If `2=x+(1)/(1+(1)/(5+1/2))`, then the value of x is equal to:

Options

A. 1
B. `(13)/(15)`
C. `(15)/(13)`
D. `(14)/(13)`

Answer

To solve the equation \(2 = x + \frac{1}{1 + \frac{1}{5 + \frac{1}{2}}}\), we will follow these steps:

### Step 1: Simplify the innermost fraction
Start with the innermost fraction \(5 + \frac{1}{2}\).

\[ 5 + \frac{1}{2} = \frac{10}{2} + \frac{1}{2} = \frac{11}{2} \]

**Hint:** When adding fractions, find a common denominator.

### Step 2: Substitute back into the equation
Now substitute \(\frac{11}{2}\) back into the equation:

\[ 2 = x + \frac{1}{1 + \frac{1}{\frac{11}{2}}} \]

### Step 3: Simplify the next fraction
Now simplify \(\frac{1}{\frac{11}{2}}\):

\[ \frac{1}{\frac{11}{2}} = \frac{2}{11} \]

So now we have:

\[ 2 = x + \frac{1}{1 + \frac{2}{11}} \]

### Step 4: Simplify \(1 + \frac{2}{11}\)
Next, simplify \(1 + \frac{2}{11}\):

\[ 1 + \frac{2}{11} = \frac{11}{11} + \frac{2}{11} = \frac{13}{11} \]

### Step 5: Substitute back into the equation
Now substitute \(\frac{13}{11}\) back into the equation:

\[ 2 = x + \frac{1}{\frac{13}{11}} \]

### Step 6: Simplify \(\frac{1}{\frac{13}{11}}\)
Now simplify \(\frac{1}{\frac{13}{11}}\):

\[ \frac{1}{\frac{13}{11}} = \frac{11}{13} \]

### Step 7: Substitute back into the equation
Now we have:

\[ 2 = x + \frac{11}{13} \]

### Step 8: Isolate \(x\)
To find \(x\), subtract \(\frac{11}{13}\) from both sides:

\[ x = 2 - \frac{11}{13} \]

### Step 9: Convert \(2\) to a fraction
Convert \(2\) to a fraction with a denominator of \(13\):

\[ 2 = \frac{26}{13} \]

### Step 10: Perform the subtraction
Now perform the subtraction:

\[ x = \frac{26}{13} - \frac{11}{13} = \frac{26 - 11}{13} = \frac{15}{13} \]

Thus, the value of \(x\) is:

\[ \boxed{\frac{15}{13}} \]

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