The associative property of addition states that the sum of 3 or more numbers provides the same result regardless of how they are grouped. It can be best understood using colored blocks.
For any three numbers a, b, and c, the formula is:
Associative Property of Multiplication
Let us group 3, 6, and 8 to prove the associative property of multiplication using an example.
Step 1: Group the 3 numbers in two ways: 3 × (6 × 8) and (3 × 6) × 8
Step 2: Add the numbers within the parenthesis of L.H.S of the equation: (6 × 8) = 48
Step 3: Add the result to the other number 48 × 3 = 144 to get the result
Step 4: Follow steps 1 to 3 for the R.H.S of the equation
As we know, according to this law: a × (b × c) = (a × b) × c
Here, a = 3, b = 6, and c = 8
Solving L.H.S
a × (b × c)
= 3 × (6 × 8)
= 3 × 48
= 144
Solving R.H.S
(a × b) × c
= (3 × 6) × 8
= 18 × 8
= 144
L.H.S = R.H.S
Thus, the associative law of multiplication holds.
Does the given equation follow the associative property of multiplication? (6 × 5) × 4 = 6 × (5 × 4)
Solution:
As we know, according to the associative property of multiplication: a × (b × c) = (a × b) × c Here, a = 12, b = 1, c = 5 L.H.S a × (b × c) => 6 × (5 × 4) => 6 × 20 => 120 R.H.S (a × b) × c => (6 × 5) × 4 => 30 × 4 => 120 L.H.S = R.H.S Thus, the associative law of multiplication holds for the given equation
Find x in given equations using the associative property of addition a) (15 × 12) × 20 = 15 × (x × 20) b) 5 × (x × 8) = 5 × (3 × 8)
Solution:
Here, a = 15, b = 12, c = 20 If the associative property of addition holds, then a × (b × c) = (a × b) × c a) Since, (15 × 12) × 20 = 180 × 20 = 3,600 and multiplication satisfies the associative property, thus 15 × (x × 20) = 3,600 => 20x = 240 => x = 12 b) Since, 5 × (3 × 8) = 120 and multiplication satisfies the associative property, thus 5 × (x × 8) = 120 => 8x = 24 => x = 3
