To simplify the expression \(3 \frac{2}{9} + 2 \frac{1}{3} + 2 \frac{7}{12}\), we will follow these steps:
### Step 1: Convert Mixed Numbers to Improper Fractions
1. **Convert \(3 \frac{2}{9}\)**:
\[
3 \frac{2}{9} = \frac{(3 \times 9) + 2}{9} = \frac{27 + 2}{9} = \frac{29}{9}
\]
2. **Convert \(2 \frac{1}{3}\)**:
\[
2 \frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}
\]
3. **Convert \(2 \frac{7}{12}\)**:
\[
2 \frac{7}{12} = \frac{(2 \times 12) + 7}{12} = \frac{24 + 7}{12} = \frac{31}{12}
\]
### Step 2: Find the Least Common Multiple (LCM) of the Denominators
The denominators are \(9\), \(3\), and \(12\). We need to find the LCM of these numbers.
- The multiples of \(9\) are \(9, 18, 27, 36, \ldots\)
- The multiples of \(3\) are \(3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, \ldots\)
- The multiples of \(12\) are \(12, 24, 36, \ldots\)
The LCM of \(9\), \(3\), and \(12\) is \(36\).
### Step 3: Convert Each Fraction to Have a Common Denominator
Now we will convert each fraction to have a denominator of \(36\).
1. **Convert \(\frac{29}{9}\)**:
\[
\frac{29}{9} = \frac{29 \times 4}{9 \times 4} = \frac{116}{36}
\]
2. **Convert \(\frac{7}{3}\)**:
\[
\frac{7}{3} = \frac{7 \times 12}{3 \times 12} = \frac{84}{36}
\]
3. **Convert \(\frac{31}{12}\)**:
\[
\frac{31}{12} = \frac{31 \times 3}{12 \times 3} = \frac{93}{36}
\]
### Step 4: Add the Fractions
Now we can add the fractions:
\[
\frac{116}{36} + \frac{84}{36} + \frac{93}{36} = \frac{116 + 84 + 93}{36} = \frac{293}{36}
\]
### Step 5: Convert the Improper Fraction to a Mixed Number
To convert \(\frac{293}{36}\) to a mixed number:
1. Divide \(293\) by \(36\):
- \(36\) goes into \(293\) \(8\) times (since \(36 \times 8 = 288\)).
- The remainder is \(293 - 288 = 5\).
Thus, we can express this as:
\[
\frac{293}{36} = 8 \frac{5}{36}
\]
### Final Answer
The simplified form of \(3 \frac{2}{9} + 2 \frac{1}{3} + 2 \frac{7}{12}\) is:
\[
8 \frac{5}{36}
\]
