Find the value of `[3(1)/(4)div{1(1)/(4)-(1)/(2)(2(1)/(2)-overline((1)/(4)-(1)/(6)))}]`

Options

A. 75
B. 78
C. `72(3)/(4)`
D. `75(2)/(3)`

Answer

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

### Step 1: Convert Mixed Numbers to Improper Fractions
Convert all mixed numbers into improper fractions:
- \(3 \frac{1}{4} = \frac{13}{4}\)
- \(1 \frac{1}{4} = \frac{5}{4}\)
- \(2 \frac{1}{2} = \frac{5}{2}\)

### Step 2: Simplify the Inner Expression
Now, we need to simplify the expression inside the denominator:
\[ \frac{5}{4} - \frac{1}{2} \left(\frac{5}{2} - \left(\frac{1}{4} - \frac{1}{6}\right)\right) \]

### Step 3: Calculate \(\frac{1}{4} - \frac{1}{6}\)
To subtract \(\frac{1}{4}\) and \(\frac{1}{6}\), we need a common denominator. The least common multiple of 4 and 6 is 12:
\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12} \]
Thus,
\[ \frac{1}{4} - \frac{1}{6} = \frac{3}{12} - \frac{2}{12} = \frac{1}{12} \]

### Step 4: Substitute Back into the Expression
Now substitute back into the expression:
\[ \frac{5}{4} - \frac{1}{2} \left(\frac{5}{2} - \frac{1}{12}\right) \]

### Step 5: Calculate \(\frac{5}{2} - \frac{1}{12}\)
Again, we need a common denominator. The least common multiple of 2 and 12 is 12:
\[ \frac{5}{2} = \frac{30}{12} \]
Thus,
\[ \frac{5}{2} - \frac{1}{12} = \frac{30}{12} - \frac{1}{12} = \frac{29}{12} \]

### Step 6: Substitute and Simplify
Now substitute this back:
\[ \frac{5}{4} - \frac{1}{2} \cdot \frac{29}{12} \]
Calculating \(\frac{1}{2} \cdot \frac{29}{12} = \frac{29}{24}\):
\[ \frac{5}{4} - \frac{29}{24} \]

### Step 7: Find a Common Denominator
The least common multiple of 4 and 24 is 24:
\[ \frac{5}{4} = \frac{30}{24} \]
Thus,
\[ \frac{30}{24} - \frac{29}{24} = \frac{1}{24} \]

### Step 8: Final Division
Now we have:
\[ \frac{13}{4} \div \frac{1}{24} \]
This is the same as multiplying by the reciprocal:
\[ \frac{13}{4} \cdot 24 = \frac{13 \cdot 24}{4} = \frac{312}{4} = 78 \]

### Final Answer
The value of the expression is \(78\).

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