The value of
`6 2/3div 2 1/2xx3 3/4-5 1/2xx 4 1/4+1 2/3(7/8+3/4xx2/3)` is:

Options

A. `-11 1/12`
B. `11 1/12`
C. `6 1/2`
D. `-6 1/2`

Answer

To solve the expression \( 6 \frac{2}{3} \div 2 \frac{1}{2} \times 3 \frac{3}{4} - 5 \frac{1}{2} \times 4 \frac{1}{4} + 1 \frac{2}{3} \left( \frac{7}{8} + 3 \frac{4}{3} \times \frac{2}{3} \right) \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert all mixed numbers to improper fractions.

- \( 6 \frac{2}{3} = \frac{20}{3} \)
- \( 2 \frac{1}{2} = \frac{5}{2} \)
- \( 3 \frac{3}{4} = \frac{15}{4} \)
- \( 5 \frac{1}{2} = \frac{11}{2} \)
- \( 4 \frac{1}{4} = \frac{17}{4} \)
- \( 1 \frac{2}{3} = \frac{5}{3} \)

Now, rewrite the expression:
\[ \frac{20}{3} \div \frac{5}{2} \times \frac{15}{4} - \frac{11}{2} \times \frac{17}{4} + \frac{5}{3} \left( \frac{7}{8} + 3 \frac{4}{3} \times \frac{2}{3} \right) \]

### Step 2: Perform Division
Calculate \( \frac{20}{3} \div \frac{5}{2} \):
\[ \frac{20}{3} \times \frac{2}{5} = \frac{20 \times 2}{3 \times 5} = \frac{40}{15} = \frac{8}{3} \]

### Step 3: Multiply
Now multiply \( \frac{8}{3} \) by \( \frac{15}{4} \):
\[ \frac{8}{3} \times \frac{15}{4} = \frac{8 \times 15}{3 \times 4} = \frac{120}{12} = 10 \]

### Step 4: Calculate the Second Term
Now calculate \( \frac{11}{2} \times \frac{17}{4} \):
\[ \frac{11 \times 17}{2 \times 4} = \frac{187}{8} \]

### Step 5: Calculate the Third Term
Now calculate \( \frac{5}{3} \left( \frac{7}{8} + 3 \frac{4}{3} \times \frac{2}{3} \right) \):
First, calculate \( 3 \frac{4}{3} \):
\[ 3 \frac{4}{3} = \frac{13}{3} \]
Now multiply:
\[ \frac{13}{3} \times \frac{2}{3} = \frac{26}{9} \]
Now add \( \frac{7}{8} + \frac{26}{9} \):
To add these fractions, find a common denominator (72):
\[ \frac{7}{8} = \frac{63}{72}, \quad \frac{26}{9} = \frac{208}{72} \]
So,
\[ \frac{63}{72} + \frac{208}{72} = \frac{271}{72} \]
Now multiply by \( \frac{5}{3} \):
\[ \frac{5}{3} \times \frac{271}{72} = \frac{1355}{216} \]

### Step 6: Combine All Parts
Now combine all parts:
\[ 10 - \frac{187}{8} + \frac{1355}{216} \]
Convert 10 to a fraction with a common denominator of 216:
\[ 10 = \frac{2160}{216} \]
Convert \( \frac{187}{8} \) to a fraction with a denominator of 216:
\[ \frac{187}{8} = \frac{187 \times 27}{216} = \frac{5049}{216} \]
Now combine:
\[ \frac{2160}{216} - \frac{5049}{216} + \frac{1355}{216} = \frac{2160 - 5049 + 1355}{216} = \frac{-1534}{216} \]

### Step 7: Simplify
Now simplify \( \frac{-1534}{216} \):
Divide both by 2:
\[ \frac{-767}{108} \]

### Final Answer
The final value is:
\[ -\frac{767}{108} \]