If `P=[5/9"of"6(3)/4div25/16+29/30]"of"[361/24div(1/4+4/3)^2]`, then the value of 5P is:

Options

A. 97
B. 101
C. 105
D. 111

Answer

To solve the problem \( P = \left[\frac{5}{9} \text{ of } 6\left(\frac{3}{4}\right) \div \frac{25}{16} + \frac{29}{30}\right] \text{ of } \left[\frac{361}{24} \div \left(\frac{1}{4} + \frac{4}{3}\right)^2\right] \), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Calculate the expression inside the first bracket:**
- Start with \( \frac{1}{4} + \frac{4}{3} \).
- To add these fractions, find a common denominator, which is 12:
\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{4}{3} = \frac{16}{12} \]
\[ \frac{1}{4} + \frac{4}{3} = \frac{3}{12} + \frac{16}{12} = \frac{19}{12} \]

2. **Square the result:**
\[ \left(\frac{19}{12}\right)^2 = \frac{361}{144} \]

3. **Calculate \( \frac{361}{24} \div \frac{361}{144} \):**
- Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{361}{24} \div \frac{361}{144} = \frac{361}{24} \times \frac{144}{361} = \frac{144}{24} = 6 \]

4. **Now substitute back into the original expression for \( P \):**
\[ P = \left[\frac{5}{9} \text{ of } 6\left(\frac{3}{4}\right) \div \frac{25}{16} + \frac{29}{30}\right] \text{ of } 6 \]

5. **Calculate \( 6 \left(\frac{3}{4}\right) \):**
\[ 6 \times \frac{3}{4} = \frac{18}{4} = \frac{9}{2} \]

6. **Now calculate \( \frac{9}{2} \div \frac{25}{16} \):**
- Again, multiply by the reciprocal:
\[ \frac{9}{2} \div \frac{25}{16} = \frac{9}{2} \times \frac{16}{25} = \frac{144}{50} = \frac{72}{25} \]

7. **Add \( \frac{72}{25} + \frac{29}{30} \):**
- Find a common denominator, which is 150:
\[ \frac{72}{25} = \frac{432}{150}, \quad \frac{29}{30} = \frac{145}{150} \]
\[ \frac{72}{25} + \frac{29}{30} = \frac{432 + 145}{150} = \frac{577}{150} \]

8. **Now calculate \( P \):**
\[ P = \left[\frac{577}{150}\right] \text{ of } 6 = \frac{577}{150} \times 6 = \frac{3462}{150} = \frac{1154}{50} = \frac{577}{25} \]

9. **Finally, calculate \( 5P \):**
\[ 5P = 5 \times \frac{577}{25} = \frac{2885}{25} = 115.4 \]

### Final Answer:
The value of \( 5P \) is \( 115.4 \).