If `2 = x + (1)/(1 + (1)/(3 + (1)/(4)))`, then the value of x is :

Options

A. `(18)/(17)`
B. `(21)/(17)`
C. `(13)/(17)`
D. `(12)/(17)`

Answer

To solve the equation \( 2 = x + \frac{1}{1 + \frac{1}{3 + \frac{1}{4}}} \), we will simplify the right-hand side step by step.

### Step 1: Simplify the innermost fraction
First, we simplify the innermost fraction \( \frac{1}{4} \):
\[ 3 + \frac{1}{4} = 3 + 0.25 = 3.25 = \frac{13}{4} \]
So, we rewrite the equation:
\[ 2 = x + \frac{1}{1 + \frac{1}{\frac{13}{4}}} \]

### Step 2: Simplify the next fraction
Now, we simplify \( \frac{1}{\frac{13}{4}} \):
\[ \frac{1}{\frac{13}{4}} = \frac{4}{13} \]
Now the equation becomes:
\[ 2 = x + \frac{1}{1 + \frac{4}{13}} \]

### Step 3: Combine the fractions in the denominator
Next, we simplify \( 1 + \frac{4}{13} \):
\[ 1 + \frac{4}{13} = \frac{13}{13} + \frac{4}{13} = \frac{17}{13} \]
Now we have:
\[ 2 = x + \frac{1}{\frac{17}{13}} \]

### Step 4: Simplify the fraction
Now, we simplify \( \frac{1}{\frac{17}{13}} \):
\[ \frac{1}{\frac{17}{13}} = \frac{13}{17} \]
So the equation now is:
\[ 2 = x + \frac{13}{17} \]

### Step 5: Isolate \( x \)
To isolate \( x \), we subtract \( \frac{13}{17} \) from both sides:
\[ x = 2 - \frac{13}{17} \]

### Step 6: Convert 2 to a fraction
We convert 2 into a fraction with a denominator of 17:
\[ 2 = \frac{34}{17} \]
Now we can write:
\[ x = \frac{34}{17} - \frac{13}{17} \]

### Step 7: Perform the subtraction
Now we perform the subtraction:
\[ x = \frac{34 - 13}{17} = \frac{21}{17} \]

### Final Answer
Thus, the value of \( x \) is:
\[ \boxed{\frac{21}{17}} \]