Question

Which property is depicted by \[\dfrac{1}{2} \times \left( {6 \times \dfrac{4}{3}} \right) = \left( {\dfrac{1}{2} \times 6} \right) \times \dfrac{4}{3}\] ?
A. Commutative
B. Closure
C. Associative
D. Distributive

Answer 1

Hint: LHS and RHS of the above equation are the same , only the position of the bracket is changed in both of them. And only multiplication signs are used between the numbers.

Complete step by step solution:
There are many number properties defined in mathematics for addition and multiplication of numbers.
Commutative property: - If this property holds for the pair of elements under certain binary operation then the two elements (numbers) are said to commute under that operation. But the commutative property can be true for only two numbers but here there are three numbers on both the sides of the equation. So, commutative must not be an answer.
Like if a and b are two elements and they hold commutative property of addition then a + b = b + a.
And if a and b also holds commutative property of multiplication then a*b = b*a.
Closure property: - If the performance of any operation on the elements of a set also produces the element of the same set then those elements must satisfy the closure property.
Like 15 and 5 both the positive numbers and 5 + 15 is also a positive number but 5 – 15 is not a positive number. So, 5 and 15 satisfies closure property for addition but not satisfies closure property for subtraction.
Associative property: - This property holds for the addition and multiplication of the numbers. If some elements hold associative property then when we multiply or add them then the result must always be the same regardless of how the numbers are grouped or in other words the result does not depend on the position of parenthesis placed.
Like if a, b and c satisfies associative property then (a + b) + c = a + (b + c) and (a*b)*c = a*(b*c).
So, if a = \[\dfrac{1}{2}\], b = 6 and c = \[\dfrac{4}{3}\], then \[\dfrac{1}{2} \times \left( {6 \times \dfrac{4}{3}} \right) = \left( {\dfrac{1}{2} \times 6} \right) \times \dfrac{4}{3}\]. So, they hold associative property.
Distributive property: - This property is of multiplication over addition or subtraction. Like when multiplication of two numbers is added or subtracted then if we take a number common from both the pairs then the result will be the same.
Like if a, b and c are three numbers then ab + ac = a(b + c) and ab – ac = a(b – c).
But in the question there is multiplication over multiplication. So, distributive will not be an answer.
So, associative property is depicted by \[\dfrac{1}{2} \times \left( {6 \times \dfrac{4}{3}} \right) = \left( {\dfrac{1}{2} \times 6} \right) \times \dfrac{4}{3}\].
Hence, the correct option is C.

Note: Whenever we come up with this type of problem then we should remember that if there are only two elements or number in both the sides of the equation then there is no need to check for associative property because associative property does not holds for operation in two numbers holds and if there are more than 2 elements (numbers) on the both sides of the equation then commutative property will not hold because it is defined for operations between two elements only. This will be the easiest and efficient way to find the solution of the problem