Question

Which property is illustrated by the following statement?
\[\left( {16 + 18} \right) + 20 = \left( {18 + 16} \right) + 20\]

Answer 1

Hint:
Here, we need to find the property illustrated in the given equation. We will use the definitions of the commutative and associative property to see which of the two is illustrated in the given statement.

Complete step by step solution:
We can observe that the given statement contains the addition operation.
First, we will define the two properties – commutative property of addition, and associative property of addition.
The commutative property of addition states that the sum of two numbers remains the same even if the numbers are swapped and their order is changed. This is true for all real numbers.
The associative property of addition states that the sum of three or more numbers remains the same even if the numbers are grouped in a different way. This is true for all real numbers.
Now, we can observe that in the statement \[\left( {16 + 18} \right) + 20 = \left( {18 + 16} \right) + 20\], the numbers grouped in the parentheses are the same.
Therefore, this does not illustrate the associative property of addition.
From the commutative property of addition, we have \[16 + 18 = 18 + 16\].
Substituting \[16 + 18 = 18 + 16\] in the expression \[\left( {16 + 18} \right) + 20\], we get
\[ \Rightarrow \left( {16 + 18} \right) + 20 = \left( {18 + 16} \right) + 20\]

Therefore, we can conclude that the given statement illustrates the commutative property of addition.

Note:
Commutative property can be represented by the equation \[a + b = b + a\].
For example: the sum of 2 and 3 is 5. The sum of 3 and 2 is also 5. Therefore, \[2 + 3 = 3 + 2\].
Associative property can be represented by the equation \[\left( {a + b} \right) + c = a + \left( {b + c} \right)\].
For example: The sum of 2 and 3 is 5. The sum of 5 and 4 is 9.
Also, the sum of 3 and 4 is 7, and the sum of 2 and 7 is 9. Therefore, \[\left( {2 + 3} \right) + 4 = 2 + \left( {3 + 4} \right)\].