Answer :
Let's analyze each expression step-by-step by using the commutative property, which allows us to change the order of the numbers when adding.
### Expression A: [tex]\(\frac{1}{7} + (-1) + \frac{2}{7}\)[/tex]
1. Rearrange the terms:
[tex]\[ \left( \frac{1}{7} + \frac{2}{7} \right) + (-1) \][/tex]
2. Add the fractions:
[tex]\[ \frac{1}{7} + \frac{2}{7} = \frac{3}{7} \][/tex]
3. Combine the result with [tex]\(-1\)[/tex]:
[tex]\[ \frac{3}{7} + (-1) = \frac{3}{7} - 1 = \frac{3}{7} - \frac{7}{7} = -\frac{4}{7} \][/tex]
Although we can simplify this expression, it involves dealing with fractions, which might be a bit cumbersome.
### Expression B: [tex]\(120 + 80 + (-65)\)[/tex]
1. Rearrange the terms:
[tex]\[ \left( 120 + 80 \right) + (-65) \][/tex]
2. Add [tex]\(120\)[/tex] and [tex]\(80\)[/tex]:
[tex]\[ 120 + 80 = 200 \][/tex]
3. Combine the result with [tex]\(-65\)[/tex]:
[tex]\[ 200 + (-65) = 200 - 65 = 135 \][/tex]
This expression simplifies directly to a single integer rather easily.
### Expression C: [tex]\(40 + 10 + (-12)\)[/tex]
1. Rearrange the terms:
[tex]\[ \left( 40 + 10 \right) + (-12) \][/tex]
2. Add [tex]\(40\)[/tex] and [tex]\(10\)[/tex]:
[tex]\[ 40 + 10 = 50 \][/tex]
3. Combine the result with [tex]\(-12\)[/tex]:
[tex]\[ 50 + (-12) = 50 - 12 = 38 \][/tex]
This expression simplifies to an integer as well, but with slightly smaller numbers than in expression B.
### Expression D: [tex]\(-15 + (-25) + 43\)[/tex]
1. Rearrange the terms:
[tex]\[ \left( -15 + 43 \right) + (-25) \][/tex]
2. Add [tex]\(-15\)[/tex] and [tex]\(43\)[/tex]:
[tex]\[ -15 + 43 = 28 \][/tex]
3. Combine the result with [tex]\(-25\)[/tex]:
[tex]\[ 28 + (-25) = 28 - 25 = 3 \][/tex]
This expression also simplifies to an integer, but again involves working with negative numbers.
### Conclusion
When considering ease of simplification using the commutative property:
- Expression A involves fractions, making it potentially more complex.
- Expressions C and D involve smaller numbers but include negative values.
- Expression B, [tex]\(120 + 80 + (-65)\)[/tex], simplifies to [tex]\(135\)[/tex] directly with straightforward steps.
Thus, Expression B is the easiest to simplify using the commutative property.
Therefore, the answer is:
[tex]\[ \boxed{2} \][/tex]
### Expression A: [tex]\(\frac{1}{7} + (-1) + \frac{2}{7}\)[/tex]
1. Rearrange the terms:
[tex]\[ \left( \frac{1}{7} + \frac{2}{7} \right) + (-1) \][/tex]
2. Add the fractions:
[tex]\[ \frac{1}{7} + \frac{2}{7} = \frac{3}{7} \][/tex]
3. Combine the result with [tex]\(-1\)[/tex]:
[tex]\[ \frac{3}{7} + (-1) = \frac{3}{7} - 1 = \frac{3}{7} - \frac{7}{7} = -\frac{4}{7} \][/tex]
Although we can simplify this expression, it involves dealing with fractions, which might be a bit cumbersome.
### Expression B: [tex]\(120 + 80 + (-65)\)[/tex]
1. Rearrange the terms:
[tex]\[ \left( 120 + 80 \right) + (-65) \][/tex]
2. Add [tex]\(120\)[/tex] and [tex]\(80\)[/tex]:
[tex]\[ 120 + 80 = 200 \][/tex]
3. Combine the result with [tex]\(-65\)[/tex]:
[tex]\[ 200 + (-65) = 200 - 65 = 135 \][/tex]
This expression simplifies directly to a single integer rather easily.
### Expression C: [tex]\(40 + 10 + (-12)\)[/tex]
1. Rearrange the terms:
[tex]\[ \left( 40 + 10 \right) + (-12) \][/tex]
2. Add [tex]\(40\)[/tex] and [tex]\(10\)[/tex]:
[tex]\[ 40 + 10 = 50 \][/tex]
3. Combine the result with [tex]\(-12\)[/tex]:
[tex]\[ 50 + (-12) = 50 - 12 = 38 \][/tex]
This expression simplifies to an integer as well, but with slightly smaller numbers than in expression B.
### Expression D: [tex]\(-15 + (-25) + 43\)[/tex]
1. Rearrange the terms:
[tex]\[ \left( -15 + 43 \right) + (-25) \][/tex]
2. Add [tex]\(-15\)[/tex] and [tex]\(43\)[/tex]:
[tex]\[ -15 + 43 = 28 \][/tex]
3. Combine the result with [tex]\(-25\)[/tex]:
[tex]\[ 28 + (-25) = 28 - 25 = 3 \][/tex]
This expression also simplifies to an integer, but again involves working with negative numbers.
### Conclusion
When considering ease of simplification using the commutative property:
- Expression A involves fractions, making it potentially more complex.
- Expressions C and D involve smaller numbers but include negative values.
- Expression B, [tex]\(120 + 80 + (-65)\)[/tex], simplifies to [tex]\(135\)[/tex] directly with straightforward steps.
Thus, Expression B is the easiest to simplify using the commutative property.
Therefore, the answer is:
[tex]\[ \boxed{2} \][/tex]
