Arrange these rational numbers in ascending order `(3)/(8), (4)/(12), (-7)/(16), (-2)/(3).`

Answer

To arrange the rational numbers \( \frac{3}{8}, \frac{4}{12}, \frac{-7}{16}, \frac{-2}{3} \) in ascending order, we will follow these steps:

### Step 1: Find the LCM of the Denominators
The denominators are 8, 12, 16, and 3. We need to find the least common multiple (LCM) of these numbers.

- The prime factorization of each number is:
- \( 8 = 2^3 \)
- \( 12 = 2^2 \times 3^1 \)
- \( 16 = 2^4 \)
- \( 3 = 3^1 \)

To find the LCM, we take the highest power of each prime:
- For \( 2 \), the highest power is \( 2^4 \) (from 16).
- For \( 3 \), the highest power is \( 3^1 \) (from 12 and 3).

Thus, the LCM is:
\[ LCM = 2^4 \times 3^1 = 16 \times 3 = 48 \]

### Step 2: Convert Each Rational Number to Have the Same Denominator
Now we will convert each rational number to have a denominator of 48.

1. **Convert \( \frac{3}{8} \)**:
\[ \frac{3}{8} = \frac{3 \times 6}{8 \times 6} = \frac{18}{48} \]

2. **Convert \( \frac{4}{12} \)**:
\[ \frac{4}{12} = \frac{4 \times 4}{12 \times 4} = \frac{16}{48} \]

3. **Convert \( \frac{-7}{16} \)**:
\[ \frac{-7}{16} = \frac{-7 \times 3}{16 \times 3} = \frac{-21}{48} \]

4. **Convert \( \frac{-2}{3} \)**:
\[ \frac{-2}{3} = \frac{-2 \times 16}{3 \times 16} = \frac{-32}{48} \]

### Step 3: List the Converted Rational Numbers
Now we have:
- \( \frac{3}{8} = \frac{18}{48} \)
- \( \frac{4}{12} = \frac{16}{48} \)
- \( \frac{-7}{16} = \frac{-21}{48} \)
- \( \frac{-2}{3} = \frac{-32}{48} \)

### Step 4: Compare the Numerators
Now we can compare the numerators:
- \( -32 \) (from \( \frac{-32}{48} \))
- \( -21 \) (from \( \frac{-21}{48} \))
- \( 16 \) (from \( \frac{16}{48} \))
- \( 18 \) (from \( \frac{18}{48} \))

### Step 5: Arrange in Ascending Order
The order from smallest to largest is:
\[ \frac{-32}{48} < \frac{-21}{48} < \frac{16}{48} < \frac{18}{48} \]

### Final Answer
Thus, the rational numbers in ascending order are:
\[ \frac{-2}{3}, \frac{-7}{16}, \frac{4}{12}, \frac{3}{8} \]