The value of `[2/3 div8/12" of "2/5+2/5" of 7"1/2 div 3/4]" of "4/19` is :

Options

A. 1
B. `1^(1/4)`
C. `1^(1/2)`
D. `26/19`

Answer

To solve the expression \(\left[\frac{2}{3} \div \frac{8}{12} \text{ of } \frac{2}{5} + \frac{2}{5} \text{ of } 7\frac{1}{2} \div \frac{3}{4}\right] \text{ of } \frac{4}{19}\), we will follow the BODMAS rule (Brackets, Off, Division, Multiplication, Addition, Subtraction).

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions:**
- \(7\frac{1}{2} = \frac{15}{2}\) (since \(7 \times 2 + 1 = 15\))

2. **Rewrite the Expression:**
- The expression now looks like:
\[ \left[\frac{2}{3} \div \frac{8}{12} \text{ of } \frac{2}{5} + \frac{2}{5} \text{ of } \frac{15}{2} \div \frac{3}{4}\right] \text{ of } \frac{4}{19} \]

3. **Calculate \(\frac{2}{3} \div \frac{8}{12}\):**
- Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{2}{3} \div \frac{8}{12} = \frac{2}{3} \times \frac{12}{8} = \frac{2 \times 12}{3 \times 8} = \frac{24}{24} = 1 \]

4. **Calculate \(\frac{2}{5} \text{ of } 1\):**
- Since \(1\) multiplied by anything is itself:
\[ 1 \text{ of } \frac{2}{5} = \frac{2}{5} \]

5. **Calculate \(\frac{2}{5} \text{ of } \frac{15}{2}\):**
\[ \frac{2}{5} \text{ of } \frac{15}{2} = \frac{2}{5} \times \frac{15}{2} = \frac{2 \times 15}{5 \times 2} = \frac{30}{10} = 3 \]

6. **Calculate \(\frac{3}{4}\):**
- Now, we need to divide \(3\) by \(\frac{3}{4}\):
\[ 3 \div \frac{3}{4} = 3 \times \frac{4}{3} = \frac{12}{3} = 4 \]

7. **Combine the Results:**
- Now we have:
\[ \frac{2}{5} + 4 \]
- To add these, convert \(\frac{2}{5}\) to a decimal or a fraction with a common denominator:
\[ 4 = \frac{20}{5} \implies \frac{2}{5} + \frac{20}{5} = \frac{22}{5} \]

8. **Final Calculation:**
- Now we need to multiply this result by \(\frac{4}{19}\):
\[ \frac{22}{5} \text{ of } \frac{4}{19} = \frac{22 \times 4}{5 \times 19} = \frac{88}{95} \]

### Final Answer:
\[ \frac{88}{95} \]