If `(1 + (1)/(2)) (1 + (1)/(4)) (1+(1)/(6))(1+(1)/(8))(1-(1)/(3))(1-(1)/(5)) (1-(1)/(7))=1 +(1)/(x)` then what is the value of x ?

Options

A. 6
B. 8
C. 5
D. 7

Answer

To solve the equation

\[ (1 + \frac{1}{2})(1 + \frac{1}{4})(1 + \frac{1}{6})(1 + \frac{1}{8})(1 - \frac{1}{3})(1 - \frac{1}{5})(1 - \frac{1}{7}) = 1 + \frac{1}{x} \]

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

### Step 1: Simplify Each Term

Start by simplifying each term in the product:

- \(1 + \frac{1}{2} = \frac{3}{2}\)
- \(1 + \frac{1}{4} = \frac{5}{4}\)
- \(1 + \frac{1}{6} = \frac{7}{6}\)
- \(1 + \frac{1}{8} = \frac{9}{8}\)
- \(1 - \frac{1}{3} = \frac{2}{3}\)
- \(1 - \frac{1}{5} = \frac{4}{5}\)
- \(1 - \frac{1}{7} = \frac{6}{7}\)

### Step 2: Write the Product

Now we can write the entire product:

\[ \frac{3}{2} \cdot \frac{5}{4} \cdot \frac{7}{6} \cdot \frac{9}{8} \cdot \frac{2}{3} \cdot \frac{4}{5} \cdot \frac{6}{7} \]

### Step 3: Cancel Out Terms

Next, we can cancel out terms in the product:

- The \(3\) in the numerator of \(\frac{3}{2}\) cancels with the \(3\) in the denominator of \(\frac{2}{3}\).
- The \(5\) in the numerator of \(\frac{5}{4}\) cancels with the \(5\) in the denominator of \(\frac{4}{5}\).
- The \(7\) in the numerator of \(\frac{7}{6}\) cancels with the \(7\) in the denominator of \(\frac{6}{7}\).

After canceling, we are left with:

\[ \frac{9}{8} \cdot \frac{2}{6} = \frac{9 \cdot 2}{8 \cdot 6} = \frac{18}{48} = \frac{3}{8} \]

### Step 4: Set the Equation

Now, we equate this to the right-hand side of the original equation:

\[ \frac{3}{8} = 1 + \frac{1}{x} \]

### Step 5: Isolate \(\frac{1}{x}\)

To isolate \(\frac{1}{x}\), we rearrange the equation:

\[ \frac{1}{x} = \frac{3}{8} - 1 \]

Convert \(1\) to a fraction with a denominator of \(8\):

\[ 1 = \frac{8}{8} \]

Thus,

\[ \frac{1}{x} = \frac{3}{8} - \frac{8}{8} = \frac{3 - 8}{8} = \frac{-5}{8} \]

### Step 6: Solve for \(x\)

Taking the reciprocal gives:

\[ x = -\frac{8}{5} \]

### Final Answer

The value of \(x\) is:

\[ x = -\frac{8}{5} \]