Which of the following options is CORRECT ?

Options

A. Additive identity if a rational number a is `-1//a`
B. For rational numbers a,b and c `a-(b-c) = (a-b)-c`
C. The value of `1/(1+(2/3)/(1+2/3+(8/9)/(1-2/3))) " is " 13/15`
D. Reciprocal of `1/x` , (where `x != 0 `) is -x .

Answer

To solve the question, we need to analyze each of the given options and determine which one is correct. Let's break it down step by step.

### Step 1: Analyze the First Option
The first option states that the additive inverse of \( A \) is \( -\frac{1}{A} \).

**Explanation**:
The additive inverse of a number \( A \) is the number that, when added to \( A \), results in zero. The correct additive inverse of \( A \) is \( -A \), not \( -\frac{1}{A} \).

**Conclusion**: The first option is incorrect.

### Step 2: Analyze the Second Option
The second option involves the expression \( A - (B - C) \).

**Explanation**:
When we simplify \( A - (B - C) \), we distribute the negative sign:
\[ A - (B - C) = A - B + C \]
The expression on the right side is \( A - B - C \).

**Conclusion**: Since \( A - B + C \) is not equal to \( A - B - C \), the second option is also incorrect.

### Step 3: Analyze the Fourth Option
The fourth option states that the reciprocal of \( \frac{1}{X} \) is \( -X \).

**Explanation**:
The reciprocal of \( \frac{1}{X} \) is \( X \), not \( -X \).

**Conclusion**: The fourth option is incorrect.

### Step 4: Analyze the Third Option
The third option involves the expression:
\[ \frac{1}{1 + \frac{2}{3}} \div \left(1 + \frac{2}{3} + \frac{8}{9}\right) \div \left(1 - \frac{2}{3}\right) \]

**Step 4.1: Simplify the Denominator**
First, simplify \( 1 - \frac{2}{3} \):
\[ 1 - \frac{2}{3} = \frac{1}{3} \]

**Step 4.2: Simplify the Expression**
Now, we need to simplify the entire expression:
\[ \frac{1}{1 + \frac{2}{3}} \div \left(1 + \frac{2}{3} + \frac{8}{9}\right) \]

**Step 4.3: Simplify \( 1 + \frac{2}{3} \)**
\[ 1 + \frac{2}{3} = \frac{3}{3} + \frac{2}{3} = \frac{5}{3} \]

**Step 4.4: Simplify \( 1 + \frac{2}{3} + \frac{8}{9} \)**
Convert \( 1 \) to a fraction with a common denominator:
\[ 1 = \frac{9}{9} \quad \text{so,} \quad 1 + \frac{2}{3} + \frac{8}{9} = \frac{9}{9} + \frac{6}{9} + \frac{8}{9} = \frac{23}{9} \]

**Step 4.5: Substitute Back**
Now we substitute back into the expression:
\[ \frac{1}{\frac{5}{3}} \div \frac{23}{9} \div \frac{1}{3} \]

**Step 4.6: Simplify Further**
The expression simplifies to:
\[ \frac{3}{5} \div \frac{23}{9} \div \frac{1}{3} \]

**Step 4.7: Final Calculation**
Calculating \( \frac{3}{5} \div \frac{23}{9} \) gives us:
\[ \frac{3}{5} \times \frac{9}{23} = \frac{27}{115} \]

Now, dividing by \( \frac{1}{3} \):
\[ \frac{27}{115} \times 3 = \frac{81}{115} \]

### Conclusion
After analyzing all options, the only correct option is the third one.

### Final Answer
The correct option is the third one.