To simplify the expression [tex]\(\frac{3}{4} - 5 + 9 - \frac{2}{4} + 2\)[/tex] using the associative and commutative properties, we need to strategically rearrange and group the terms to make the computation easier.
Let's examine each option:
Option A: [tex]\(\left[\frac{3}{4}+\left(-\frac{2}{4}\right)\right]+[9+2+(-5)]\)[/tex]
1. Group the fractions: [tex]\(\left[\frac{3}{4} + \left(-\frac{2}{4}\right)\right]\)[/tex]
- This simplifies to [tex]\(\frac{1}{4}\)[/tex].
2. Group the integers: [tex]\([9 + 2 + (-5)]\)[/tex]
- This simplifies to [tex]\(6\)[/tex].
Combining these results, the expression simplifies to [tex]\(\frac{1}{4} + 6\)[/tex], which is easier to compute.
Option B: [tex]\(\left[2+\frac{3}{4}+(-5)\right]+\left[\left(-\frac{2}{4}\right)+9\right]\)[/tex]
1. Group terms inside the first bracket: [tex]\(\left[2 + \frac{3}{4} + (-5)\right]\)[/tex]
- We do have mixed numbers and an integer, which may not be as straightforward to simplify.
2. Group terms inside the second bracket: [tex]\(\left[(-\frac{2}{4}) + 9\right]\)[/tex]
- This suggests subtracting a fraction from an integer, which might be less straightforward than needed.
Option C: [tex]\(\left[\frac{3}{4}+9+(-5)\right]+\left[2+\left(-\frac{2}{4}\right)\right]\)[/tex]
1. Group terms inside the first bracket: [tex]\(\left[\frac{3}{4} + 9 + (-5)\right]\)[/tex]
- This combines a fraction with integers, leading to more steps.
2. Group terms inside the second bracket: [tex]\(\left[2 + (-\frac{2}{4})\right]\)[/tex]
- Again, it combines a fraction and an integer, which complicates the simplification.
Option D: [tex]\(\left[\frac{3}{4}+(-5)\right]+\left[9+\left(-\frac{2}{4}\right)+2\right]\)[/tex]
1. Group terms inside the first bracket: [tex]\(\left[\frac{3}{4} + (-5)\right]\)[/tex]
- This combines a fraction with a negative integer, again leading to more steps.
2. Group terms inside the second bracket: [tex]\(\left[9 + (-\frac{2}{4}) + 2\right]\)[/tex]
- Combining larger integers and fractions could be slightly complex and need more simplification steps.
From these analyses, Option A emerges as the best use of the associative and commutative properties to simplify the expression. It involves straightforward grouping and calculation:
[tex]\[
\left[\frac{3}{4} + \left(-\frac{2}{4}\right)\right] + [9 + 2 + (-5)] = \left[\frac{1}{4}\right] + [6]
\][/tex]
This simplifies directly to [tex]\(\frac{1}{4} + 6\)[/tex], making Option A the most efficient and simplest method to simplify the expression.
Therefore, Option A is the correct answer.
