The value of $6\frac{2}{3}\div 2\frac{1}{2} imes 3\frac{3}{4} - 5\frac{1}{2} imes 4\frac{1}{4}+1\frac{2}{3}(\frac{7}{8}+\frac{3}{4} imes\frac{2}{3}$ is

Q: The value of $6\frac{2}{3}\div 2\frac{1}{2} \times 3\frac{3}{4} - 5\frac{1}{2} \times 4\frac{1}{4}+1\frac{2}{3}(\frac{7}{8}+\frac{3}{4}\times\frac{2}{3}$ is

The value of $6\frac{2}{3}\div 2\frac{1}{2} \times 3\frac{3}{4} - 5\frac{1}{2} \times 4\frac{1}{4}+1\frac{2}{3}(\frac{7}{8}+\frac{3}{4}\times\frac{2}{3}$ is

Question

The value of $6\frac{2}{3}\div 2\frac{1}{2} 	imes 3\frac{3}{4} - 5\frac{1}{2} 	imes 4\frac{1}{4}+1\frac{2}{3}(\frac{7}{8}+\frac{3}{4}	imes\frac{2}{3}$ is

Accepted Answer

To solve the expression $6\frac{2}{3}\div 2\frac{1}{2} \times 3\frac{3}{4} - 5\frac{1}{2} \times 4\frac{1}{4}+1\frac{2}{3}(\frac{7}{8}+\frac{3}{4}\times\frac{2}{3})$, we need to follow the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

1. **Convert Mixed Numbers to Improper Fractions:**

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

2. **Evaluate the Expression Inside the Parentheses:**

- $\frac{3}{4} \times \frac{2}{3} = \frac{6}{12} = \frac{1}{2}$
   - $\frac{7}{8} + \frac{1}{2} = \frac{7}{8} + \frac{4}{8} = \frac{11}{8}$

3. **Substitute Back into the Expression:**

$$
   \frac{20}{3} \div \frac{5}{2} \times \frac{15}{4} - \frac{11}{2} \times \frac{17}{4} + \frac{5}{3} \times \frac{11}{8}
   $$

4. **Perform Division and Multiplication:**

- $\frac{20}{3} \div \frac{5}{2} = \frac{20}{3} \times \frac{2}{5} = \frac{40}{15} = \frac{8}{3}$
   - $\frac{8}{3} \times \frac{15}{4} = \frac{120}{12} = 10$
   - $\frac{11}{2} \times \frac{17}{4} = \frac{187}{8}$
   - $\frac{5}{3} \times \frac{11}{8} = \frac{55}{24}$

5. **Combine All Parts:**

$$
   10 - \frac{187}{8} + \frac{55}{24}
   $$

6. **Convert to a Common Denominator:**

- Convert 10 to a fraction with a denominator of 24: $10 = \frac{240}{24}$
   - Convert $\frac{187}{8}$ to a denominator of 24: $\frac{187}{8} = \frac{561}{24}$

7. **Perform the Final Subtraction and Addition:**

$$
   \frac{240}{24} - \frac{561}{24} + \frac{55}{24} = \frac{240 - 561 + 55}{24} = \frac{-266}{24} = -11\frac{1}{12}
   $$

Thus, the correct answer is $-11\frac{1}{12}$, which corresponds to Option 1.

**Explanation of Incorrect Options:**

- **Option 2: $11\frac{1}{12}$** is incorrect because the calculations clearly show a negative result, not a positive one.
- **Option 3: $6\frac{1}{2}$** is incorrect because the calculations result in a value significantly different from this positive number.
- **Option 4: $-6\frac{1}{2}$** is incorrect because the calculations show a more negative result than this option suggests.

Therefore, the correct answer is indeed Option 1: $-11\frac{1}{12}$.