The value of blank th will be 1+`(1)/(1+(1)/(?))=8/5`

Options

A. `1/3`
B. `1/5`
C. `3/2`
D. `1/4`

Answer

To solve the equation \( 1 + \frac{1}{1 + \frac{1}{?}} = \frac{8}{5} \), we will follow these steps:

### Step 1: Set up the equation
Let \( x = ? \). The equation can be rewritten as:
\[ 1 + \frac{1}{1 + \frac{1}{x}} = \frac{8}{5} \]

### Step 2: Simplify the inner fraction
First, simplify the inner fraction \( \frac{1}{1 + \frac{1}{x}} \):
\[ 1 + \frac{1}{x} = \frac{x + 1}{x} \]
Thus,
\[ \frac{1}{1 + \frac{1}{x}} = \frac{x}{x + 1} \]

### Step 3: Substitute back into the equation
Now substitute this back into the equation:
\[ 1 + \frac{x}{x + 1} = \frac{8}{5} \]

### Step 4: Combine the left side
The left side can be combined:
\[ 1 = \frac{x + 1}{x + 1} \quad \text{(to have a common denominator)} \]
So,
\[ 1 + \frac{x}{x + 1} = \frac{x + 1 + x}{x + 1} = \frac{2x + 1}{x + 1} \]
Now the equation becomes:
\[ \frac{2x + 1}{x + 1} = \frac{8}{5} \]

### Step 5: Cross-multiply
Cross-multiply to eliminate the fractions:
\[ 5(2x + 1) = 8(x + 1) \]

### Step 6: Expand both sides
Expanding both sides gives:
\[ 10x + 5 = 8x + 8 \]

### Step 7: Rearrange the equation
Rearranging the equation to isolate \( x \):
\[ 10x - 8x = 8 - 5 \]
\[ 2x = 3 \]

### Step 8: Solve for \( x \)
Now, divide by 2:
\[ x = \frac{3}{2} \]

### Conclusion
Thus, the value of \( ? \) is:
\[ \boxed{\frac{3}{2}} \]