Arrangement of the fractions `(4)/(3) , (2)/(9) , - (7)/(8) , (5)/(12)` into ascending order is

Options

A. `- (7)/(8), (2)/(9), (5)/(12), (4)/(3)`
B. `- (7)/(8) , - (2)/(9), (4)/(3), (5)/(12)`
C. `- (2)/(9), - (7)/(8) , (5)/(12) , (4)/(3)`
D. `- (2)/(9) ,- (7)/(8), (4)/(3) , (5)/(12)`

Answer

To arrange the fractions \( \frac{4}{3}, \frac{2}{9}, -\frac{7}{8}, \frac{5}{12} \) into ascending order, we will follow these steps:

### Step 1: Identify the fractions
We have the following fractions:
- \( \frac{4}{3} \)
- \( \frac{2}{9} \)
- \( -\frac{7}{8} \)
- \( \frac{5}{12} \)

### Step 2: Find the LCM of the denominators
The denominators are \( 3, 9, 8, \) and \( 12 \). We need to find the least common multiple (LCM) of these numbers.

1. **Prime factorization**:
- \( 3 = 3^1 \)
- \( 9 = 3^2 \)
- \( 8 = 2^3 \)
- \( 12 = 2^2 \times 3^1 \)

2. **Taking the highest powers**:
- For \( 2 \): highest power is \( 2^3 \) from \( 8 \)
- For \( 3 \): highest power is \( 3^2 \) from \( 9 \)

3. **Calculating LCM**:
\[ \text{LCM} = 2^3 \times 3^2 = 8 \times 9 = 72 \]

### Step 3: Convert each fraction to have the common denominator
Now we will convert each fraction to have a denominator of \( 72 \):

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

2. **Convert \( \frac{2}{9} \)**:
\[ \frac{2}{9} \times \frac{8}{8} = \frac{16}{72} \]

3. **Convert \( -\frac{7}{8} \)**:
\[ -\frac{7}{8} \times \frac{9}{9} = -\frac{63}{72} \]

4. **Convert \( \frac{5}{12} \)**:
\[ \frac{5}{12} \times \frac{6}{6} = \frac{30}{72} \]

### Step 4: List the converted fractions
Now we have:
- \( \frac{96}{72} \)
- \( \frac{16}{72} \)
- \( -\frac{63}{72} \)
- \( \frac{30}{72} \)

### Step 5: Arrange the fractions in ascending order
Now we can compare the numerators:
- \( -63 \) (from \( -\frac{63}{72} \))
- \( 16 \) (from \( \frac{16}{72} \))
- \( 30 \) (from \( \frac{30}{72} \))
- \( 96 \) (from \( \frac{96}{72} \))

Thus, the order from smallest to largest is:
1. \( -\frac{7}{8} \)
2. \( \frac{2}{9} \)
3. \( \frac{5}{12} \)
4. \( \frac{4}{3} \)

### Final Answer
The fractions in ascending order are:
\[ -\frac{7}{8}, \frac{2}{9}, \frac{5}{12}, \frac{4}{3} \]

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