Certainly! Let's understand why both Jenna and Mia arrived at the same answer and analyze the advantages of their methods.
### Jenna's Method
Jenna used the distributive property of multiplication over addition. The distributive property states that [tex]\( a(b + c) = ab + ac \)[/tex].
Here are the steps Jenna took:
1. Expression Setup: Start with the expression [tex]\( 5(30 + 4) \)[/tex].
2. Apply Distributive Property: Distribute 5 to both 30 and 4.
[tex]\[
5(30 + 4) = (5 \cdot 30) + (5 \cdot 4)
\][/tex]
3. Compute Each Term:
[tex]\[
5 \cdot 30 = 150
\][/tex]
[tex]\[
5 \cdot 4 = 20
\][/tex]
4. Add the Results:
[tex]\[
150 + 20 = 170
\][/tex]
So, Jenna's method involves breaking the multiplication down into two simpler multiplications and then adding the results.
### Mia's Method
Mia combined the terms inside the parentheses first and then performed the multiplication.
Here are the steps Mia took:
1. Expression Setup: Start with the expression [tex]\( 5(30 + 4) \)[/tex].
2. Combine the Terms in Parentheses:
[tex]\[
30 + 4 = 34
\][/tex]
3. Multiply:
[tex]\[
5 \cdot 34 = 170
\][/tex]
Mia’s method is more straightforward as she combines the terms first and then performs a single multiplication.
### Why Both Methods Yield the Same Result
Both Jenna’s and Mia’s methods yield the same result because they are different applications of basic arithmetic principles. Jenna uses the distributive property explicitly, while Mia simplifies the expression within the parentheses first and then multiplies. Mathematically, both paths are valid and lead to the same end result due to the inherent properties of numbers and operations.
### Advantages of Each Method
- Jenna's Method:
- Visualization: This method can help visualize and understand the distribution of multiplication over addition, which is particularly useful when learning the distributive property.
- Step-by-Step Process: It breaks down the problem into smaller, more manageable parts, which can help in understanding and verifying each part of the process.
- Mia's Method:
- Simplicity: This method is more straightforward and efficient, especially when dealing with basic additions and smaller numbers.
- Less Computational Steps: Fewer steps are involved, which can save time and make calculations easier for someone comfortable with basic arithmetic.
In summary, while both methods are correct and yield the same result of [tex]\( 170 \)[/tex], Jenna’s method is beneficial for step-by-step understanding of the distributive property, and Mia’s method is advantageous for its simplicity and efficiency in straightforward calculations.
