To find the product of [tex]\(4 \times 27\)[/tex] using mental math, we can use various strategies, such as breaking the number into more manageable parts or using properties of operations. Here are the detailed explanations for each option:
1. [tex]\((4 \times 20) + (4 \times 7)\)[/tex]:
- This uses the distributive property.
- Break 27 into 20 and 7.
- Calculate [tex]\(4 \times 20 = 80\)[/tex].
- Calculate [tex]\(4 \times 7 = 28\)[/tex].
- Add the results: [tex]\(80 + 28 = 108\)[/tex].
This correctly represents [tex]\(4 \times 27\)[/tex] using the distributive property.
2. [tex]\(4 \times (20 \times 7)\)[/tex]:
- This suggests multiplying 20 and 7 first and then multiplying by 4.
- However, multiplying 20 and 7 gives 140.
- So, this becomes [tex]\(4 \times 140\)[/tex], which is not equal to [tex]\(4 \times 27\)[/tex].
This is incorrect for finding [tex]\(4 \times 27\)[/tex].
3. [tex]\((4 \times 30) - (4 \times 3)\)[/tex]:
- This uses the distributive property with subtraction.
- Break 27 into 30 and 3.
- Calculate [tex]\(4 \times 30 = 120\)[/tex].
- Calculate [tex]\(4 \times 3 = 12\)[/tex].
- Subtract the results: [tex]\(120 - 12 = 108\)[/tex].
This correctly represents [tex]\(4 \times 27\)[/tex] using the distributive property.
4. [tex]\((4 \times 25) + (4 \times 2)\)[/tex]:
- This again uses the distributive property.
- Break 27 into 25 and 2.
- Calculate [tex]\(4 \times 25 = 100\)[/tex].
- Calculate [tex]\(4 \times 2 = 8\)[/tex].
- Add the results: [tex]\(100 + 8 = 108\)[/tex].
This correctly represents [tex]\(4 \times 27\)[/tex] using the distributive property.
5. [tex]\(4 \times 2 \times 7\)[/tex]:
- This suggests reordering the multiplication due to the associative property.
- However, this would be interpreted as [tex]\(4 \times 2 \times 7 = (4 \times 2) \times 7 = 8 \times 7 = 56\)[/tex], which is not equal to [tex]\(4 \times 27\)[/tex].
This is incorrect for finding [tex]\(4 \times 27\)[/tex].
### Correct Options:
- [tex]\((4 \times 20) + (4 \times 7)\)[/tex]
- [tex]\((4 \times 30) - (4 \times 3)\)[/tex]
- [tex]\((4 \times 25) + (4 \times 2)\)[/tex]
