The simplified of `((1)/(3)-:(1)/(3) xx(1)/(3))/((1)/(3)-:(1)/(3)"of"(1)/3)-(1)/(9)` is

Options

A. 0
B. `(1)/(9)`
C. `(1)/(3)`
D. 1

Answer

To simplify the expression \(\frac{\frac{1}{3} \div \frac{1}{3} \times \frac{1}{3}}{\frac{1}{3} \div \frac{1}{3} \text{ of } \frac{1}{3} - \frac{1}{9}}\), we will follow these steps:

### Step 1: Simplify the numerator
The numerator is \(\frac{1}{3} \div \frac{1}{3} \times \frac{1}{3}\).

- First, calculate \(\frac{1}{3} \div \frac{1}{3}\):
\[ \frac{1}{3} \div \frac{1}{3} = 1 \]
- Now, multiply by \(\frac{1}{3}\):
\[ 1 \times \frac{1}{3} = \frac{1}{3} \]

### Step 2: Simplify the denominator
The denominator is \(\frac{1}{3} \div \frac{1}{3} \text{ of } \frac{1}{3} - \frac{1}{9}\).

- First, calculate \(\frac{1}{3} \div \frac{1}{3}\):
\[ \frac{1}{3} \div \frac{1}{3} = 1 \]
- Now, calculate \(1 \text{ of } \frac{1}{3}\):
\[ 1 \text{ of } \frac{1}{3} = \frac{1}{3} \]
- Now, subtract \(\frac{1}{9}\):
\[ \frac{1}{3} - \frac{1}{9} \]

To subtract these fractions, we need a common denominator. The least common multiple of 3 and 9 is 9.

- Convert \(\frac{1}{3}\) to have a denominator of 9:
\[ \frac{1}{3} = \frac{3}{9} \]
- Now, perform the subtraction:
\[ \frac{3}{9} - \frac{1}{9} = \frac{2}{9} \]

### Step 3: Combine the results
Now we can substitute the simplified numerator and denominator back into the expression:
\[ \frac{\frac{1}{3}}{\frac{2}{9}} \]

### Step 4: Simplify the fraction
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{1}{3} \div \frac{2}{9} = \frac{1}{3} \times \frac{9}{2} = \frac{9}{6} \]

### Step 5: Simplify \(\frac{9}{6}\)
Now simplify \(\frac{9}{6}\):
\[ \frac{9}{6} = \frac{3}{2} \]

### Step 6: Final Calculation
Now we have:
\[ \frac{3}{2} - \frac{1}{9} \]

To subtract these fractions, we need a common denominator. The least common multiple of 2 and 9 is 18.

- Convert \(\frac{3}{2}\) to have a denominator of 18:
\[ \frac{3}{2} = \frac{27}{18} \]
- Convert \(\frac{1}{9}\) to have a denominator of 18:
\[ \frac{1}{9} = \frac{2}{18} \]
- Now, perform the subtraction:
\[ \frac{27}{18} - \frac{2}{18} = \frac{25}{18} \]

### Conclusion
The simplified value of the expression is \(\frac{25}{18}\).