The value of ` (2 (1)/(3) - 1(2)/(11) )/(3 + (1)/(3 + (1)/(3 + 1/3) ))`

Options

A. `38/109`
B. `109/38`
C. 1
D. `116/109`

Answer

To solve the expression \( \frac{2 \frac{1}{3} - 1 \frac{2}{11}}{3 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3}}}} \), we will break it down step by step.

### Step 1: Convert mixed numbers to improper fractions
The numerator consists of two mixed numbers: \( 2 \frac{1}{3} \) and \( 1 \frac{2}{11} \).

1. Convert \( 2 \frac{1}{3} \):
\[ 2 \frac{1}{3} = 2 + \frac{1}{3} = \frac{6}{3} + \frac{1}{3} = \frac{7}{3} \]

2. Convert \( 1 \frac{2}{11} \):
\[ 1 \frac{2}{11} = 1 + \frac{2}{11} = \frac{11}{11} + \frac{2}{11} = \frac{13}{11} \]

### Step 2: Substitute back into the numerator
Now, substitute these values back into the numerator:
\[ \frac{7}{3} - \frac{13}{11} \]

### Step 3: Find a common denominator for the numerator
The least common multiple (LCM) of 3 and 11 is 33. We will convert both fractions:
\[ \frac{7}{3} = \frac{7 \times 11}{3 \times 11} = \frac{77}{33} \]
\[ \frac{13}{11} = \frac{13 \times 3}{11 \times 3} = \frac{39}{33} \]

### Step 4: Subtract the fractions
Now, subtract the two fractions:
\[ \frac{77}{33} - \frac{39}{33} = \frac{77 - 39}{33} = \frac{38}{33} \]

### Step 5: Simplify the denominator
Next, we simplify the denominator \( 3 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3}}} \).

1. Start from the innermost fraction:
\[ 3 + \frac{1}{3} = \frac{9}{3} + \frac{1}{3} = \frac{10}{3} \]

2. Now substitute this back into the next layer:
\[ 3 + \frac{1}{\frac{10}{3}} = 3 + \frac{3}{10} = \frac{30}{10} + \frac{3}{10} = \frac{33}{10} \]

3. Finally, substitute this back into the outer layer:
\[ 3 + \frac{1}{\frac{33}{10}} = 3 + \frac{10}{33} = \frac{99}{33} + \frac{10}{33} = \frac{109}{33} \]

### Step 6: Combine the results
Now, we have:
\[ \frac{\frac{38}{33}}{\frac{109}{33}} \]

### Step 7: Simplify the fraction
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{38}{33} \times \frac{33}{109} = \frac{38}{109} \]

### Final Answer
Thus, the value of the given expression is:
\[ \frac{38}{109} \]