To determine which equation demonstrates the Associative Property of Addition, let's examine each option carefully.
Definition: The Associative Property of Addition states that the way in which numbers are grouped when adding does not change their sum. Specifically, for any numbers [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex], the property can be written as:
[tex]\[ (a + b) + c = a + (b + c) \][/tex]
Now, let's analyze each option one by one:
Option A: [tex]\(3 + (17 + 11) = (3 + 17) + 11\)[/tex]
- This expression rearranges the grouping of the numbers being added without changing the order of the numbers themselves.
- It clearly shows that changing the grouping does not affect the sum:
[tex]\[3 + 28 = 20 + 11 \][/tex]
[tex]\[31 = 31\][/tex]
This is a valid demonstration of the Associative Property of Addition.
Option B: [tex]\(\frac{1}{4} \times 7 \times 8 = \frac{1}{4} \times 8 \times 7\)[/tex]
- This expression involves multiplication, not addition.
- Therefore, it cannot demonstrate the Associative Property of Addition.
Option C: [tex]\(\frac{1}{4} \times (16 \times 11) = \left(\frac{1}{4} \times 16\right) \times 11\)[/tex]
- This expression also involves multiplication rather than addition.
- Hence, it does not demonstrate the Associative Property of Addition.
Option D: [tex]\(12 + 17 + 8 = 12 + 8 + 17\)[/tex]
- This expression changes the order of the numbers being added (from [tex]\(17 + 8\)[/tex] to [tex]\(8 + 17\)[/tex]), which is the Commutative Property.
- The Associative Property only changes the grouping, not the order of the numbers.
Option E: [tex]\(4(x + 3) = 4x + 12\)[/tex]
- This expression is an application of the distributive property, not the associative property, since it distributes 4 over [tex]\(x + 3\)[/tex].
From our analysis, we see that only Option A correctly demonstrates the Associative Property of Addition. Therefore, the correct option is:
A: [tex]\(3 + (17 + 11) = (3 + 17) + 11\)[/tex].
