# Commutative, Associative and Distributive Laws

*Wow! What a mouthful of words! But the ideas are simple.*

## Commutative Laws

The "Commutative Laws" say we can **swap numbers** over and still get the same answer ...

... when we **add**:

a + b** = **b + a

### Example:

... or when we **multiply**:

a × b** = **b × a

### Example:

Why **"commutative**" ... ?

Because the numbers can travel back and forth like a **commuter**.

## Associative Laws

The "Associative Laws" say that it doesn't matter how we group the numbers (i.e. which we calculate first) ...

... when we **add**:

(a + b) + c** = **a + (b + c)

... or when we **multiply**:

(a × b) × c** = **a × (b × c)

### Examples:

This: | (2 + 4) + 5 = 6 + 5 = 11 |

Has the same answer as this: | 2 + (4 + 5) = 2 + 9 = 11 |

This: | (3 × 4) × 5 = 12 × 5 = 60 |

Has the same answer as this: | 3 × (4 × 5) = 3 × 20 = 60 |

### Uses:

Sometimes it is easier to add or multiply in a different order:

### What is 19 + 36 + 4?

19 + 36 + 4 = 19 + **(36 + 4)**

= 19 + **40** = 59

Or to rearrange a little:

### What is 2 × 16 × 5?

2 × 16 × 5** = (2 × 5)** × 16**
= 10** × 16 = 160

## Distributive Law

The "Distributive Law" is the BEST one of all, but needs careful attention.

This is what it lets us do:

3 lots of **(2+4)** is the same as **3 lots of 2** plus **3 lots of 4**

So, the **3×** can be "distributed" across the **2+4**, into **3×2** and **3×4**

And we write it like this:

a × (b + c) = a × b + a × c

Try the calculations yourself:

- 3 × (
**2 + 4**) = 3 ×**6**= 18 - 3×2 + 3×4 = 6 + 12 = 18

Either way gets the same answer.

In English we can say:

We get the same answer when we:

- multiply a number by a
**group of numbers added together**, or - do each
**multiply**separately then**add**them

### Uses:

Sometimes it is easier to break up a difficult multiplication:

### Example: What is 6 × 204 ?

6 × 204 = 6×200 + 6×4

= 1,200 + 24

= 1,224

Or to combine:

### Example: What is 16 × 6 + 16 × 4?

16 × 6 + 16 × 4 = 16 × **(6+4)**

= 16 × **10**

= 160

We can use it in subtraction too:

### Example: 26×3 - 24×3

**(26 - 24)**× 3

= 2 × 3

= 6

We could use it for a long list of additions, too:

### Example: 6×7 + 2×7 + 3×7 + 5×7 + 4×7

**6**×7 + **2**×7 + **3**×7 + **5**×7 + **4**×7

= **(6+2+3+5+4)** × 7

= **20** × 7

= **140**

## And those are the Laws . . .

## . . . but don't go too far!

The Commutative Law does **not** work for subtraction or division:

### Example:

- 12 / 3 =
**4**, but - 3 / 12 =
**¼**

The Associative Law does **not** work for subtraction or division:

### Example:

- (9 – 4) – 3 = 5 – 3 =
**2**, but - 9 – (4 – 3) = 9 – 1 =
**8**

The Distributive Law does **not** work for division:

### Example:

- 24 / (4 + 8) = 24 / 12 =
**2**, but - 24 / 4 + 24 / 8 = 6 + 3 =
**9**

## Summary

Commutative Laws: | a + b = b + aa × b = b × a |

Associative Laws: | (a + b) + c = a + (b + c)(a × b) × c = a × (b × c) |

Distributive Law: | a × (b + c) = a × b + a × c |