What is the value of `((1)/(3) + (1)/(3) xx (1)/(3))/((1)/(3) xx (1)/(3) + (1)/(3))` ?

Options

A. 1
B. 3
C. `(1)/(3)`
D. `(1)/(2)`

Answer

To solve the expression \(\frac{\frac{1}{3} + \frac{1}{3} \times \frac{1}{3}}{\frac{1}{3} \times \frac{1}{3} + \frac{1}{3}}\), we will follow these steps:

### Step 1: Simplify the numerator
The numerator is \(\frac{1}{3} + \frac{1}{3} \times \frac{1}{3}\).
- First, calculate \(\frac{1}{3} \times \frac{1}{3}\):
\[ \frac{1}{3} \times \frac{1}{3} = \frac{1}{9} \]
- Now, substitute this back into the numerator:
\[ \frac{1}{3} + \frac{1}{9} \]
- To add these fractions, find a common denominator, which is 9:
\[ \frac{1}{3} = \frac{3}{9} \]
- Now, add:
\[ \frac{3}{9} + \frac{1}{9} = \frac{4}{9} \]

### Step 2: Simplify the denominator
The denominator is \(\frac{1}{3} \times \frac{1}{3} + \frac{1}{3}\).
- We already know that \(\frac{1}{3} \times \frac{1}{3} = \frac{1}{9}\).
- Substitute this back into the denominator:
\[ \frac{1}{9} + \frac{1}{3} \]
- Again, find a common denominator, which is 9:
\[ \frac{1}{3} = \frac{3}{9} \]
- Now, add:
\[ \frac{1}{9} + \frac{3}{9} = \frac{4}{9} \]

### Step 3: Combine the results
Now we have:
\[ \frac{\frac{4}{9}}{\frac{4}{9}} \]
This simplifies to:
\[ 1 \]

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

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