If `(1)/(1+(1)/(1+(1)/(1+(1)/(x))))= (5)/(8)` then what is the value of x ?

Options

A. 2
B. 3
C. 1
D. 4

Answer

To solve the equation

\[ \frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{x}}}} = \frac{5}{8}, \]

we will simplify the left-hand side step by step.

### Step 1: Start with the innermost fraction

Let’s denote the innermost fraction as \( y \):

\[ y = \frac{1}{x}. \]

Then, we can rewrite the equation as:

\[ \frac{1}{1+\frac{1}{1+\frac{1}{1+y}}} = \frac{5}{8}. \]

### Step 2: Simplify the next layer

Now, we simplify \( \frac{1}{1+y} \):

\[ \frac{1}{1+y} = \frac{1}{1+\frac{1}{x}} = \frac{x}{x+1}. \]

So now we have:

\[ \frac{1}{1+\frac{x}{x+1}}. \]

### Step 3: Combine the fractions

Next, we simplify \( 1+\frac{x}{x+1} \):

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

Thus, we have:

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

### Step 4: Substitute back into the equation

Now, substituting back, our equation becomes:

\[ \frac{x+1}{2x+1} = \frac{5}{8}. \]

### Step 5: Cross-multiply

Cross-multiplying gives:

\[ 8(x+1) = 5(2x+1). \]

### Step 6: Expand both sides

Expanding both sides results in:

\[ 8x + 8 = 10x + 5. \]

### Step 7: Rearrange the equation

Rearranging gives:

\[ 8 - 5 = 10x - 8x, \]

which simplifies to:

\[ 3 = 2x. \]

### Step 8: Solve for x

Dividing both sides by 2 gives:

\[ x = \frac{3}{2}. \]

### Final Answer

Thus, the value of \( x \) is

\[ \boxed{\frac{3}{2}}. \]