Question

Fill in the blank.
$\dfrac{{ - 3}}{8} + \dfrac{1}{7} = \dfrac{1}{7} + \left( {\dfrac{{ - 3}}{8}} \right)$ is an example to show that _____.

Answer 1

Hint: The given expression suggests the addition of two terms on both the sides of the equation. Here, we will check the value for the terms on the left and the right hand side to show which property it follows.

Complete step by step solution:
Take the left hand side of the given expression,
LHS $ = \dfrac{{ - 3}}{8} + \dfrac{1}{7}$
Take LCM (Least common multiple) in the above expression –
LHS $ = \dfrac{{ - 3(7) + 1(8)}}{{8(7)}}$
Simplify the above expression –
LHS $ = \dfrac{{ - 21 + 8}}{{56}}$
When you combine two terms with different signs you have to do subtraction and give signs of the bigger term to the resultant value.
LHS$ = \dfrac{{ - 13}}{{56}}$ ….. (A)
Now, similarly take right hand side of the equation –
RHS $ = \dfrac{1}{7} + \left( {\dfrac{{ - 3}}{8}} \right)$
Product of one negative sign and one positive term gives us the negative term as the value.
Take LCM (Least common multiple) in the above expression –
RHS $ = \dfrac{{1(8) + ( - 3)(7)}}{{8(7)}}$
Simplify the above expression –
RHS $ = \dfrac{{8 - 21}}{{56}}$
When you combine two terms with different signs you have to do subtraction and give signs of the bigger term to the resultant value.
RHS$ = \dfrac{{ - 13}}{{56}}$ ….. (B)
From equation (A) and (B) we can state that the value remains the same irrespective of the order of the addition and it follows the commutative property.
Commutative property states that when we add two or more integers their resultant sum remains the same irrespective of the order in which they add.

Therefore, $\dfrac{{ - 3}}{8} + \dfrac{1}{7} = \dfrac{1}{7} + \left( {\dfrac{{ - 3}}{8}} \right)$ is an example to show that “Commutative Property of Rational Numbers”.

Note:
Basically, there are four properties of the addition for the integers. They are-
Closure property, commutative property, associative property and the additive identity property. Go through it and remember its concepts nicely and also know the difference between them.