To determine which expression illustrates the associative property of addition, let's evaluate each option and identify if any of them demonstrate this property.
The associative property of addition states that the way in which numbers are grouped in addition does not change their sum. Mathematically, this property can be written as:
[tex]\[(a + b) + c = a + (b + c)\][/tex]
Let's examine each given expression step-by-step:
1. [tex]\((3 + 19) - 12 = (3 + 12) - 19\)[/tex]
- First, calculate [tex]\((3 + 19) - 12\)[/tex]:
[tex]\[3 + 19 = 22\][/tex]
[tex]\[22 - 12 = 10\][/tex]
- Now, calculate [tex]\((3 + 12) - 19\)[/tex]:
[tex]\[3 + 12 = 15\][/tex]
[tex]\[15 - 19 = -4\][/tex]
- Comparing the results:
[tex]\[10 \neq -4\][/tex]
- Therefore, this is not an example of the associative property.
2. [tex]\(3 + (19 - 12) = 3 + (19 + 12)\)[/tex]
- First, calculate [tex]\(3 + (19 - 12)\)[/tex]:
[tex]\[19 - 12 = 7\][/tex]
[tex]\[3 + 7 = 10\][/tex]
- Now, calculate [tex]\(3 + (19 + 12)\)[/tex]:
[tex]\[19 + 12 = 31\][/tex]
[tex]\[3 + 31 = 34\][/tex]
- Comparing the results:
[tex]\[10 \neq 34\][/tex]
- Therefore, this is not an example of the associative property.
3. [tex]\((3 + 19) - 12 = 3 + (19 - 12)\)[/tex]
- Calculate [tex]\((3 + 19) - 12\)[/tex] as previously:
[tex]\[3 + 19 = 22\][/tex]
[tex]\[22 - 12 = 10\][/tex]
- Now, calculate [tex]\(3 + (19 - 12)\)[/tex] as previously:
[tex]\[19 - 12 = 7\][/tex]
[tex]\[3 + 7 = 10\][/tex]
- Comparing the results:
[tex]\[10 = 10\][/tex]
- Even though the results are equal, this is not an example of the associative property because it involves a subtraction operation and changes the grouping of terms.
4. [tex]\(3 + (19 - 12) = 3 - (19 + 12)\)[/tex]
- First, calculate [tex]\(3 + (19 - 12)\)[/tex] as previously:
[tex]\[19 - 12 = 7\][/tex]
[tex]\[3 + 7 = 10\][/tex]
- Now, calculate [tex]\(3 - (19 + 12)\)[/tex]:
[tex]\[19 + 12 = 31\][/tex]
[tex]\[3 - 31 = -28\][/tex]
- Comparing the results:
[tex]\[10 \neq -28\][/tex]
- Therefore, this is not an example of the associative property.
After examining all the options, none of the provided expressions correctly illustrate the associative property of addition. The expressions involve either subtraction or grouping changes that affect the operations, and the sums do not match when evaluated under the associative property criteria.
