`2 (5)/(4) - 4 (7)/(6) + 3 (1)/(3)`

Answer

To solve the expression \(2 \frac{5}{4} - 4 \frac{7}{6} + 3 \frac{1}{3}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert each mixed number into an improper fraction.

1. **For \(2 \frac{5}{4}\)**:
\[ 2 \frac{5}{4} = \frac{2 \times 4 + 5}{4} = \frac{8 + 5}{4} = \frac{13}{4} \]

2. **For \(4 \frac{7}{6}\)**:
\[ 4 \frac{7}{6} = \frac{4 \times 6 + 7}{6} = \frac{24 + 7}{6} = \frac{31}{6} \]

3. **For \(3 \frac{1}{3}\)**:
\[ 3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{9 + 1}{3} = \frac{10}{3} \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression using the improper fractions:
\[ \frac{13}{4} - \frac{31}{6} + \frac{10}{3} \]

### Step 3: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators \(4\), \(6\), and \(3\).

- The LCM of \(4\), \(6\), and \(3\) is \(12\).

### Step 4: Convert Each Fraction to Have the Same Denominator
Now we will convert each fraction to have a denominator of \(12\).

1. **For \(\frac{13}{4}\)**:
\[ \frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12} \]

2. **For \(\frac{31}{6}\)**:
\[ \frac{31}{6} = \frac{31 \times 2}{6 \times 2} = \frac{62}{12} \]

3. **For \(\frac{10}{3}\)**:
\[ \frac{10}{3} = \frac{10 \times 4}{3 \times 4} = \frac{40}{12} \]

### Step 5: Combine the Fractions
Now we can combine the fractions:
\[ \frac{39}{12} - \frac{62}{12} + \frac{40}{12} = \frac{39 - 62 + 40}{12} \]

### Step 6: Simplify the Numerator
Now simplify the numerator:
\[ 39 - 62 + 40 = 39 - 62 = -23 \quad \text{and then} \quad -23 + 40 = 17 \]

So we have:
\[ \frac{17}{12} \]

### Final Answer
Thus, the simplified result of the expression \(2 \frac{5}{4} - 4 \frac{7}{6} + 3 \frac{1}{3}\) is:
\[ \frac{17}{12} \]

---