Simplify `[3 (1)/(4) div {1 (1)/(4) - (1)/(2) (2 (1)/(2) - bar((1)/(4) - (1)/(6)))}] div ((1)/(2)"of 4" (1)/(3))`

Options

A. 18
B. 36
C. 39
D. 78

Answer

To simplify the expression

\[ \left[ 3 \frac{1}{4} \div \left\{ 1 \frac{1}{4} - \frac{1}{2} \left( 2 \frac{1}{2} - \bar{\left( \frac{1}{4} - \frac{1}{6} \right)} \right) \right\} \right] \div \left( \frac{1}{2} \text{ of } 4 \frac{1}{3} \right) \]

we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions

Convert all mixed numbers to improper fractions:

- \( 3 \frac{1}{4} = \frac{13}{4} \)
- \( 1 \frac{1}{4} = \frac{5}{4} \)
- \( 2 \frac{1}{2} = \frac{5}{2} \)
- \( 4 \frac{1}{3} = \frac{13}{3} \)

Now the expression becomes:

\[ \left[ \frac{13}{4} \div \left\{ \frac{5}{4} - \frac{1}{2} \left( \frac{5}{2} - \bar{\left( \frac{1}{4} - \frac{1}{6} \right)} \right) \right\} \right] \div \left( \frac{1}{2} \text{ of } \frac{13}{3} \right) \]

### Step 2: Solve the Inner Bracket

First, simplify \( \frac{1}{4} - \frac{1}{6} \):

To find the LCM of 4 and 6, we get 12. Thus,

\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12} \implies \frac{1}{4} - \frac{1}{6} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12} \]

Now, substitute back into the expression:

\[ \left[ \frac{13}{4} \div \left\{ \frac{5}{4} - \frac{1}{2} \left( \frac{5}{2} - \frac{1}{12} \right) \right\} \right] \div \left( \frac{1}{2} \text{ of } \frac{13}{3} \right) \]

### Step 3: Simplify \( \frac{5}{2} - \frac{1}{12} \)

Finding the LCM of 2 and 12 gives us 12:

\[ \frac{5}{2} = \frac{30}{12} \implies \frac{5}{2} - \frac{1}{12} = \frac{30}{12} - \frac{1}{12} = \frac{29}{12} \]

### Step 4: Substitute Back

Now substitute back into the expression:

\[ \left[ \frac{13}{4} \div \left\{ \frac{5}{4} - \frac{1}{2} \cdot \frac{29}{12} \right\} \right] \div \left( \frac{1}{2} \cdot \frac{13}{3} \right) \]

### Step 5: Calculate \( \frac{1}{2} \cdot \frac{29}{12} \)

\[ \frac{1}{2} \cdot \frac{29}{12} = \frac{29}{24} \]

### Step 6: Solve \( \frac{5}{4} - \frac{29}{24} \)

Finding the LCM of 4 and 24 gives us 24:

\[ \frac{5}{4} = \frac{30}{24} \implies \frac{5}{4} - \frac{29}{24} = \frac{30}{24} - \frac{29}{24} = \frac{1}{24} \]

### Step 7: Substitute Back Again

Now substitute back into the expression:

\[ \left[ \frac{13}{4} \div \frac{1}{24} \right] \div \left( \frac{1}{2} \cdot \frac{13}{3} \right) \]

### Step 8: Calculate \( \frac{13}{4} \div \frac{1}{24} \)

Dividing by a fraction is the same as multiplying by its reciprocal:

\[ \frac{13}{4} \div \frac{1}{24} = \frac{13}{4} \cdot 24 = \frac{13 \cdot 24}{4} = \frac{312}{4} = 78 \]

### Step 9: Calculate \( \frac{1}{2} \cdot \frac{13}{3} \)

\[ \frac{1}{2} \cdot \frac{13}{3} = \frac{13}{6} \]

### Step 10: Final Division

Now we have:

\[ 78 \div \frac{13}{6} = 78 \cdot \frac{6}{13} = \frac{468}{13} = 36 \]

### Final Answer

The simplified result is:

\[ \boxed{36} \]