Arrange the given ratios as indicated:
a. In ascending order:
In descending order:
i. 2:3, 3:5, 5:6, 2:9 ii. 1:4, 3:5, 5:6, 7:10

Answer

To solve the problem of arranging the given ratios in ascending and descending order, we will follow these steps:

### Step-by-Step Solution:

**Part a: Ratios 2:3, 3:5, 5:6, 2:9**

1. **Convert Ratios to Decimal Form:**
- \( \frac{2}{3} = 0.6667 \)
- \( \frac{3}{5} = 0.6 \)
- \( \frac{5}{6} = 0.8333 \)
- \( \frac{2}{9} = 0.2222 \)

2. **Arrange in Ascending Order:**
- The smallest value is \( 0.2222 \) (which is \( \frac{2}{9} \)).
- Next is \( 0.6 \) (which is \( \frac{3}{5} \)).
- Then \( 0.6667 \) (which is \( \frac{2}{3} \)).
- Finally, \( 0.8333 \) (which is \( \frac{5}{6} \)).
- **Ascending Order:** \( \frac{2}{9} < \frac{3}{5} < \frac{2}{3} < \frac{5}{6} \)

3. **Arrange in Descending Order:**
- Start with the largest value \( 0.8333 \) (which is \( \frac{5}{6} \)).
- Next is \( 0.6667 \) (which is \( \frac{2}{3} \)).
- Then \( 0.6 \) (which is \( \frac{3}{5} \)).
- Finally, \( 0.2222 \) (which is \( \frac{2}{9} \)).
- **Descending Order:** \( \frac{5}{6} > \frac{2}{3} > \frac{3}{5} > \frac{2}{9} \)

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**Part b: Ratios 1:4, 3:5, 5:6, 7:10**

1. **Convert Ratios to Decimal Form:**
- \( \frac{1}{4} = 0.25 \)
- \( \frac{3}{5} = 0.6 \)
- \( \frac{5}{6} = 0.8333 \)
- \( \frac{7}{10} = 0.7 \)

2. **Arrange in Ascending Order:**
- The smallest value is \( 0.25 \) (which is \( \frac{1}{4} \)).
- Next is \( 0.6 \) (which is \( \frac{3}{5} \)).
- Then \( 0.7 \) (which is \( \frac{7}{10} \)).
- Finally, \( 0.8333 \) (which is \( \frac{5}{6} \)).
- **Ascending Order:** \( \frac{1}{4} < \frac{3}{5} < \frac{7}{10} < \frac{5}{6} \)

3. **Arrange in Descending Order:**
- Start with the largest value \( 0.8333 \) (which is \( \frac{5}{6} \)).
- Next is \( 0.7 \) (which is \( \frac{7}{10} \)).
- Then \( 0.6 \) (which is \( \frac{3}{5} \)).
- Finally, \( 0.25 \) (which is \( \frac{1}{4} \)).
- **Descending Order:** \( \frac{5}{6} > \frac{7}{10} > \frac{3}{5} > \frac{1}{4} \)

### Final Answers:
- **Part a Ascending Order:** \( \frac{2}{9} < \frac{3}{5} < \frac{2}{3} < \frac{5}{6} \)
- **Part a Descending Order:** \( \frac{5}{6} > \frac{2}{3} > \frac{3}{5} > \frac{2}{9} \)
- **Part b Ascending Order:** \( \frac{1}{4} < \frac{3}{5} < \frac{7}{10} < \frac{5}{6} \)
- **Part b Descending Order:** \( \frac{5}{6} > \frac{7}{10} > \frac{3}{5} > \frac{1}{4} \)