The simplification of `(1.bar3xx1.bar3xx1.bar3-1)/(1.bar3xx1.bar3+1.bar3+1)` is

Options

A. `1/3`
B. `1(1)/3`
C. `37/91`
D. `27/91`

Answer

To simplify the expression \((1.\overline{3} \times 1.\overline{3} \times 1.\overline{3} - 1) / (1.\overline{3} \times 1.\overline{3} + 1.\overline{3} + 1)\), we will follow these steps:

### Step 1: Convert \(1.\overline{3}\) to a Fraction
Let \(x = 1.\overline{3}\).
Then, we can express \(x\) as:
\[ x = 1.3333\ldots \]
Multiplying both sides by 10:
\[ 10x = 13.3333\ldots \]
Now, we can set up the equation:
\[ 10x - x = 13.3333\ldots - 1.3333\ldots \]
This simplifies to:
\[ 9x = 12 \]
Thus, we find:
\[ x = \frac{12}{9} = \frac{4}{3} \]

### Step 2: Substitute \(1.\overline{3}\) with \(\frac{4}{3}\)
Now, we substitute \(1.\overline{3}\) with \(\frac{4}{3}\) in the original expression:
\[ \frac{\left(\frac{4}{3} \times \frac{4}{3} \times \frac{4}{3}\right) - 1}{\left(\frac{4}{3} \times \frac{4}{3}\right) + \frac{4}{3} + 1} \]

### Step 3: Simplify the Numerator
Calculating the numerator:
\[ \frac{4}{3} \times \frac{4}{3} \times \frac{4}{3} = \frac{64}{27} \]
Thus, the numerator becomes:
\[ \frac{64}{27} - 1 = \frac{64}{27} - \frac{27}{27} = \frac{64 - 27}{27} = \frac{37}{27} \]

### Step 4: Simplify the Denominator
Calculating the denominator:
\[ \frac{4}{3} \times \frac{4}{3} = \frac{16}{9} \]
Now adding \(\frac{4}{3}\) and \(1\):
\[ \frac{4}{3} + 1 = \frac{4}{3} + \frac{3}{3} = \frac{7}{3} \]
Now, we find the common denominator for \(\frac{16}{9}\) and \(\frac{7}{3}\):
\[ \frac{7}{3} = \frac{21}{9} \]
Thus, the denominator becomes:
\[ \frac{16}{9} + \frac{21}{9} = \frac{37}{9} \]

### Step 5: Combine the Results
Now we can combine the results:
\[ \frac{\frac{37}{27}}{\frac{37}{9}} = \frac{37}{27} \times \frac{9}{37} = \frac{9}{27} = \frac{1}{3} \]

### Final Answer
Thus, the simplification of the expression is:
\[ \frac{1}{3} \]