Number & the laws of arithmetic
The laws of arithmetic
Three laws make calculations easier: the commutative, associative and distributive laws. They work for adding and multiplying (but not for subtracting or dividing).
The three laws
ā¢ Commutative: you can swap the order ā a + b = b + a, and a Ć b = b Ć a. ā¢ Associative: you can group differently ā 25 Ć 9 Ć 4 = 25 Ć 4 Ć 9. ā¢ Distributive: you can split one factor ā 36 Ć 8 = (30 Ć 8) + (6 Ć 8).
Worked examples
Commutative law
- 108 Ć 6 gives the same answer as 6 Ć 108.
- Swapping the order does not change a multiplication.
Answer: 108 Ć 6 = 6 Ć 108
Associative law
- Work out 25 Ć 9 Ć 4 by regrouping.
- Do 25 Ć 4 = 100 first, then Ć 9.
Answer: 25 Ć 9 Ć 4 = 100 Ć 9 = 900
Distributive law
- Work out 36 Ć 8 by splitting 36 into 30 + 6.
- (30 Ć 8) + (6 Ć 8) = 240 + 48.
Answer: 36 Ć 8 = 288
Where you see this in real life
- Adding a shopping list in whatever order is easiest.
- Grouping numbers that make 10 or 100 to add quickly.
- Splitting a tricky times-table fact into two easier ones.
Key words
- Commutative law
- You can swap the order when adding or multiplying.
- Associative law
- You can group numbers differently when adding or multiplying.
- Distributive law
- You can split a multiplication into the sum of two easier multiplications.