(-3) \times 4 = -12$ .

Show worked solution

Solution

1 $(-3) \times 4 = -12$ .
2 $-8 - (-12) = -8 + 12 = 4$ .

Answer: $4$

Order of operations

$\textbf{B}$ rackets, $\textbf{I}$ ndices, $\textbf{D}$ ivision and $\textbf{M}$ ultiplication (left to right), $\textbf{A}$ ddition and $\textbf{S}$ ubtraction (left to right).

A fraction line acts like brackets: in $\frac{3 + 9}{4}$ , work out the top first.

Example 1Work out

20 - 3 \times 2^2

1Indices:

2^2 = 4

2Multiplication:

3 \times 4 = 12

3Subtraction:

20 - 12 = 8

Answer

8

Your turnSame method, new numbers — try it before moving on.

Work out $(15 - 3) \div 4 + 2 \times 5$ .

Your answer

Show worked solution

Solution

1Brackets: $12$ .
2 $12 \div 4 = 3$ and $2 \times 5 = 10$ .
3 $3 + 10 = 13$ .

Answer: $13$

Example 2Insert brackets to make

4 + 6 \times 3 - 1 = 20

true.

1Try

4 + 6 \times (3 - 1) = 4 + 12 = 16

— no.

2Try

(4 + 6) \times (3 - 1) = 10 \times 2 = 20

✓

Answer

(4 + 6) \times (3 - 1) = 20

Left to right

Division and multiplication have equal priority, so work left to right: $24 \div 4 \times 2 = 6 \times 2 = 12$ , not $24 \div 8 = 3$ . The same applies to addition and subtraction.

Inverse operations

Addition and subtraction undo each other; so do multiplication and division. Use them to check: if $1584 \div 6 = 264$ , then $264 \times 6$ should give $1584$ .

Inverse operations also solve "missing number" problems: $\square \times 7 = 91$ means $\square = 91 \div 7 = 13$ .

Think like an examiner

On the non-calculator paper, show your written method. A correct method with one slip can still earn most of the marks.
For decimal multiplication, the method mark is often for the whole-number product — write it down.
In problem questions, state what each calculation represents: "cost of 26 tickets = £9022".

Rules

-a \times -b = +ab

Same signs: positive.

a - (-b) = a + b

Subtracting a negative adds.

\text{B} \to \text{I} \to \text{DM} \to \text{AS}

Order of operations.

Watch out for these

2 + 3 \times 4 = 20

Multiply first:

2 + 12 = 14

-3 - 5 = 2

8

Start at

-3

and move 5 left:

-8

-5^2 = 25

on a calculator.

The calculator squares first, giving

-25

. Type

(-5)^2

for 25.

Forgetting the placeholder zero in long multiplication:

347 \times 20 = 694

347 \times 20 = 6940

Stretch yourself

A school orders 38 boxes of pencils. Each box holds 144 pencils. The pencils are shared equally between 16 classes. How many pencils does each class get, and how many are left over?

Hint — Multiply to find the total, then divide by 16. The remainder is what is left over.

Reveal solution

Solution

144 \times 38 = 5472

pencils.

5472 \div 16 = 342

exactly, since

16 \times 342 = 5472

. Each class gets 342 pencils, with none left over.

\blacksquare

Questions students ask

Do I have to use the column method?

No. Any correct written method (grid, partitioning, column) is accepted. What matters is that the working is clear and reliable.

Is BIDMAS the same as BODMAS?

Yes. The O stands for "orders", another word for indices.

What happens when I multiply three negative numbers?

Pair them up: two negatives make a positive, then positive times negative is negative. An odd number of negative factors gives a negative answer.

Key takeaways

Line up place values; carry and exchange carefully.
Long multiplication: multiply by each digit, remember place-holder zeros, then add.
Negatives: same signs positive, different signs negative; subtracting a negative adds.
BIDMAS, working left to right within each level.
Check answers with inverse operations and estimation.

Exam-style questions

Practise it the way it is examined

3 original questions · 6 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 22 marks

(a)Work out $6.4 - 2.75$ .[1]
Show mark scheme (a)
- B1 $3.65$
Answer: $3.65$
(b)Work out $-5 \times -8 + 3$ .[1]
Show mark scheme (b)
- B1 $43$
Answer: $43$

Question 2Grade 32 marks

A coach costs £1,265 to hire. 23 people share the cost equally.

How much does each person pay? You must show your working.[2]
Show mark scheme
- M1A complete division method, e.g. long division or repeated subtraction
- A1£55
Answer: £55

Question 3Grade 42 marks

Work out $\dfrac{4.2 \times 30}{0.6}$ without a calculator.[2]
Show mark scheme
- M1Correct numerator $126$ , or simplifies, e.g. $4.2 \div 0.6 = 7$
- A1 $210$
Answer: $210$

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Work out $458 + 376$ .
- $834$
- $724$
- $824$
- $934$
Show answer and explanation
Answer: $834$
Column addition with carrying.
Work out $-7 + 12$ .
- $5$
- $-5$
- $19$
- $-19$
Show answer and explanation
Answer: $5$
Start at $-7$ and move 12 right.
Work out $6 + 4 \times 5$ .
- $26$
- $50$
- $30$
- $15$
Show answer and explanation
Answer: $26$
Multiply first.
Work out $-45 \div -9$ .
- $5$
- $-5$
- $36$
- $-36$
Show answer and explanation
Answer: $5$
Same signs give a positive.
Work out $56 \times 23$ .
- $1288$
- $1268$
- $280$
- $1388$
Show answer and explanation
Answer: $1288$
$1120 + 168$ .
Work out $(2 + 3)^2 - 4 \times 3$ .
- $13$
- $63$
- $1$
- $3$
Show answer and explanation
Answer: $13$
$25 - 12$ .
Work out $0.3 \times 0.07$ .
- $0.021$
- $0.21$
- $2.1$
- $0.0021$
Show answer and explanation
Answer: $0.021$
$3 \times 7 = 21$ , with 3 decimal places.
Work out $36 \div 4 \times 3$ .
- $27$
- $3$
- $12$
- $108$
Show answer and explanation
Answer: $27$
Left to right: $9 \times 3$ .

Next step

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