Order of Operations, Brackets and Laws of Arithmetic

British & IGCSE › Checkpoint 6 › Checkpoint 6 Maths — Complete Course

Is 2 + 3 × 4 equal to 20 or 14? Everyone in the world must get the same answer, so mathematicians agree on an order of operations. On this page you will learn that order, how brackets change it, and three laws of arithmetic that let you do hard-looking calculations in your head.

Key words

1. The order of operations

1. Brackets first. Work out anything inside ( ).
2. Then × and ÷, working from left to right. Neither one comes before the other — whichever is further left is done first.
3. Then + and −, also from left to right.

2 + 3 × 4 → multiply first: 3 × 4 = 12 → 2 + 12 = 14 (not 20).

2. Brackets change the order

Brackets tell you: do this part first, even if it is an addition or subtraction. The same numbers and signs can give different answers when the brackets move.

Brackets in word problems. "A family of 4 each buy a cinema ticket for E£45 and popcorn for E£15." Each person spends 45 + 15, so the total is 4 × (45 + 15) = 4 × 60 = E£240. Without brackets, 4 × 45 + 15 = 195 would only pay for one popcorn.

Brackets that do not change anything. In 3 + (4 × 5) the brackets are not needed, because × is done first anyway. Brackets around a multiplication or division in a sum do not change the answer — but they can make it clearer.

3. The laws of arithmetic

These laws are not new rules to learn by heart — they describe what you already know about numbers. They help you choose an easier order to calculate.

Commutative law: swap the order

For addition and multiplication, the order does not matter: 3 + 8 = 8 + 3 and 3 × 5 = 5 × 3.

It does not work for subtraction or division: 10 − 4 = 6 but 4 − 10 = −6; 12 ÷ 3 = 4 but 3 ÷ 12 = ¼.

Associative law: group in a helpful way

When you add or multiply three or more numbers, you can choose which pair to do first: (17 + 26) + 4 = 17 + (26 + 4) = 17 + 30 = 47. (25 × 7) × 4 = 7 × (25 × 4) = 7 × 100 = 700. Look for pairs that make 10, 100, 1000 (like 25 × 4, 5 × 20, 38 + 62).

Distributive law: split a number

To multiply a number by a sum, multiply each part and add: 6 × 27 = 6 × (20 + 7) = 6 × 20 + 6 × 7 = 120 + 42 = 162. The area model shows why.

It works with subtraction too, which is great for numbers just below a tens number: 8 × 19 = 8 × (20 − 1) = 160 − 8 = 152.

Division can be split in the same way, as long as you split the number being divided: 96 ÷ 4 = 80 ÷ 4 + 16 ÷ 4 = 20 + 4 = 24.

Worked examples

Worked example 1 — Work out 18 − 3 × 4. [1 mark]

1. No brackets. Look for × or ÷: 3 × 4 = 12.
2. Then subtract: 18 − 12 = 6.
3. Answer: 6

Common mistake: 18 − 3 = 15, then 15 × 4 = 60. Multiplication must come before subtraction.

Worked example 2 — (a) Work out 40 ÷ (2 + 6). (b) Work out 40 ÷ 2 + 6. [2 marks]

1. (a) Brackets first: 2 + 6 = 8. Then 40 ÷ 8 = 5.
2. (b) No brackets, so divide first: 40 ÷ 2 = 20. Then 20 + 6 = 26.
3. The brackets changed the answer from 26 to 5.

Worked example 3 — Work out 7 + 24 ÷ 6 − 2 × 3. [1 mark]

1. Do every × and ÷ first: 24 ÷ 6 = 4 and 2 × 3 = 6.
2. Rewrite: 7 + 4 − 6.
3. Left to right: 7 + 4 = 11, 11 − 6 = 5.
4. Answer: 5

Tip: underline each × and ÷ part first, then rewrite the line. This stops you losing track.

Worked example 4 — Put brackets in this calculation to make it correct: 5 + 3 × 6 − 2 = 32 [1 mark]

1. Without brackets: 5 + 18 − 2 = 21. Too small, so we need to add or subtract before multiplying.
2. Try (5 + 3) × 6 − 2 = 48 − 2 = 46. No.
3. Try 5 + 3 × (6 − 2) = 5 + 12 = 17. No.
4. Try (5 + 3) × (6 − 2) = 8 × 4 = 32 ✓
5. Answer: (5 + 3) × (6 − 2) = 32

Technique: think about which two numbers multiply to give the target: 32 = 8 × 4, and 8 = 5 + 3, 4 = 6 − 2.

Worked example 5 — Use the laws of arithmetic to work out these mentally. Show how. (a) 4 × 37 × 25 (b) 8 × 49 [2 marks]

1. (a) Change the order and group: (4 × 25) × 37 = 100 × 37 = 3700.
2. (b) 49 = 50 − 1, so 8 × 49 = 8 × 50 − 8 × 1 = 400 − 8 = 392.

Worked example 6 — Mona buys 3 notebooks at E£28 each and pays with a E£100 note. Write one calculation for her change, then work it out. [2 marks]

1. Cost of notebooks: 3 × 28. Change = 100 minus the cost.
2. One calculation: 100 − 3 × 28 (the brackets 100 − (3 × 28) are allowed but not needed).
3. 3 × 28 = 84 (3 × 20 + 3 × 8 = 60 + 24).
4. 100 − 84 = 16. Answer: E£16

Common mistake: writing (100 − 3) × 28 — that would be 2716!

Practice

All pages in this course

Start here

• Checkpoint 6 Maths: The Test and How to Prepare

Number: integers and calculation

• Counting On and Back and Negative Numbers
• Sequences, Letters and Position-to-Term Rules
• Order of Operations, Brackets and Laws of Arithmetic
• Multiplying Whole Numbers (up to 10 000 × 2-digit)
• Dividing Whole Numbers (up to 1000 ÷ 2-digit)
• Factors, Multiples and Tests of Divisibility
• Square Numbers and Cube Numbers

Number: place value, decimals, fractions, percentages and ratio

• Place Value and Decimals to Thousandths
• Rounding Decimals
• Fractions: Division, Operators and Simplest Form
• Adding and Subtracting Fractions
• Multiplying and Dividing Fractions by Whole Numbers
• Fractions, Decimals and Percentages: Equivalence and Ordering
• Adding, Subtracting, Multiplying and Dividing Decimals
• Ratio and Direct Proportion

Geometry and measure

• Time: Decimals and Mixed Units
• Quadrilaterals: Properties and Classification
• Circles: Parts and Constructing Circles
• Angles: Measuring, Drawing and Angles in a Triangle
• Area of Rectangles and Right-angled Triangles
• 3D Shapes, Compound Shapes and Nets
• Capacity, Volume and Surface Area
• Coordinates in Four Quadrants
• Rotational Symmetry, Translation, Reflection and Rotation

Statistics and probability

• Collecting and Representing Data
• Mode, Median, Mean and Range, and Interpreting Data
• Probability

Practice tests

• Checkpoint 6 Maths — Practice Test 1
• Checkpoint 6 Maths — Practice Test 2
• Checkpoint 6 Maths — Practice Test 3