Simplify: `(2 3/4)/( 1 5/6)-:7/8xx(1/3+1/4)+5/7-:3/4` of `3/7`

Options

A. `56/77`
B. `49/80`
C. `2/3`
D. `3 2/9`

Answer

To simplify the expression `(2 3/4)/(1 5/6) - (7/8) * (1/3 + 1/4) + (5/7) - (3/4) * (3/7)`, we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions**:
- Convert `2 3/4` to an improper fraction:
\[ 2 \times 4 + 3 = 8 + 3 = 11 \Rightarrow \frac{11}{4} \]
- Convert `1 5/6` to an improper fraction:
\[ 1 \times 6 + 5 = 6 + 5 = 11 \Rightarrow \frac{11}{6} \]

2. **Rewrite the Expression**:
The expression now looks like:
\[ \frac{11/4}{11/6} - \frac{7}{8} \times \left(\frac{1}{3} + \frac{1}{4}\right) + \frac{5}{7} - \frac{3}{4} \times \frac{3}{7} \]

3. **Simplify the Division of Fractions**:
\[ \frac{11/4}{11/6} = \frac{11}{4} \times \frac{6}{11} = \frac{6}{4} = \frac{3}{2} \]

4. **Calculate the Sum Inside the Parentheses**:
\[ \frac{1}{3} + \frac{1}{4} = \frac{4 + 3}{12} = \frac{7}{12} \]

5. **Multiply by \(\frac{7}{8}\)**:
\[ \frac{7}{8} \times \frac{7}{12} = \frac{49}{96} \]

6. **Calculate the Product \(\frac{3}{4} \times \frac{3}{7}\)**:
\[ \frac{3}{4} \times \frac{3}{7} = \frac{9}{28} \]

7. **Rewrite the Expression**:
The expression now looks like:
\[ \frac{3}{2} - \frac{49}{96} + \frac{5}{7} - \frac{9}{28} \]

8. **Convert \(\frac{3}{2}\) to a Common Denominator**:
The common denominator for \(2, 96, 7, 28\) is \(336\):
\[ \frac{3}{2} = \frac{3 \times 168}{2 \times 168} = \frac{504}{336} \]
\[ \frac{49}{96} = \frac{49 \times 7}{96 \times 7} = \frac{343}{336} \]
\[ \frac{5}{7} = \frac{5 \times 48}{7 \times 48} = \frac{240}{336} \]
\[ \frac{9}{28} = \frac{9 \times 12}{28 \times 12} = \frac{108}{336} \]

9. **Combine All Fractions**:
\[ \frac{504}{336} - \frac{343}{336} + \frac{240}{336} - \frac{108}{336} \]
Combine the numerators:
\[ 504 - 343 + 240 - 108 = 293 \]
So we have:
\[ \frac{293}{336} \]

10. **Final Answer**:
The simplified expression is:
\[ \frac{293}{336} \]