Arrange the rational numbers `(-3)/(7), (5)/(-14) , (-7)/(12)` in ascending order.

Answer

To arrange the rational numbers \(-\frac{3}{7}\), \(\frac{5}{-14}\), and \(-\frac{7}{12}\) in ascending order, we will follow these steps:

### Step 1: Identify the Rational Numbers
The given rational numbers are:
1. \(-\frac{3}{7}\)
2. \(\frac{5}{-14}\) (which is the same as \(-\frac{5}{14}\))
3. \(-\frac{7}{12}\)

### Step 2: Convert to a Common Denominator
To compare these fractions, we need to convert them to a common denominator. The denominators are 7, 14, and 12.

**Finding the Least Common Multiple (LCM):**
- The LCM of 7, 14, and 12 can be calculated:
- The prime factorization of 7 is \(7^1\)
- The prime factorization of 14 is \(2^1 \times 7^1\)
- The prime factorization of 12 is \(2^2 \times 3^1\)

The LCM is obtained by taking the highest power of each prime number:
- \(2^2\) from 12
- \(3^1\) from 12
- \(7^1\) from 14

Thus, \(LCM = 2^2 \times 3^1 \times 7^1 = 4 \times 3 \times 7 = 84\).

### Step 3: Rewrite Each Fraction with the Common Denominator
Now we will convert each fraction to have a denominator of 84.

1. For \(-\frac{3}{7}\):
\[ -\frac{3}{7} = -\frac{3 \times 12}{7 \times 12} = -\frac{36}{84} \]

2. For \(-\frac{5}{14}\):
\[ -\frac{5}{14} = -\frac{5 \times 6}{14 \times 6} = -\frac{30}{84} \]

3. For \(-\frac{7}{12}\):
\[ -\frac{7}{12} = -\frac{7 \times 7}{12 \times 7} = -\frac{49}{84} \]

### Step 4: Compare the Numerators
Now we have:
1. \(-\frac{36}{84}\)
2. \(-\frac{30}{84}\)
3. \(-\frac{49}{84}\)

To arrange these in ascending order, we compare the numerators:
- \(-49\) (smallest)
- \(-36\) (middle)
- \(-30\) (largest)

### Step 5: Write in Ascending Order
Thus, the order from smallest to largest is:
\[ -\frac{49}{84}, -\frac{36}{84}, -\frac{30}{84} \]
Converting back to the original fractions, we have:
\[ -\frac{7}{12}, -\frac{3}{7}, \frac{5}{-14} \]

### Final Answer
The ascending order of the rational numbers is:
\[ -\frac{7}{12}, -\frac{3}{7}, \frac{5}{-14} \]

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