Arrange the rational numbers `3/(-4), 9/16, -11/12` and `23/(-32)` in ascending order.

Answer

To arrange the rational numbers \( \frac{3}{-4}, \frac{9}{16}, \frac{-11}{12}, \frac{23}{-32} \) in ascending order, we can follow these steps:

### Step 1: Identify the Rational Numbers
The given rational numbers are:
- \( \frac{3}{-4} \) (which is equivalent to \( -\frac{3}{4} \))
- \( \frac{9}{16} \)
- \( \frac{-11}{12} \)
- \( \frac{23}{-32} \) (which is equivalent to \( -\frac{23}{32} \))

### Step 2: Find the LCM of the Denominators
The denominators are \( 4, 16, 12, \) and \( 32 \). We need to find the least common multiple (LCM) of these numbers.

- **Factors:**
- \( 4 = 2^2 \)
- \( 16 = 2^4 \)
- \( 12 = 2^2 \times 3 \)
- \( 32 = 2^5 \)

The LCM will take the highest power of each prime factor:
- For \( 2 \): highest power is \( 2^5 \) from \( 32 \)
- For \( 3 \): highest power is \( 3^1 \) from \( 12 \)

Thus, the LCM is:
\[ LCM = 2^5 \times 3^1 = 32 \times 3 = 96 \]

### Step 3: Convert Each Rational Number to Have a Common Denominator
Now, we will convert each rational number to have a denominator of 96.

1. **For \( -\frac{3}{4} \)**:
\[ -\frac{3}{4} \times \frac{24}{24} = -\frac{72}{96} \]

2. **For \( \frac{9}{16} \)**:
\[ \frac{9}{16} \times \frac{6}{6} = \frac{54}{96} \]

3. **For \( -\frac{11}{12} \)**:
\[ -\frac{11}{12} \times \frac{8}{8} = -\frac{88}{96} \]

4. **For \( -\frac{23}{32} \)**:
\[ -\frac{23}{32} \times \frac{3}{3} = -\frac{69}{96} \]

### Step 4: Arrange the Converted Rational Numbers
Now we have:
- \( -\frac{72}{96} \)
- \( \frac{54}{96} \)
- \( -\frac{88}{96} \)
- \( -\frac{69}{96} \)

Next, we can arrange these fractions in ascending order. Remember, for negative fractions, a larger absolute value means a smaller value.

1. \( -\frac{88}{96} \) (smallest)
2. \( -\frac{72}{96} \)
3. \( -\frac{69}{96} \)
4. \( \frac{54}{96} \) (largest)

### Step 5: Write the Original Rational Numbers in Order
Now we can write the original rational numbers in ascending order:
1. \( -\frac{11}{12} \)
2. \( -\frac{3}{4} \)
3. \( -\frac{23}{32} \)
4. \( \frac{9}{16} \)

### Final Answer
The ascending order of the rational numbers is:
\[ -\frac{11}{12}, -\frac{3}{4}, -\frac{23}{32}, \frac{9}{16} \]