Simplify : `((5)/(6) + (7)/(8)"of" (4)/(5) div (3)/(4) "of" (9)/(10))/(8 (1)/(3) - ((4)/(1 - (7)/(8))"of 2" (1)/(4)) div (7)/(9) "of 12") "of 6" (1)/(2) + 5(1)/(9)`

Options

A. `24 (1)/(4)`
B. `24 (3)/(4)`
C. `22 (1)/(2)`
D. `23 (1)/(3)`

Answer

To simplify the expression

\[ \frac{\left(\frac{5}{6} + \frac{7}{8} \text{ of } \frac{4}{5} \div \frac{3}{4} \text{ of } \frac{9}{10}\right)}{\left(8 \frac{1}{3} - \left(\frac{4}{1 - \frac{7}{8}} \text{ of } 2 \frac{1}{4}\right) \div \frac{7}{9} \text{ of } 12\right) \text{ of } 6 \frac{1}{2}} + 5 \frac{1}{9} \]

we will follow the order of operations (BODMAS/BIDMAS).

### Step 1: Simplify the Numerator

1. **Calculate \(\frac{7}{8} \text{ of } \frac{4}{5}\)**:
\[ \frac{7}{8} \times \frac{4}{5} = \frac{7 \times 4}{8 \times 5} = \frac{28}{40} = \frac{7}{10} \]

**Hint**: Remember that "of" means multiplication.

2. **Calculate \(\frac{3}{4} \text{ of } \frac{9}{10}\)**:
\[ \frac{3}{4} \times \frac{9}{10} = \frac{27}{40} \]

3. **Now, calculate \(\frac{7}{10} \div \frac{27}{40}\)**:
\[ \frac{7}{10} \div \frac{27}{40} = \frac{7}{10} \times \frac{40}{27} = \frac{280}{270} = \frac{28}{27} \]

4. **Now, add \(\frac{5}{6} + \frac{28}{27}\)**:
- Find LCM of 6 and 27, which is 54.
- Convert \(\frac{5}{6}\) to \(\frac{45}{54}\) and \(\frac{28}{27}\) to \(\frac{56}{54}\).
\[ \frac{45}{54} + \frac{56}{54} = \frac{101}{54} \]

### Step 2: Simplify the Denominator

1. **Convert \(8 \frac{1}{3}\) to an improper fraction**:
\[ 8 \frac{1}{3} = \frac{25}{3} \]

2. **Calculate \(1 - \frac{7}{8}\)**:
\[ 1 - \frac{7}{8} = \frac{1}{8} \]

3. **Calculate \(\frac{4}{\frac{1}{8}} \text{ of } 2 \frac{1}{4}\)**:
- Convert \(2 \frac{1}{4}\) to \(\frac{9}{4}\).
\[ \frac{4}{\frac{1}{8}} = 32 \]
\[ 32 \text{ of } \frac{9}{4} = 32 \times \frac{9}{4} = 72 \]

4. **Now calculate \(72 \div \frac{7}{9} \text{ of } 12\)**:
- Calculate \(\frac{7}{9} \text{ of } 12\):
\[ \frac{7}{9} \times 12 = \frac{84}{9} = \frac{28}{3} \]
- Now divide \(72\) by \(\frac{28}{3}\):
\[ 72 \div \frac{28}{3} = 72 \times \frac{3}{28} = \frac{216}{28} = \frac{54}{7} \]

5. **Now calculate \(\frac{25}{3} - \frac{54}{7}\)**:
- Find LCM of 3 and 7, which is 21.
- Convert \(\frac{25}{3}\) to \(\frac{175}{21}\) and \(\frac{54}{7}\) to \(\frac{162}{21}\).
\[ \frac{175}{21} - \frac{162}{21} = \frac{13}{21} \]

6. **Now multiply by \(6 \frac{1}{2}\)**:
- Convert \(6 \frac{1}{2}\) to \(\frac{13}{2}\).
\[ \frac{13}{21} \text{ of } \frac{13}{2} = \frac{169}{42} \]

### Step 3: Combine the Numerator and Denominator

1. **Now we have**:
\[ \frac{\frac{101}{54}}{\frac{169}{42}} = \frac{101}{54} \times \frac{42}{169} = \frac{4242}{9126} \]

2. **Simplifying**:
- Find GCD of 4242 and 9126, which is 42.
\[ \frac{101}{217} \]

### Step 4: Add \(5 \frac{1}{9}\) to the result

1. **Convert \(5 \frac{1}{9}\) to an improper fraction**:
\[ 5 \frac{1}{9} = \frac{46}{9} \]

2. **Find LCM of 9 and 217**:
- LCM is \(1953\).
- Convert \(\frac{101}{217}\) to \(\frac{468}{1953}\) and \(\frac{46}{9}\) to \(\frac{10406}{1953}\).
\[ \frac{468 + 10406}{1953} = \frac{10874}{1953} \]

### Final Answer

The final simplified result is:

\[ \frac{10874}{1953} \]