Question

Which property is being used in the following:
 $ \dfrac{2}{3}+\dfrac{4}{7}=\dfrac{4}{7}+\dfrac{2}{3} $
(a) Additive identity
(b) Commutative property of addition
(c) Commutative property of multiplication
(d) Associative property of addition

Answer 1

Hint: We will check the value of the left-hand side and the right-hand side of the given expression. We will look at the definitions of the terms given in the options. Then we will verify the property that is satisfied by the given expression from among the given options. We will eliminate options as we verify the definitions with respect to the given expression to find the correct option.

Complete step by step answer:
The left hand side of the given expression can be simplified as follows,
\[\begin{align}
  & \dfrac{2}{3}+\dfrac{4}{7}=\dfrac{2\times 7+4\times 3}{7\times 3} \\
 & \Rightarrow \dfrac{2}{3}+\dfrac{4}{7}=\dfrac{14+12}{21} \\
 & \therefore \dfrac{2}{3}+\dfrac{4}{7}=\dfrac{26}{21} \\
\end{align}\]
Similarly, the right hand side of the given expression can be simplified as follows,
\[\begin{align}
  & \dfrac{4}{7}+\dfrac{2}{3}=\dfrac{4\times 3+2\times 7}{7\times 3} \\
 & \Rightarrow \dfrac{4}{7}+\dfrac{2}{3}=\dfrac{12+14}{21} \\
 & \therefore \dfrac{4}{7}+\dfrac{2}{3}=\dfrac{26}{21} \\
\end{align}\]
We can see that the values on both sides of the expression are same.
We define the additive identity as the number, which when added to another number, gives the sum as the number itself. This means that the number 0 is the additive identity. Since any number added to 0 gives the sum as the number itself.
The commutative property of addition states that the order of terms being added does not affect the sum of the terms. Similarly, the commutative property of multiplication states that the order of the numbers being multiplied does not affect the product.
The associative property of addition states that if there are more than two terms to be added and we group the terms to perform the addition, the sum of the terms does not change even if the groups of terms are changed.
Since 0, the additive identity is not involved in the terms being added in the given expression, we can eliminate the first option.
While calculating the value of the left-hand side and the right-hand side of the given expression, we saw that the order of the terms was changed and the sum of the terms was the same. This implies that the given expression is an example of the commutative property of addition.
Hence, the correct option is (b).

Note:
The third option involves binary operation multiplication. Since the given expression has addition, we can eliminate the third option. The associative property of addition talks about more than two terms being added. As there are only two terms involved in the given expression, we can eliminate the fourth option as well.