Which of the following options shows the numbers arranged in ascending order ?

Options

A. `- (6)/(7) lt - (8)/(7) lt (4)/(7) lt (5)/(7)`
B. `- (5)/(2) lt - (5)/(4) lt -(5)/(7) lt - (5)/(9)`
C. `- (2)/(9) lt - (3)/(7) lt - (4)/(5) lt - (6)/(5)`
D. `- (5)/(4) lt - (7)/(3) lt - (2)/(9) lt - (6)/(7)`

Answer

To determine which option shows the numbers arranged in ascending order, we will convert each rational number into its decimal form and then compare them. Let's analyze the options step by step.

### Step-by-Step Solution:

**Step 1: Convert each rational number to decimal form.**

1. **Option 1:**
- \( \frac{5}{7} \) = 0.714
- \( \frac{2}{7} \) = 0.2857 (approximately)
- \( -\frac{8}{7} \) = -1.142857 (approximately)
- \( -\frac{6}{7} \) = -0.857142 (approximately)

2. **Option 2:**
- \( -\frac{5}{9} \) = -0.5555 (approximately)
- \( -\frac{5}{7} \) = -0.714285 (approximately)
- \( -\frac{5}{4} \) = -1.25
- \( -\frac{5}{2} \) = -2.5

3. **Option 3:**
- \( -\frac{6}{5} \) = -1.2
- \( -\frac{4}{5} \) = -0.8

4. **Option 4:**
- \( -\frac{6}{7} \) = -0.857142 (approximately)
- \( -\frac{2}{9} \) = -0.2222 (approximately)

**Step 2: Compare the decimal values.**

1. **Option 1:**
- Order: \( -\frac{8}{7} < -\frac{6}{7} < \frac{2}{7} < \frac{5}{7} \)
- This order is not correct as \( -1.142857 < -0.857142 < 0.2857 < 0.714 \)

2. **Option 2:**
- Order: \( -\frac{5}{2} < -\frac{5}{4} < -\frac{5}{7} < -\frac{5}{9} \)
- This order is correct as \( -2.5 < -1.25 < -0.714285 < -0.5555 \)

3. **Option 3:**
- Order: \( -\frac{6}{5} < -\frac{4}{5} \)
- This order is not correct as \( -1.2 < -0.8 \)

4. **Option 4:**
- Order: \( -\frac{6}{7} < -\frac{2}{9} \)
- This order is not correct as \( -0.857142 < -0.2222 \)

**Conclusion:**
Only **Option 2** shows the numbers arranged in ascending order.