Find the value of : $\left[ 3\frac{1}{4} \div \left\{ 1\frac{1}{4} - \frac{1}{2} \left( 2\frac{1}{2} - \frac{1}{4} - \frac{1}{6} \right) \right\} \right]$

Question

Accepted Answer

To solve the given expression $\left[ 3\frac{1}{4} \div \left\{ 1\frac{1}{4} - \frac{1}{2} \left( 2\frac{1}{2} - \frac{1}{4} - \frac{1}{6} \right) \right\} \right]$, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

1. **Simplify the innermost parentheses:**
   $$
   2\frac{1}{2} - \frac{1}{4} - \frac{1}{6}
   $$
   Convert mixed numbers and fractions to improper fractions:
   $$
   2\frac{1}{2} = \frac{5}{2}, \quad \frac{1}{4} = \frac{1}{4}, \quad \frac{1}{6} = \frac{1}{6}
   $$
   Find a common denominator for $\frac{5}{2}$, $\frac{1}{4}$, and $\frac{1}{6}$, which is 12:
   $$
   \frac{5}{2} = \frac{30}{12}, \quad \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}
   $$
   Subtract the fractions:
   $$
   \frac{30}{12} - \frac{3}{12} - \frac{2}{12} = \frac{25}{12}
   $$

2. **Continue with the expression:**
   $$
   1\frac{1}{4} - \frac{1}{2} \times \frac{25}{12}
   $$
   Convert $1\frac{1}{4}$ to an improper fraction:
   $$
   1\frac{1}{4} = \frac{5}{4}
   $$
   Calculate $\frac{1}{2} \times \frac{25}{12}$:
   $$
   \frac{1}{2} \times \frac{25}{12} = \frac{25}{24}
   $$
   Subtract from $\frac{5}{4}$:
   $$
   \frac{5}{4} = \frac{30}{24}
   $$
   $$
   \frac{30}{24} - \frac{25}{24} = \frac{5}{24}
   $$

3. **Divide $3\frac{1}{4}$ by the result:**
   Convert $3\frac{1}{4}$ to an improper fraction:
   $$
   3\frac{1}{4} = \frac{13}{4}
   $$
   Divide by $\frac{5}{24}$:
   $$
   \frac{13}{4} \div \frac{5}{24} = \frac{13}{4} \times \frac{24}{5} = \frac{312}{20} = 78
   $$

Thus, the value of the expression is 78, which corresponds to Option 2.

**Explanation of Incorrect Options:**

- **Option 1 (75):** This result might arise from a miscalculation in the division or subtraction steps.
- **Option 3 ($72\frac{3}{4}$):** This result could occur from an error in converting fractions or in the multiplication/division steps.
- **Option 4 ($75\frac{2}{3}$):** This result might be due to an incorrect simplification or arithmetic mistake in the process.

Therefore, the correct answer is indeed Option 2, which is 78.