Commutative, Associative and Distributive Laws

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

Video

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:

Commutative Law Addition

 

... or when we multiply:

a × b  =  b × a

Example:

Commutative Law multiplication

 

Percentages too!

Because a × b  =  b × a it is also true that:

a% of b  =  b% of a

Example: what is 8% of 50 ?

8% of 50 = 50% of 8
  = 4

 

commute

Why "commutative" ... ?

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

Mathopolis:Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8

 

Video

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)

Associative Law addition

... or when we multiply:

(a × b) × c  =  a × (b × c)

Associative Law multiplication

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

 

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Video

Distributive Law

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

This is what it lets us do:

Distributive Law

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

So, the 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:

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

26×3 - 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

 

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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 + a
a × 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