If `(1+(1)/(3)) (1+(1)/(5)) (1+(1)/(7)) (1+(1)/(9)) (1-(1)/(4)) (1-(1)/(6)) (1-(1)/(8))=1 +(1)/(x)` then what is the value of x ?

Options

A. 9
B. 8
C. 6
D. 4

Answer

To solve the equation

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

we will simplify each term step by step.

### Step 1: Simplify each term

1. **Calculate \(1 + \frac{1}{3}\)**:
\[ 1 + \frac{1}{3} = \frac{3}{3} + \frac{1}{3} = \frac{4}{3} \]

2. **Calculate \(1 + \frac{1}{5}\)**:
\[ 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5} \]

3. **Calculate \(1 + \frac{1}{7}\)**:
\[ 1 + \frac{1}{7} = \frac{7}{7} + \frac{1}{7} = \frac{8}{7} \]

4. **Calculate \(1 + \frac{1}{9}\)**:
\[ 1 + \frac{1}{9} = \frac{9}{9} + \frac{1}{9} = \frac{10}{9} \]

5. **Calculate \(1 - \frac{1}{4}\)**:
\[ 1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4} \]

6. **Calculate \(1 - \frac{1}{6}\)**:
\[ 1 - \frac{1}{6} = \frac{6}{6} - \frac{1}{6} = \frac{5}{6} \]

7. **Calculate \(1 - \frac{1}{8}\)**:
\[ 1 - \frac{1}{8} = \frac{8}{8} - \frac{1}{8} = \frac{7}{8} \]

### Step 2: Combine all the simplified terms

Now we can combine all the simplified fractions:

\[ \frac{4}{3} \cdot \frac{6}{5} \cdot \frac{8}{7} \cdot \frac{10}{9} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdot \frac{7}{8} \]

### Step 3: Cancel out common terms

When we multiply these fractions, we can see that many terms will cancel out:

- \(4\) in the numerator of \(\frac{4}{3}\) cancels with \(4\) in the denominator of \(\frac{3}{4}\).
- \(6\) in the numerator of \(\frac{6}{5}\) cancels with \(6\) in the denominator of \(\frac{5}{6}\).
- \(8\) in the numerator of \(\frac{8}{7}\) cancels with \(8\) in the denominator of \(\frac{7}{8}\).
- The \(3\) in the numerator of \(\frac{3}{4}\) cancels with the \(3\) in the denominator of \(\frac{4}{3}\).
- The \(5\) in the numerator of \(\frac{5}{6}\) cancels with the \(5\) in the denominator of \(\frac{6}{5}\).
- The \(7\) in the numerator of \(\frac{7}{8}\) cancels with the \(7\) in the denominator of \(\frac{8}{7}\).

After canceling, we are left with:

\[ \frac{10}{9} \]

### Step 4: Set the equation equal to \(1 + \frac{1}{x}\)

Now we set the left-hand side equal to the right-hand side:

\[ \frac{10}{9} = 1 + \frac{1}{x} \]

### Step 5: Solve for \(x\)

Subtract \(1\) from both sides:

\[ \frac{10}{9} - 1 = \frac{1}{x} \]

Convert \(1\) to a fraction with a denominator of \(9\):

\[ \frac{10}{9} - \frac{9}{9} = \frac{1}{x} \]

This simplifies to:

\[ \frac{1}{9} = \frac{1}{x} \]

Thus, we find:

\[ x = 9 \]

### Final Answer

The value of \(x\) is \(9\).

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