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

Options

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

Answer

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

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers into improper fractions.

1. \(2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}\)
2. \(1 \frac{2}{11} = \frac{1 \times 11 + 2}{11} = \frac{11 + 2}{11} = \frac{13}{11}\)

Now, we rewrite the expression:
\[ \frac{\frac{7}{3} - \frac{13}{11}}{3 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3}}}} \]

### Step 2: Find a Common Denominator for the Numerator
Next, we need to subtract the fractions in the numerator. The common denominator for \(3\) and \(11\) is \(33\).

1. Convert \(\frac{7}{3}\) to have a denominator of \(33\):
\[ \frac{7}{3} = \frac{7 \times 11}{3 \times 11} = \frac{77}{33} \]
2. Convert \(\frac{13}{11}\) to have a denominator of \(33\):
\[ \frac{13}{11} = \frac{13 \times 3}{11 \times 3} = \frac{39}{33} \]

Now we can subtract:
\[ \frac{77}{33} - \frac{39}{33} = \frac{77 - 39}{33} = \frac{38}{33} \]

### Step 3: Simplify the Denominator
Now we simplify the denominator \(3 + \frac{1}{3 + \frac{1}{3 + \frac{1}{3}}}\).

1. Start from the innermost fraction:
\[ 3 + \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{9 + 1}{3} = \frac{10}{3} \]
2. Now substitute back:
\[ 3 + \frac{1}{\frac{10}{3}} = 3 + \frac{3}{10} = \frac{3 \times 10 + 3}{10} = \frac{30 + 3}{10} = \frac{33}{10} \]

### Step 4: Rewrite the Expression
Now we can rewrite the entire expression:
\[ \frac{\frac{38}{33}}{\frac{33}{10}} = \frac{38}{33} \times \frac{10}{33} = \frac{38 \times 10}{33 \times 33} = \frac{380}{1089} \]

### Final Answer
Thus, the value of the expression is:
\[ \frac{380}{1089} \]