Which ratio is greater ?
(i) `(5 : 6)` or `(7: 9)`
(ii) `(2 : 3)` or `(4 : 7)`
(iii) `(1 : 2)` or `(4 : 7)`
(iv) `(3 : 5)` or `(8 : 13)`

Answer

To determine which ratio is greater for each pair given in the question, we will follow these steps:

### Step 1: Compare the ratios (5:6) and (7:9)

1. **Find the LCM of the denominators (6 and 9)**:
- The LCM of 6 and 9 is 18.

2. **Convert the ratios to have the same denominator**:
- For (5:6), multiply both parts by 3:
\[ 5 \times 3 = 15 \quad \text{and} \quad 6 \times 3 = 18 \quad \Rightarrow \quad 5:6 = 15:18 \]
- For (7:9), multiply both parts by 2:
\[ 7 \times 2 = 14 \quad \text{and} \quad 9 \times 2 = 18 \quad \Rightarrow \quad 7:9 = 14:18 \]

3. **Compare the numerators**:
- Since 15 > 14, we conclude that:
\[ 5:6 \text{ is greater than } 7:9 \]

### Step 2: Compare the ratios (2:3) and (4:7)

1. **Find the LCM of the denominators (3 and 7)**:
- The LCM of 3 and 7 is 21.

2. **Convert the ratios to have the same denominator**:
- For (2:3), multiply both parts by 7:
\[ 2 \times 7 = 14 \quad \text{and} \quad 3 \times 7 = 21 \quad \Rightarrow \quad 2:3 = 14:21 \]
- For (4:7), multiply both parts by 3:
\[ 4 \times 3 = 12 \quad \text{and} \quad 7 \times 3 = 21 \quad \Rightarrow \quad 4:7 = 12:21 \]

3. **Compare the numerators**:
- Since 14 > 12, we conclude that:
\[ 2:3 \text{ is greater than } 4:7 \]

### Step 3: Compare the ratios (1:2) and (4:7)

1. **Find the LCM of the denominators (2 and 7)**:
- The LCM of 2 and 7 is 14.

2. **Convert the ratios to have the same denominator**:
- For (1:2), multiply both parts by 7:
\[ 1 \times 7 = 7 \quad \text{and} \quad 2 \times 7 = 14 \quad \Rightarrow \quad 1:2 = 7:14 \]
- For (4:7), multiply both parts by 2:
\[ 4 \times 2 = 8 \quad \text{and} \quad 7 \times 2 = 14 \quad \Rightarrow \quad 4:7 = 8:14 \]

3. **Compare the numerators**:
- Since 8 > 7, we conclude that:
\[ 4:7 \text{ is greater than } 1:2 \]

### Step 4: Compare the ratios (3:5) and (8:13)

1. **Find the LCM of the denominators (5 and 13)**:
- The LCM of 5 and 13 is 65.

2. **Convert the ratios to have the same denominator**:
- For (3:5), multiply both parts by 13:
\[ 3 \times 13 = 39 \quad \text{and} \quad 5 \times 13 = 65 \quad \Rightarrow \quad 3:5 = 39:65 \]
- For (8:13), multiply both parts by 5:
\[ 8 \times 5 = 40 \quad \text{and} \quad 13 \times 5 = 65 \quad \Rightarrow \quad 8:13 = 40:65 \]

3. **Compare the numerators**:
- Since 40 > 39, we conclude that:
\[ 8:13 \text{ is greater than } 3:5 \]

### Summary of Results:
1. (5:6) > (7:9)
2. (2:3) > (4:7)
3. (4:7) > (1:2)
4. (8:13) > (3:5)