`4(1)/(2)xx4(1)/(3)-8(1)/(3)div5(2)/(3)=?`

Options

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

Explanation

To solve the 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
First, we convert all mixed numbers into improper fractions.

1. \( 4\frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2} \)
2. \( 4\frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3} \)
3. \( 8\frac{1}{3} = \frac{8 \times 3 + 1}{3} = \frac{24 + 1}{3} = \frac{25}{3} \)
4. \( 5\frac{2}{3} = \frac{5 \times 3 + 2}{3} = \frac{15 + 2}{3} = \frac{17}{3} \)

Now, we can rewrite the expression as:
\[ \frac{9}{2} \times \frac{13}{3} - \frac{25}{3} \div \frac{17}{3} \]

### Step 2: Perform the Division
Next, we handle the division:
\[ \frac{25}{3} \div \frac{17}{3} = \frac{25}{3} \times \frac{3}{17} = \frac{25 \times 3}{3 \times 17} = \frac{25}{17} \]

### Step 3: Rewrite the Expression
Now, substitute back into the expression:
\[ \frac{9}{2} \times \frac{13}{3} - \frac{25}{17} \]

### Step 4: Perform the Multiplication
Now, we calculate the multiplication:
\[ \frac{9}{2} \times \frac{13}{3} = \frac{9 \times 13}{2 \times 3} = \frac{117}{6} \]

### Step 5: Find a Common Denominator
To subtract \( \frac{25}{17} \) from \( \frac{117}{6} \), we need a common denominator. The least common multiple of 6 and 17 is 102.

Convert both fractions:
1. \( \frac{117}{6} = \frac{117 \times 17}{6 \times 17} = \frac{1989}{102} \)
2. \( \frac{25}{17} = \frac{25 \times 6}{17 \times 6} = \frac{150}{102} \)

### Step 6: Perform the Subtraction
Now we can perform the subtraction:
\[ \frac{1989}{102} - \frac{150}{102} = \frac{1989 - 150}{102} = \frac{1839}{102} \]

### Step 7: Simplify the Result
Now we simplify \( \frac{1839}{102} \):
1. Find the GCD of 1839 and 102. The GCD is 3.
2. Divide both the numerator and denominator by 3:
\[ \frac{1839 \div 3}{102 \div 3} = \frac{613}{34} \]

### Final Answer
Thus, the final answer is:
\[ \frac{613}{34} \]