Arrange `-(5)/(9), (7)/(12), -(2)/(3)` and `(11)/(18)` in the ascending order of their magnitudes. Also, find the difference between the largest and the smallest of these rational numbers. Express this difference as a decimal fraction correct to one decimal place.

Answer

To solve the problem of arranging the rational numbers \(-\frac{5}{9}\), \(\frac{7}{12}\), \(-\frac{2}{3}\), and \(\frac{11}{18}\) in ascending order of their magnitudes and finding the difference between the largest and smallest, we will follow these steps:

### Step 1: Identify the Rational Numbers
The given rational numbers are:
- \(-\frac{5}{9}\)
- \(\frac{7}{12}\)
- \(-\frac{2}{3}\)
- \(\frac{11}{18}\)

### Step 2: Find the LCM of the Denominators
The denominators are \(9\), \(12\), \(3\), and \(18\). We need to find the least common multiple (LCM) of these numbers.

- The prime factorization of the denominators:
- \(9 = 3^2\)
- \(12 = 2^2 \times 3\)
- \(3 = 3^1\)
- \(18 = 2^1 \times 3^2\)

The LCM is obtained by taking the highest power of each prime:
- For \(2\), the highest power is \(2^2\) (from \(12\)).
- For \(3\), the highest power is \(3^2\) (from \(9\) and \(18\)).

Thus, the LCM is:
\[ LCM = 2^2 \times 3^2 = 4 \times 9 = 36 \]

### Step 3: Convert Each Rational Number to Have a Common Denominator
Now we convert each rational number to have a denominator of \(36\):

1. \(-\frac{5}{9} = -\frac{5 \times 4}{9 \times 4} = -\frac{20}{36}\)
2. \(\frac{7}{12} = \frac{7 \times 3}{12 \times 3} = \frac{21}{36}\)
3. \(-\frac{2}{3} = -\frac{2 \times 12}{3 \times 12} = -\frac{24}{36}\)
4. \(\frac{11}{18} = \frac{11 \times 2}{18 \times 2} = \frac{22}{36}\)

### Step 4: List the Converted Rational Numbers
Now we have:
- \(-\frac{24}{36}\)
- \(-\frac{20}{36}\)
- \(\frac{21}{36}\)
- \(\frac{22}{36}\)

### Step 5: Arrange in Ascending Order of Magnitudes
To arrange these in ascending order, we compare the numerators (since the denominators are the same):

- \(-\frac{24}{36}\) (smallest)
- \(-\frac{20}{36}\)
- \(\frac{21}{36}\)
- \(\frac{22}{36}\) (largest)

Thus, the ascending order of the rational numbers based on their magnitudes is:
\[ -\frac{2}{3}, -\frac{5}{9}, \frac{7}{12}, \frac{11}{18} \]

### Step 6: Find the Difference Between the Largest and Smallest
The largest rational number is \(\frac{11}{18}\) and the smallest is \(-\frac{2}{3}\).

To find the difference:
\[ \text{Difference} = \frac{11}{18} - \left(-\frac{2}{3}\right) = \frac{11}{18} + \frac{2}{3} \]

Now, convert \(\frac{2}{3}\) to have a denominator of \(18\):
\[ \frac{2}{3} = \frac{2 \times 6}{3 \times 6} = \frac{12}{18} \]

So now we can add:
\[ \text{Difference} = \frac{11}{18} + \frac{12}{18} = \frac{23}{18} \]

### Step 7: Convert the Difference to Decimal
To express \(\frac{23}{18}\) as a decimal:
\[ \frac{23}{18} = 1.2777... \approx 1.3 \text{ (to one decimal place)} \]

### Final Answer
The ascending order of the rational numbers is:
\[ -\frac{2}{3}, -\frac{5}{9}, \frac{7}{12}, \frac{11}{18} \]
The difference between the largest and smallest rational numbers is:
\[ 1.3 \]