Question

According to the Associative Property of Addition, which of the following is equivalent to the expression (b + c) + a + d?
d + (b + c) + a;
b + (c + a) + d;
(b + c) + d + a;
d + a + (b + c)

Answer 1

The Associative Property of Addition states that the way in which numbers are grouped in addition does not change their sum. This means that if you have a + (b + c) or (a + b) + c, both expressions will yield the same result.

Given the expression \((b + c) + a + d\), we can regroup the terms in several ways without changing the value of the expression:

1. \(d + (b + c) + a\)
2. \(b + (c + a) + d\)
3. \((b + c) + d + a\)
4. \(d + a + (b + c)\)

Let's analyze the options:

- **Option 1**: \(d + (b + c) + a\) – This reorders the terms under the associative property and is valid.
- **Option 2**: \(b + (c + a) + d\) – This also reorders the terms, but changes the grouping, which is not equivalent.
- **Option 3**: \((b + c) + d + a\) – This simply rearranges the addition but keeps the two groupings (b + c) together, so it is valid.
- **Option 4**: \(d + a + (b + c)\) – This also rearranges terms, maintaining the group (b + c), so it is valid.

Thus, the expressions that are equivalent to \((b + c) + a + d\) are options 1, 3, and 4.

To conclude, any expression that maintains the sum without changing its internal grouping is equivalent, which means:

- **Equivalent expressions**:
- **d + (b + c) + a**
- **(b + c) + d + a**
- **d + a + (b + c)**

Therefore, options 1, 3, and 4 are correct, while option 2 is not equivalent.