`4(1)/(2) xx 4(1)/(3) -8(1)/(3) -: 5(2)/(3)` =?

Options

A. `(7)/(17)`
B. `1 (33)/(34)`
C. 8
D. `18 (1)/(34)`

Answer

To solve the given expression \( 4\frac{1}{2} \times 4\frac{1}{3} - 8\frac{1}{3} \div 5\frac{2}{3} \), we will follow these steps:

### Step 1: Convert mixed numbers to improper fractions
1. **Convert \( 4\frac{1}{2} \)**:
\[ 4\frac{1}{2} = 4 \times 2 + 1 = 8 + 1 = \frac{9}{2} \]

2. **Convert \( 4\frac{1}{3} \)**:
\[ 4\frac{1}{3} = 4 \times 3 + 1 = 12 + 1 = \frac{13}{3} \]

3. **Convert \( 8\frac{1}{3} \)**:
\[ 8\frac{1}{3} = 8 \times 3 + 1 = 24 + 1 = \frac{25}{3} \]

4. **Convert \( 5\frac{2}{3} \)**:
\[ 5\frac{2}{3} = 5 \times 3 + 2 = 15 + 2 = \frac{17}{3} \]

### Step 2: Substitute the improper fractions into the expression
Now, we can rewrite the expression:
\[ \frac{9}{2} \times \frac{13}{3} - \frac{25}{3} \div \frac{17}{3} \]

### Step 3: Perform the multiplication and division
1. **Multiply \( \frac{9}{2} \) and \( \frac{13}{3} \)**:
\[ \frac{9}{2} \times \frac{13}{3} = \frac{9 \times 13}{2 \times 3} = \frac{117}{6} \]

2. **Divide \( \frac{25}{3} \) by \( \frac{17}{3} \)**:
\[ \frac{25}{3} \div \frac{17}{3} = \frac{25}{3} \times \frac{3}{17} = \frac{25 \times 3}{3 \times 17} = \frac{25}{17} \]

### Step 4: Rewrite the expression with the results
Now, we have:
\[ \frac{117}{6} - \frac{25}{17} \]

### Step 5: Find a common denominator and subtract
1. **Find the LCM of 6 and 17**: The LCM is \( 102 \).
2. **Convert each fraction**:
- For \( \frac{117}{6} \):
\[ \frac{117}{6} = \frac{117 \times 17}{6 \times 17} = \frac{1989}{102} \]
- For \( \frac{25}{17} \):
\[ \frac{25}{17} = \frac{25 \times 6}{17 \times 6} = \frac{150}{102} \]

3. **Subtract the fractions**:
\[ \frac{1989}{102} - \frac{150}{102} = \frac{1989 - 150}{102} = \frac{1839}{102} \]

### Step 6: Simplify the result
1. **Simplify \( \frac{1839}{102} \)**:
- Divide both the numerator and denominator by 3:
\[ \frac{1839 \div 3}{102 \div 3} = \frac{613}{34} \]

### Step 7: Convert back to a mixed number
1. **Convert \( \frac{613}{34} \)** to a mixed number:
- Divide \( 613 \) by \( 34 \):
\[ 613 \div 34 = 18 \quad \text{(whole number)} \]
- Remainder:
\[ 613 - (34 \times 18) = 613 - 612 = 1 \]
- So, \( \frac{613}{34} = 18\frac{1}{34} \).

### Final Answer
Thus, the final answer is:
\[ \boxed{18\frac{1}{34}} \]