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

Answer

To solve the expression \[6 \frac{1}{4} \div \left\{2 \frac{1}{4} - \frac{1}{2}\left(2 \frac{1}{2} - \bar{\left(1 \frac{1}{4} - \frac{1}{6}\right)}\right)\right\}\], we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Solve the innermost bar**:
\[ 1 \frac{1}{4} - \frac{1}{6} \]
Convert \(1 \frac{1}{4}\) to an improper fraction:
\[ 1 \frac{1}{4} = \frac{5}{4} \]
Now, find a common denominator to subtract:
\[ \frac{5}{4} - \frac{1}{6} = \frac{5 \times 3}{4 \times 3} - \frac{1 \times 2}{6 \times 2} = \frac{15}{12} - \frac{2}{12} = \frac{13}{12} \]

2. **Substitute back into the expression**:
\[ 2 \frac{1}{2} - \frac{1}{2}\left(2 \frac{1}{2} - \frac{13}{12}\right) \]
Convert \(2 \frac{1}{2}\) to an improper fraction:
\[ 2 \frac{1}{2} = \frac{5}{2} \]
Substitute:
\[ \frac{5}{2} - \frac{1}{2}\left(\frac{5}{2} - \frac{13}{12}\right) \]

3. **Calculate the expression inside the parentheses**:
Find a common denominator:
\[ \frac{5}{2} = \frac{30}{12} \quad \text{(convert to 12 as the common denominator)} \]
Now, perform the subtraction:
\[ \frac{30}{12} - \frac{13}{12} = \frac{17}{12} \]

4. **Multiply by \(\frac{1}{2}\)**:
\[ \frac{1}{2} \times \frac{17}{12} = \frac{17}{24} \]

5. **Substitute back into the expression**:
\[ \frac{5}{2} - \frac{17}{24} \]
Convert \(\frac{5}{2}\) to a fraction with a denominator of 24:
\[ \frac{5}{2} = \frac{60}{24} \]
Now perform the subtraction:
\[ \frac{60}{24} - \frac{17}{24} = \frac{43}{24} \]

6. **Now substitute back into the original expression**:
\[ 6 \frac{1}{4} \div \frac{43}{24} \]
Convert \(6 \frac{1}{4}\) to an improper fraction:
\[ 6 \frac{1}{4} = \frac{25}{4} \]

7. **Perform the division**:
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{25}{4} \div \frac{43}{24} = \frac{25}{4} \times \frac{24}{43} = \frac{25 \times 24}{4 \times 43} \]
Simplify:
\[ = \frac{600}{172} = \frac{150}{43} \]

8. **Final Result**:
The final value of the expression is:
\[ \frac{150}{43} \approx 3.49 \quad \text{(or as a mixed number, } 3 \frac{31}{43}\text{)} \]