Simplify:
(i) `5/12 xx (-36)`, (ii) `-17/18 xx 12`, (iii) `-5/6 xx 6/5`, (iv) `-14 xx 9/28`, (v) `-12/5 xx (-15)`, (vi) `-3/4` by `8/7`, (v) `-16/9` by `12/-5`, (vi) `35/(-8)` by `12/(-5)`
(vii) `-3/10` by `-40/9`, (viii) `-32/5` by `15/(-16)`, (ix) `-8/15` by `-25/32`

Answer

Let's simplify each of the given expressions step by step.

### (i) \( \frac{5}{12} \times (-36) \)

1. **Simplify the fraction and the integer:**
- \( -36 \) can be rewritten as \( \frac{-36}{1} \).
- Multiply the numerators: \( 5 \times (-36) = -180 \).
- Multiply the denominators: \( 12 \times 1 = 12 \).
- So, we have \( \frac{-180}{12} \).

2. **Simplify \( \frac{-180}{12} \):**
- Divide both numerator and denominator by 12:
- \( -180 \div 12 = -15 \).

**Final answer:** \( -15 \)

### (ii) \( -\frac{17}{18} \times 12 \)

1. **Rewrite 12 as a fraction:**
- \( 12 = \frac{12}{1} \).
- Multiply the numerators: \( -17 \times 12 = -204 \).
- Multiply the denominators: \( 18 \times 1 = 18 \).
- So, we have \( \frac{-204}{18} \).

2. **Simplify \( \frac{-204}{18} \):**
- Divide both numerator and denominator by 6:
- \( -204 \div 6 = -34 \) and \( 18 \div 6 = 3 \).
- Thus, \( \frac{-204}{18} = \frac{-34}{3} \).

**Final answer:** \( -\frac{34}{3} \)

### (iii) \( -\frac{5}{6} \times \frac{6}{5} \)

1. **Multiply the fractions:**
- Multiply the numerators: \( -5 \times 6 = -30 \).
- Multiply the denominators: \( 6 \times 5 = 30 \).
- So, we have \( \frac{-30}{30} \).

2. **Simplify \( \frac{-30}{30} \):**
- This simplifies to \( -1 \).

**Final answer:** \( -1 \)

### (iv) \( -14 \times \frac{9}{28} \)

1. **Rewrite -14 as a fraction:**
- \( -14 = \frac{-14}{1} \).
- Multiply the numerators: \( -14 \times 9 = -126 \).
- Multiply the denominators: \( 28 \times 1 = 28 \).
- So, we have \( \frac{-126}{28} \).

2. **Simplify \( \frac{-126}{28} \):**
- Divide both numerator and denominator by 14:
- \( -126 \div 14 = -9 \) and \( 28 \div 14 = 2 \).
- Thus, \( \frac{-126}{28} = \frac{-9}{2} \).

**Final answer:** \( -\frac{9}{2} \)

### (v) \( -\frac{12}{5} \times (-15) \)

1. **Rewrite -15 as a fraction:**
- \( -15 = \frac{-15}{1} \).
- Multiply the numerators: \( -12 \times -15 = 180 \).
- Multiply the denominators: \( 5 \times 1 = 5 \).
- So, we have \( \frac{180}{5} \).

2. **Simplify \( \frac{180}{5} \):**
- Divide both numerator and denominator by 5:
- \( 180 \div 5 = 36 \).

**Final answer:** \( 36 \)

### (vi) \( -\frac{3}{4} \times \frac{8}{7} \)

1. **Multiply the fractions:**
- Multiply the numerators: \( -3 \times 8 = -24 \).
- Multiply the denominators: \( 4 \times 7 = 28 \).
- So, we have \( \frac{-24}{28} \).

2. **Simplify \( \frac{-24}{28} \):**
- Divide both numerator and denominator by 4:
- \( -24 \div 4 = -6 \) and \( 28 \div 4 = 7 \).
- Thus, \( \frac{-24}{28} = \frac{-6}{7} \).

**Final answer:** \( -\frac{6}{7} \)

### (vii) \( -\frac{16}{9} \times \frac{12}{-5} \)

1. **Multiply the fractions:**
- Multiply the numerators: \( -16 \times 12 = -192 \).
- Multiply the denominators: \( 9 \times -5 = -45 \).
- So, we have \( \frac{-192}{-45} \).

2. **Simplify \( \frac{-192}{-45} \):**
- The negatives cancel out, so we have \( \frac{192}{45} \).
- This fraction cannot be simplified further.

**Final answer:** \( \frac{192}{45} \)

### (viii) \( \frac{35}{-8} \times \frac{12}{-5} \)

1. **Multiply the fractions:**
- Multiply the numerators: \( 35 \times 12 = 420 \).
- Multiply the denominators: \( -8 \times -5 = 40 \).
- So, we have \( \frac{420}{40} \).

2. **Simplify \( \frac{420}{40} \):**
- Divide both numerator and denominator by 20:
- \( 420 \div 20 = 21 \) and \( 40 \div 20 = 2 \).
- Thus, \( \frac{420}{40} = \frac{21}{2} \).

**Final answer:** \( \frac{21}{2} \)

### (ix) \( -\frac{3}{10} \times \frac{-40}{9} \)

1. **Multiply the fractions:**
- Multiply the numerators: \( -3 \times -40 = 120 \).
- Multiply the denominators: \( 10 \times 9 = 90 \).
- So, we have \( \frac{120}{90} \).

2. **Simplify \( \frac{120}{90} \):**
- Divide both numerator and denominator by 30:
- \( 120 \div 30 = 4 \) and \( 90 \div 30 = 3 \).
- Thus, \( \frac{120}{90} = \frac{4}{3} \).

**Final answer:** \( \frac{4}{3} \)

### (x) \( -\frac{32}{5} \times \frac{15}{-16} \)

1. **Multiply the fractions:**
- Multiply the numerators: \( -32 \times 15 = -480 \).
- Multiply the denominators: \( 5 \times -16 = -80 \).
- So, we have \( \frac{-480}{-80} \).

2. **Simplify \( \frac{-480}{-80} \):**
- The negatives cancel out, so we have \( \frac{480}{80} \).
- This simplifies to \( 6 \).

**Final answer:** \( 6 \)

### (xi) \( -\frac{8}{15} \times \frac{-25}{32} \)

1. **Multiply the fractions:**
- Multiply the numerators: \( -8 \times -25 = 200 \).
- Multiply the denominators: \( 15 \times 32 = 480 \).
- So, we have \( \frac{200}{480} \).

2. **Simplify \( \frac{200}{480} \):**
- Divide both numerator and denominator by 40:
- \( 200 \div 40 = 5 \) and \( 480 \div 40 = 12 \).
- Thus, \( \frac{200}{480} = \frac{5}{12} \).

**Final answer:** \( \frac{5}{12} \)

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