The value of `(2)/(5) div (3)/(10)` of `(4)/(9) - (4)/(5) xx 1 (1)/(9) div (8)/(15) - (3)/(4) + (3)/(4) div (1)/(2)` is:
(a)7/9
(b)23/6
(c)4/3
(d)25/12

Options

A. `(7)/(9)`
B. `(23)/(6)`
C. `(4)/(3)`
D. `(25)/(12)`

Answer

To solve the expression \(\frac{2}{5} \div \frac{3}{10} \text{ of } \frac{4}{9} - \frac{4}{5} \times 1 \frac{1}{9} \div \frac{8}{15} - \frac{3}{4} + \frac{3}{4} \div \frac{1}{2}\), we will follow the order of operations (BODMAS/BIDMAS).

### Step 1: Rewrite the Mixed Number
Convert the mixed number \(1 \frac{1}{9}\) into an improper fraction:
\[ 1 \frac{1}{9} = 1 + \frac{1}{9} = \frac{9}{9} + \frac{1}{9} = \frac{10}{9} \]
Now, the expression becomes:
\[ \frac{2}{5} \div \frac{3}{10} \text{ of } \frac{4}{9} - \frac{4}{5} \times \frac{10}{9} \div \frac{8}{15} - \frac{3}{4} + \frac{3}{4} \div \frac{1}{2} \]

### Step 2: Solve the "of" (Multiplication)
The term "of" means multiplication. Therefore, we calculate:
\[ \frac{2}{5} \div \frac{3}{10} \text{ of } \frac{4}{9} = \frac{2}{5} \div \frac{3}{10} \times \frac{4}{9} \]
First, solve \(\frac{2}{5} \div \frac{3}{10}\):
\[ \frac{2}{5} \div \frac{3}{10} = \frac{2}{5} \times \frac{10}{3} = \frac{2 \times 10}{5 \times 3} = \frac{20}{15} = \frac{4}{3} \]
Now, multiply by \(\frac{4}{9}\):
\[ \frac{4}{3} \times \frac{4}{9} = \frac{16}{27} \]

### Step 3: Solve the Division and Multiplication
Now we need to solve the next part:
\[ -\frac{4}{5} \times \frac{10}{9} \div \frac{8}{15} \]
First, calculate \(\frac{10}{9} \div \frac{8}{15}\):
\[ \frac{10}{9} \div \frac{8}{15} = \frac{10}{9} \times \frac{15}{8} = \frac{10 \times 15}{9 \times 8} = \frac{150}{72} = \frac{25}{12} \]
Now, multiply by \(-\frac{4}{5}\):
\[ -\frac{4}{5} \times \frac{25}{12} = -\frac{100}{60} = -\frac{5}{3} \]

### Step 4: Solve the Remaining Terms
Next, we calculate:
\[ -\frac{3}{4} + \frac{3}{4} \div \frac{1}{2} \]
First, calculate \(\frac{3}{4} \div \frac{1}{2}\):
\[ \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times 2 = \frac{3 \times 2}{4} = \frac{6}{4} = \frac{3}{2} \]
Now, substitute this back into the expression:
\[ -\frac{3}{4} + \frac{3}{2} \]
Convert \(\frac{3}{2}\) to have a common denominator of 4:
\[ \frac{3}{2} = \frac{6}{4} \]
So,
\[ -\frac{3}{4} + \frac{6}{4} = \frac{3}{4} \]

### Step 5: Combine All Parts
Now, we combine all parts:
\[ \frac{16}{27} - \frac{5}{3} + \frac{3}{4} \]
Convert \(-\frac{5}{3}\) and \(\frac{3}{4}\) to have a common denominator of 108:
\[ -\frac{5}{3} = -\frac{180}{108}, \quad \frac{3}{4} = \frac{81}{108} \]
Now, convert \(\frac{16}{27}\):
\[ \frac{16}{27} = \frac{64}{108} \]
Now combine:
\[ \frac{64}{108} - \frac{180}{108} + \frac{81}{108} = \frac{64 - 180 + 81}{108} = \frac{-35}{108} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{25}{12} \]