Which of the following rational numbers satisfies the given property ?
`a + (b+c) = (a + b) + c`

Options

A. `a=-2/3, b = 5/6` and `c = -3/4`
B. `a=1/5, b = 3/5` and `c = -2/7`
C. `a =-5/7, b=-11/13` and `c = 17/21`
D. All of these

Answer

To solve the problem, we need to verify which of the given rational numbers satisfies the property \( a + (b + c) = (a + b) + c \). This property is known as the associative property of addition.

Let's denote the three rational numbers as follows:
- \( a = -\frac{2}{3} \)
- \( b = \frac{5}{6} \)
- \( c = -\frac{3}{4} \)

### Step 1: Calculate \( b + c \)
First, we need to calculate \( b + c \):
\[ b + c = \frac{5}{6} + \left(-\frac{3}{4}\right) \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 6 and 4 is 12.

Convert each fraction:
\[ \frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12} \]
\[ -\frac{3}{4} = -\frac{3 \times 3}{4 \times 3} = -\frac{9}{12} \]

Now, add them:
\[ b + c = \frac{10}{12} - \frac{9}{12} = \frac{1}{12} \]

### Step 2: Calculate \( a + (b + c) \)
Now we calculate \( a + (b + c) \):
\[ a + (b + c) = -\frac{2}{3} + \frac{1}{12} \]

Convert \( -\frac{2}{3} \) to have a denominator of 12:
\[ -\frac{2}{3} = -\frac{2 \times 4}{3 \times 4} = -\frac{8}{12} \]

Now, add:
\[ a + (b + c) = -\frac{8}{12} + \frac{1}{12} = -\frac{7}{12} \]

### Step 3: Calculate \( a + b \)
Next, we calculate \( a + b \):
\[ a + b = -\frac{2}{3} + \frac{5}{6} \]

Convert \( -\frac{2}{3} \) to have a denominator of 6:
\[ -\frac{2}{3} = -\frac{2 \times 2}{3 \times 2} = -\frac{4}{6} \]

Now, add:
\[ a + b = -\frac{4}{6} + \frac{5}{6} = \frac{1}{6} \]

### Step 4: Calculate \( (a + b) + c \)
Now we calculate \( (a + b) + c \):
\[ (a + b) + c = \frac{1}{6} + \left(-\frac{3}{4}\right) \]

Convert \( \frac{1}{6} \) to have a denominator of 12:
\[ \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \]
\[ -\frac{3}{4} = -\frac{3 \times 3}{4 \times 3} = -\frac{9}{12} \]

Now, add:
\[ (a + b) + c = \frac{2}{12} - \frac{9}{12} = -\frac{7}{12} \]

### Conclusion
We have:
\[ a + (b + c) = -\frac{7}{12} \]
\[ (a + b) + c = -\frac{7}{12} \]

Since both sides are equal, we can conclude that the property \( a + (b + c) = (a + b) + c \) holds true for the given rational numbers.

### Final Answer
Thus, the rational numbers \( a = -\frac{2}{3}, b = \frac{5}{6}, c = -\frac{3}{4} \) satisfy the property.