Which of the following rational number are equal?
(i) `frac(8)(-12)("and")frac(-10)(15)`
(ii) `frac(-3)(9)("and")frac(7)(-21)`
(iii) `frac(-8)(-14)("and")frac(15)(21)`

Answer

To determine which of the following rational numbers are equal, we will simplify each pair of fractions step by step.

### Step 1: Simplifying the first pair of fractions
**Fractions:** \( \frac{8}{-12} \) and \( \frac{-10}{15} \)

1. **Simplify \( \frac{8}{-12} \):**
- The greatest common divisor (GCD) of 8 and 12 is 4.
- Simplifying gives:
\[ \frac{8 \div 4}{-12 \div 4} = \frac{2}{-3} = -\frac{2}{3} \]

2. **Simplify \( \frac{-10}{15} \):**
- The GCD of -10 and 15 is 5.
- Simplifying gives:
\[ \frac{-10 \div 5}{15 \div 5} = \frac{-2}{3} \]

**Conclusion for Step 1:** Both fractions simplify to \( -\frac{2}{3} \), so they are equal.

### Step 2: Simplifying the second pair of fractions
**Fractions:** \( \frac{-3}{9} \) and \( \frac{7}{-21} \)

1. **Simplify \( \frac{-3}{9} \):**
- The GCD of -3 and 9 is 3.
- Simplifying gives:
\[ \frac{-3 \div 3}{9 \div 3} = \frac{-1}{3} \]

2. **Simplify \( \frac{7}{-21} \):**
- The GCD of 7 and -21 is 7.
- Simplifying gives:
\[ \frac{7 \div 7}{-21 \div 7} = \frac{1}{-3} = -\frac{1}{3} \]

**Conclusion for Step 2:** Both fractions simplify to \( -\frac{1}{3} \), so they are equal.

### Step 3: Simplifying the third pair of fractions
**Fractions:** \( \frac{-8}{-14} \) and \( \frac{15}{21} \)

1. **Simplify \( \frac{-8}{-14} \):**
- The negatives cancel out, so we have:
\[ \frac{8}{14} \]
- The GCD of 8 and 14 is 2.
- Simplifying gives:
\[ \frac{8 \div 2}{14 \div 2} = \frac{4}{7} \]

2. **Simplify \( \frac{15}{21} \):**
- The GCD of 15 and 21 is 3.
- Simplifying gives:
\[ \frac{15 \div 3}{21 \div 3} = \frac{5}{7} \]

**Conclusion for Step 3:** The fractions simplify to \( \frac{4}{7} \) and \( \frac{5}{7} \), which are not equal.

### Final Conclusion:
- The first pair \( \frac{8}{-12} \) and \( \frac{-10}{15} \) are equal.
- The second pair \( \frac{-3}{9} \) and \( \frac{7}{-21} \) are equal.
- The third pair \( \frac{-8}{-14} \) and \( \frac{15}{21} \) are not equal.