The value of `3 1/9 -: 4/9 of (7 3/10 + 3 1/5) + 4/5 -: 2 2/5` is :

Options

A. 5
B. 1
C. 2
D. 10

Answer

To solve the expression \(3 \frac{1}{9} \div \frac{4}{9} \text{ of } (7 \frac{3}{10} + 3 \frac{1}{5}) + \frac{4}{5} \div 2 \frac{2}{5}\), we will follow the order of operations (BODMAS/BIDMAS).

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions**:
- \(3 \frac{1}{9} = \frac{28}{9}\)
- \(7 \frac{3}{10} = \frac{73}{10}\)
- \(3 \frac{1}{5} = \frac{16}{5}\)
- \(2 \frac{2}{5} = \frac{12}{5}\)

2. **Calculate the Sum Inside the Bracket**:
\[ 7 \frac{3}{10} + 3 \frac{1}{5} = \frac{73}{10} + \frac{16}{5} \]
To add these fractions, we need a common denominator. The least common multiple of 10 and 5 is 10.
\[ \frac{16}{5} = \frac{32}{10} \]
Now, add:
\[ \frac{73}{10} + \frac{32}{10} = \frac{105}{10} = \frac{21}{2} \]

3. **Calculate the 'of' Operation**:
We need to calculate \(\frac{4}{9} \text{ of } \frac{21}{2}\):
\[ \frac{4}{9} \times \frac{21}{2} = \frac{84}{18} = \frac{14}{3} \]

4. **Calculate the Division**:
Now we have:
\[ \frac{28}{9} \div \frac{14}{3} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{28}{9} \times \frac{3}{14} = \frac{28 \times 3}{9 \times 14} = \frac{84}{126} = \frac{2}{3} \]

5. **Calculate the Second Division**:
Now calculate \(\frac{4}{5} \div \frac{12}{5}\):
\[ \frac{4}{5} \times \frac{5}{12} = \frac{4 \times 5}{5 \times 12} = \frac{4}{12} = \frac{1}{3} \]

6. **Combine the Results**:
Now we combine the results:
\[ \frac{2}{3} + \frac{1}{3} = \frac{3}{3} = 1 \]

### Final Answer:
The value of the expression is \(1\).