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 question, we need to demonstrate that the associative property holds true for rational numbers. The associative property states that for any three numbers \(a\), \(b\), and \(c\):

\[ a + (b + c) = (a + b) + c \]

This property is valid for all rational numbers. Let's verify this by using specific rational numbers.

### Step 1: Choose Rational Numbers
Let's choose three rational numbers:
- \(a = -\frac{2}{3}\)
- \(b = \frac{5}{6}\)
- \(c = -\frac{3}{4}\)

### Step 2: Calculate \(b + c\)
First, we 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 of 6 and 4 is 12.

Convert the fractions:
\[ \frac{5}{6} = \frac{10}{12} \quad \text{and} \quad -\frac{3}{4} = -\frac{9}{12} \]
Now add them:
\[ b + c = \frac{10}{12} - \frac{9}{12} = \frac{1}{12} \]

### Step 3: Calculate \(a + (b + c)\)
Now calculate \(a + (b + c)\):
\[ a + (b + c) = -\frac{2}{3} + \frac{1}{12} \]
Convert \(-\frac{2}{3}\) to have a denominator of 12:
\[ -\frac{2}{3} = -\frac{8}{12} \]
Now add:
\[ a + (b + c) = -\frac{8}{12} + \frac{1}{12} = -\frac{7}{12} \]

### Step 4: Calculate \(a + b\)
Next, calculate \(a + b\):
\[ a + b = -\frac{2}{3} + \frac{5}{6} \]
Convert \(-\frac{2}{3}\) to have a denominator of 6:
\[ -\frac{2}{3} = -\frac{4}{6} \]
Now add:
\[ a + b = -\frac{4}{6} + \frac{5}{6} = \frac{1}{6} \]

### Step 5: Calculate \((a + b) + c\)
Now calculate \((a + b) + c\):
\[ (a + b) + c = \frac{1}{6} + \left(-\frac{3}{4}\right) \]
Convert \(-\frac{3}{4}\) to have a denominator of 12:
\[ -\frac{3}{4} = -\frac{9}{12} \]
Convert \(\frac{1}{6}\) to have a denominator of 12:
\[ \frac{1}{6} = \frac{2}{12} \]
Now add:
\[ (a + b) + c = \frac{2}{12} - \frac{9}{12} = -\frac{7}{12} \]

### Step 6: Compare Results
Now we compare the two results:
- \(a + (b + c) = -\frac{7}{12}\)
- \((a + b) + c = -\frac{7}{12}\)

Since both sides are equal, we have verified the associative property for the chosen rational numbers.

### Conclusion
The associative property \(a + (b + c) = (a + b) + c\) holds true for all rational numbers.