Study the statements carefully.
(i) Every integer is a rational number and every fraction is a rational number.
(ii) A rational number `(p)/(q)` is positive, if p and q are either both positive or both negative.
(iii) A rational number `(p)/(q)` is negative, if one of p and q is positive and other is negative.
(iv) If there are two rational numbers with common denominator, then the one with the larger numerator is large than the other.
Which of the following options hold ?

Options

A. Both (i) and (iv) are incorrect
B. Both (ii) and (iii) are incorrect
C. Only (i) is inocrrect
D. All are correct

Answer

To solve the question, we need to analyze each of the four statements provided and determine whether they are true or false. Let's go through them step by step.

### Step 1: Analyze the First Statement
**Statement:** Every integer is a rational number and every fraction is a rational number.

**Explanation:**
- A rational number is defined as any number that can be expressed in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- Any integer \( n \) can be expressed as \( \frac{n}{1} \), which fits the definition of a rational number.
- A fraction, by definition, is already in the form \( \frac{p}{q} \).

**Conclusion:** This statement is true.

### Step 2: Analyze the Second Statement
**Statement:** A rational number \( \frac{p}{q} \) is positive if \( p \) and \( q \) are either both positive or both negative.

**Explanation:**
- If both \( p \) and \( q \) are positive, the fraction \( \frac{p}{q} \) is positive.
- If both \( p \) and \( q \) are negative, the negative signs cancel out, making the fraction positive.

**Conclusion:** This statement is true.

### Step 3: Analyze the Third Statement
**Statement:** A rational number \( \frac{p}{q} \) is negative if one of \( p \) and \( q \) is positive and the other is negative.

**Explanation:**
- If \( p \) is positive and \( q \) is negative, then \( \frac{p}{q} \) will be negative.
- Conversely, if \( p \) is negative and \( q \) is positive, \( \frac{p}{q} \) will also be negative.

**Conclusion:** This statement is true.

### Step 4: Analyze the Fourth Statement
**Statement:** If there are two rational numbers with a common denominator, then the one with the larger numerator is larger than the other.

**Explanation:**
- If we have two rational numbers \( \frac{a}{c} \) and \( \frac{b}{c} \) (where \( c \) is the common denominator), then:
- If \( a > b \), then \( \frac{a}{c} > \frac{b}{c} \).
- This is because dividing by a positive number (the denominator) does not change the inequality.

**Conclusion:** This statement is true.

### Final Conclusion
All four statements are true. Therefore, the answer to the question is that all options hold true.

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