Which of the following are positive rational numbers?
(i) `frac(3)(-5)`
(ii) `frac(-11)(15)`
(iii) `frac(-5)(-8)`
(iv) `frac(37)(53)`
(v) `frac(0)(3)`
(vi) 8

Answer

To determine which of the given numbers are positive rational numbers, we will analyze each option step by step.

### Step 1: Analyze `frac(3)(-5)`
- The number is `3 / -5`.
- The numerator (3) is positive, but the denominator (-5) is negative.
- A rational number is positive if both the numerator and denominator are either both positive or both negative.
- Since we have a positive numerator and a negative denominator, this number is **not a positive rational number**.

### Step 2: Analyze `frac(-11)(15)`
- The number is `-11 / 15`.
- The numerator (-11) is negative, and the denominator (15) is positive.
- Since the numerator is negative, this number is **not a positive rational number**.

### Step 3: Analyze `frac(-5)(-8)`
- The number is `-5 / -8`.
- Both the numerator (-5) and the denominator (-8) are negative.
- Since both are negative, they will cancel each other out, making the overall fraction positive.
- Therefore, this number is a **positive rational number**.

### Step 4: Analyze `frac(37)(53)`
- The number is `37 / 53`.
- Both the numerator (37) and the denominator (53) are positive.
- Since both are positive, this number is a **positive rational number**.

### Step 5: Analyze `frac(0)(3)`
- The number is `0 / 3`.
- The numerator (0) is zero, and the denominator (3) is positive.
- Zero is considered a rational number, but it is **not a positive rational number**. It is classified as a non-negative rational number.

### Step 6: Analyze `8`
- The number is `8`.
- We can express this as `8 / 1`.
- Both the numerator (8) and the denominator (1) are positive.
- Therefore, this number is a **positive rational number**.

### Summary of Positive Rational Numbers
From the analysis:
- (i) `frac(3)(-5)` - Not a positive rational number
- (ii) `frac(-11)(15)` - Not a positive rational number
- (iii) `frac(-5)(-8)` - Positive rational number
- (iv) `frac(37)(53)` - Positive rational number
- (v) `frac(0)(3)` - Not a positive rational number
- (vi) `8` - Positive rational number

### Final Answer
The positive rational numbers from the options are:
- (iii) `frac(-5)(-8)`
- (iv) `frac(37)(53)`
- (vi) `8`