Find the sum `(-4/9 + (-5)/12 + 1/18)`

Answer

To find the sum of the rational numbers \(-\frac{4}{9} + \left(-\frac{5}{12}\right) + \frac{1}{18}\), we will follow these steps:

### Step 1: Identify the rational numbers
We have three rational numbers:
- \(-\frac{4}{9}\)
- \(-\frac{5}{12}\)
- \(\frac{1}{18}\)

### Step 2: Find the Least Common Multiple (LCM) of the denominators
The denominators are 9, 12, and 18. We need to find the LCM of these numbers.

- **Prime factorization**:
- \(9 = 3^2\)
- \(12 = 2^2 \times 3\)
- \(18 = 2 \times 3^2\)

- **Taking the highest powers of each prime**:
- For 2: highest power is \(2^2\) from 12
- For 3: highest power is \(3^2\) from 9 or 18

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

### Step 3: Convert each fraction to have the common denominator
Now we convert each fraction to have a denominator of 36.

1. For \(-\frac{4}{9}\):
\[ -\frac{4}{9} = -\frac{4 \times 4}{9 \times 4} = -\frac{16}{36} \]

2. For \(-\frac{5}{12}\):
\[ -\frac{5}{12} = -\frac{5 \times 3}{12 \times 3} = -\frac{15}{36} \]

3. For \(\frac{1}{18}\):
\[ \frac{1}{18} = \frac{1 \times 2}{18 \times 2} = \frac{2}{36} \]

### Step 4: Add the fractions
Now we can add the fractions:
\[ -\frac{16}{36} + -\frac{15}{36} + \frac{2}{36} = \frac{-16 - 15 + 2}{36} \]
Calculating the numerator:
\[ -16 - 15 + 2 = -31 + 2 = -29 \]
So, we have:
\[ \frac{-29}{36} \]

### Final Answer
The sum of the rational numbers is:
\[ -\frac{29}{36} \]

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