Question: Vance used the associative property to find the equivalent expression of $\left(4.5 m + \frac{7}{8}\right) - 9$. His work is shown below.
$\left(4.5 m + \frac{7}{8}\right) - 9 \quad \to \quad 4.5 m \left(\frac{7}{8} - 9\right)$
Did Vance apply the associative property correctly? Why or why not?
Yes, the terms with variables should be written first.
Yes, the terms can be regrouped and still be equivalent.
No, the order of the terms should have been changed.
No, the addition symbol was dropped in the second expression.
1. **State the problem:**
We are given the expression $\left(4.5 m + \frac{7}{8}\right) - 9$ and Vance's attempt to use the associative property to rewrite it as $4.5 m \left(\frac{7}{8} - 9\right)$. We need to determine if this is a correct application of the associative property.
2. **Recall the associative property:**
The associative property states that for addition or multiplication, the grouping of terms can be changed without changing the result. For example, for addition:
$$ (a + b) + c = a + (b + c) $$
and for multiplication:
$$ (a \times b) \times c = a \times (b \times c) $$
3. **Analyze the original expression:**
The original expression is:
$$ \left(4.5 m + \frac{7}{8}\right) - 9 $$
This is addition inside the parentheses and then subtraction outside.
4. **Check Vance's rewritten expression:**
Vance wrote:
$$ 4.5 m \left(\frac{7}{8} - 9\right) $$
Here, the addition symbol between $4.5 m$ and $\frac{7}{8}$ is replaced by multiplication, and the subtraction is moved inside the parentheses.
5. **Is this valid?**
- The associative property only allows regrouping of terms with the same operation (addition or multiplication), not changing addition to multiplication.
- Vance changed the addition to multiplication, which is not allowed.
- Also, the subtraction outside the parentheses was moved inside and combined with $\frac{7}{8}$, which is not justified by the associative property.
6. **Conclusion:**
Vance did not apply the associative property correctly because he changed the operation from addition to multiplication and dropped the addition symbol.
**Final answer:**
No, the addition symbol was dropped in the second expression, so the associative property was not applied correctly.