The laws of arithmetic
The laws of arithmetic
The laws of arithmetic are rules that tell you the order to do calculations and let you rearrange numbers to make them easier. You will use the order of operations, brackets, and the commutative, associative and distributive laws.
The order of operations
You must do multiplication and division BEFORE addition and subtraction. For example, in 6 + 9 × 5 you do 9 × 5 = 45 first, then 6 + 45 = 51. When only adding and subtracting (or only multiplying and dividing), work from left to right.
Brackets change the order
Always do the calculation inside brackets first, even if it is an addition or subtraction. So 3 × 4 + 6 = 18, but 3 × (4 + 6) = 3 × 10 = 30. The brackets change which operation is done first.
The commutative and associative laws
The commutative law says you can add or multiply numbers in ANY order: 5 + 6 = 6 + 5 and 5 × 6 = 6 × 5. The associative law says you can GROUP them in any way: 7 × 5 × 8 × 2 can be grouped as (7 × 8) × (5 × 2) = 56 × 10 = 560. These laws do not work for subtraction or division.
The distributive law
The distributive law lets you split a number to multiply more easily: a × (b + c) = a × b + a × c. For example 7 × 48 = 7 × (40 + 8) = 280 + 56 = 336. You share the multiplication over each part.
Worked examples
Using the order of operations
- Calculate 6 + 9 × 5.
- Multiplication comes before addition, so work out 9 × 5 = 45 first.
- Then add: 6 + 45.
Answer: 6 + 9 × 5 = 51
Brackets change the order
- Compare 3 × 4 + 6 with 3 × (4 + 6).
- In the first, multiply first: 3 × 4 = 12, then + 6 = 18.
- In the second, do the brackets first: 4 + 6 = 10, then 3 × 10.
Answer: 3 × 4 + 6 = 18, but 3 × (4 + 6) = 30
The commutative law
- Compare 5 × 6 and 6 × 5.
- The commutative law says the order does not change the answer.
- Both give the same product.
Answer: 5 × 6 = 6 × 5 = 30
The associative law
- Work out 7 × 5 × 8 × 2 by choosing an easy grouping.
- Group the pairs that make round numbers: 7 × 8 = 56 and 5 × 2 = 10.
- Then multiply the results: 56 × 10.
Answer: 7 × 5 × 8 × 2 = 560
The distributive law
- Work out 7 × 48 by splitting 48 into 40 + 8.
- Multiply each part: 7 × 40 = 280 and 7 × 8 = 56.
- Add the two products: 280 + 56.
Answer: 7 × 48 = 336
Where you see this in real life
- Working out 7 lots of 48 in your head by splitting it into 7 × 40 and 7 × 8.
- Multiplying 7 × 5 × 8 × 2 quickly by grouping into 56 × 10.
- Following the rules a calculator uses so multiplication is done before addition.
- Using brackets to make sure a total is worked out in the right order.
Key words
- Order of operations
- The rule to do multiplication and division before addition and subtraction.
- Brackets
- Symbols ( ) that show which part of a calculation to do first.
- Commutative law
- You can add or multiply numbers in any order without changing the answer.
- Associative law
- You can group numbers being added or multiplied in any way without changing the answer.
- Distributive law
- Multiplying over a bracket: a × (b + c) = a × b + a × c.