Arrange the following ratios in acending order:
(i) `( 5 : 6), (8 : 9), (11 : 18)`
(ii) `(11 : 14), (17 : 21) : (5 : 7)` and `(2 : 3)`

Answer

To arrange the given ratios in ascending order, we will follow these steps:

### Part (i): Ratios `(5 : 6), (8 : 9), (11 : 18)`

1. **Convert Ratios to Fractions**:
- \( \frac{5}{6} \)
- \( \frac{8}{9} \)
- \( \frac{11}{18} \)

2. **Find the LCM of the Denominators**:
- The denominators are 6, 9, and 18.
- The LCM of 6, 9, and 18 is 18.

3. **Convert Each Fraction to Have the Same Denominator**:
- \( \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \)
- \( \frac{8}{9} = \frac{8 \times 2}{9 \times 2} = \frac{16}{18} \)
- \( \frac{11}{18} = \frac{11 \times 1}{18 \times 1} = \frac{11}{18} \)

4. **List the Converted Fractions**:
- \( \frac{15}{18}, \frac{16}{18}, \frac{11}{18} \)

5. **Arrange the Fractions in Ascending Order**:
- The fractions in order are \( \frac{11}{18}, \frac{15}{18}, \frac{16}{18} \).
- Therefore, in terms of ratios: \( 11 : 18 < 5 : 6 < 8 : 9 \).

### Part (ii): Ratios `(11 : 14), (17 : 21), (5 : 7), (2 : 3)`

1. **Convert Ratios to Fractions**:
- \( \frac{11}{14} \)
- \( \frac{17}{21} \)
- \( \frac{5}{7} \)
- \( \frac{2}{3} \)

2. **Find the LCM of the Denominators**:
- The denominators are 14, 21, 7, and 3.
- The LCM of 14, 21, 7, and 3 is 42.

3. **Convert Each Fraction to Have the Same Denominator**:
- \( \frac{11}{14} = \frac{11 \times 3}{14 \times 3} = \frac{33}{42} \)
- \( \frac{17}{21} = \frac{17 \times 2}{21 \times 2} = \frac{34}{42} \)
- \( \frac{5}{7} = \frac{5 \times 6}{7 \times 6} = \frac{30}{42} \)
- \( \frac{2}{3} = \frac{2 \times 14}{3 \times 14} = \frac{28}{42} \)

4. **List the Converted Fractions**:
- \( \frac{33}{42}, \frac{34}{42}, \frac{30}{42}, \frac{28}{42} \)

5. **Arrange the Fractions in Ascending Order**:
- The fractions in order are \( \frac{28}{42}, \frac{30}{42}, \frac{33}{42}, \frac{34}{42} \).
- Therefore, in terms of ratios: \( 2 : 3 < 5 : 7 < 11 : 14 < 17 : 21 \).

### Final Answer:
- For part (i): \( 11 : 18 < 5 : 6 < 8 : 9 \)
- For part (ii): \( 2 : 3 < 5 : 7 < 11 : 14 < 17 : 21 \)