Which of the following are the pairs of equivalent fractions?
(i) `(5)/(6)and (20)/(24)`
(ii) `(3)/(8)and (15)/(40)`
(iii) `(4)/(7)and (16)/(21)`
(iv) `(2)/(9)and (14)/(63)`
(v) `(1)/(3)and (9)/(24)`
(vi) `(2)/(3)and (33)/(22)`

Answer

To determine which pairs of fractions are equivalent, we will check each pair by cross-multiplying or simplifying them. Equivalent fractions are fractions that represent the same value or proportion.

Let's analyze each pair step by step:

### Step 1: Check Pair (i) \( \frac{5}{6} \) and \( \frac{20}{24} \)

1. **Cross-multiply**: \( 5 \times 24 \) and \( 6 \times 20 \)
- \( 5 \times 24 = 120 \)
- \( 6 \times 20 = 120 \)
2. Since both products are equal, \( \frac{5}{6} = \frac{20}{24} \).

**Conclusion**: They are equivalent.

### Step 2: Check Pair (ii) \( \frac{3}{8} \) and \( \frac{15}{40} \)

1. **Cross-multiply**: \( 3 \times 40 \) and \( 8 \times 15 \)
- \( 3 \times 40 = 120 \)
- \( 8 \times 15 = 120 \)
2. Since both products are equal, \( \frac{3}{8} = \frac{15}{40} \).

**Conclusion**: They are equivalent.

### Step 3: Check Pair (iii) \( \frac{4}{7} \) and \( \frac{16}{21} \)

1. **Cross-multiply**: \( 4 \times 21 \) and \( 7 \times 16 \)
- \( 4 \times 21 = 84 \)
- \( 7 \times 16 = 112 \)
2. Since the products are not equal, \( \frac{4}{7} \neq \frac{16}{21} \).

**Conclusion**: They are not equivalent.

### Step 4: Check Pair (iv) \( \frac{2}{9} \) and \( \frac{14}{63} \)

1. **Cross-multiply**: \( 2 \times 63 \) and \( 9 \times 14 \)
- \( 2 \times 63 = 126 \)
- \( 9 \times 14 = 126 \)
2. Since both products are equal, \( \frac{2}{9} = \frac{14}{63} \).

**Conclusion**: They are equivalent.

### Step 5: Check Pair (v) \( \frac{1}{3} \) and \( \frac{9}{24} \)

1. **Cross-multiply**: \( 1 \times 24 \) and \( 3 \times 9 \)
- \( 1 \times 24 = 24 \)
- \( 3 \times 9 = 27 \)
2. Since the products are not equal, \( \frac{1}{3} \neq \frac{9}{24} \).

**Conclusion**: They are not equivalent.

### Step 6: Check Pair (vi) \( \frac{2}{3} \) and \( \frac{33}{22} \)

1. **Cross-multiply**: \( 2 \times 22 \) and \( 3 \times 33 \)
- \( 2 \times 22 = 44 \)
- \( 3 \times 33 = 99 \)
2. Since the products are not equal, \( \frac{2}{3} \neq \frac{33}{22} \).

**Conclusion**: They are not equivalent.

### Final Results:
- **Equivalent Fractions**: (i) \( \frac{5}{6} \) and \( \frac{20}{24} \), (ii) \( \frac{3}{8} \) and \( \frac{15}{40} \), (iv) \( \frac{2}{9} \) and \( \frac{14}{63} \)
- **Not Equivalent Fractions**: (iii) \( \frac{4}{7} \) and \( \frac{16}{21} \), (v) \( \frac{1}{3} \) and \( \frac{9}{24} \), (vi) \( \frac{2}{3} \) and \( \frac{33}{22} \)