The Commutative Law states that even if we switch the order of the numbers, the resulting answer is the same. The commutative law holds for addition and multiplication.
The Commutative Law of Addition: a + b = b + a
For example:
The Commutative Law of Multiplication: a × b = b × a
For example:
The commutative law does not hold for subtraction or division.
For instance:
Uses of Commutative Law
It is sometimes easier to compute the answer to a multiplication or addition question by switching the order of numbers.
Example for addition question: In the question 3 + 44 + 17, it is easier if we add 17 and 3 first, so 3 + 44 + 17 = 44 + 3 + 17 = 44+20 = 64
Example for multiplication question: In the question, 5 × 10 × 6, it is easier if we multiply 6 and 5 first, so 5 × 10 × 6 = 10 × 5 × 6 = 10 × 30 = 300
The Associative Law states that even if we group numbers differently, the answer is still the same. The associative law holds for addition and multiplication.
The Associative Law of Addition: (a + b) + c = a + (b + c)
Example:
The Associative Law of Multiplication: (a × b) × c = a × (b × c)
Example:
The associative law does not hold for subtraction or division.
For instance,
Uses of Associative Law
It is sometimes easier to add or multiply if we group numbers differently.
Example for addition: In the question, 23 + 45 + 5, it is easier to add 45 and 5 first, rather than adding 23 and 45 first. So, 23 + 45 + 5 = 23 + (45 + 5) = 23 + 50 = 73
Example for multiplication: In the question, 15 × 5 × 2, it is easier to first multiply 5 and 2 rather than 15 and 5. So, 15 × 5 × 2 = 15 × (5 × 2) = 15 × 10 = 150
The Distributive Law: a × (b + c) = a × b + a × c and a × (b - c) = a × b - a × c.
Example:
_{The distributive law }_{does not hold }_{for division. }
_{For instance, 16 ÷ (8 + 2) = 16 ÷ 10 = 1.6 but 16 ÷ 8 + 16 ÷ 2 = 2 + 8 = 10. }
_{The }_{correct way}_{ is 16 ÷ (8 + 2) = 16 ÷ 10 = 1.6.}
Uses of Distributive Law
A difficult multiplication question’s numbers can be broken up or combined, resulting in an easier multiplication question using the distributive law.
Example where we break up a number: In the question, 3 × 502, 502 can be broken up into (500 + 2) and so the question becomes, 3 × (500 + 2) and using the distributive law, this is equal to 3 × 500 + 3 × 2 = 1500 + 6 = 1506
Example where we combine numbers: In the question, 15 × 3 + 15 × 7, we can combine 3 and 7, so 15 × 3 + 15 × 7 = 15 × (3 + 7) = 15 × 10 = 150