`(1)/(20)+(1)/(30)+(1)/(42)+(1)/(56)+(1)/(72)+(1)/(90)+(1)/(110)+(1)/(132)` is equal to :

Options

A. `(1)/(8)`
B. `(1)/(7)`
C. `(1)/(6)`
D. `(1)/(10)`

Answer

To solve the expression

\[ \frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} + \frac{1}{132} \]

we can rewrite each term in a form that allows us to see a pattern.

### Step 1: Rewrite each term
We will express each fraction in terms of differences of fractions:

1. \(\frac{1}{20} = \frac{1}{4} - \frac{1}{5}\)
2. \(\frac{1}{30} = \frac{1}{5} - \frac{1}{6}\)
3. \(\frac{1}{42} = \frac{1}{6} - \frac{1}{7}\)
4. \(\frac{1}{56} = \frac{1}{7} - \frac{1}{8}\)
5. \(\frac{1}{72} = \frac{1}{8} - \frac{1}{9}\)
6. \(\frac{1}{90} = \frac{1}{9} - \frac{1}{10}\)
7. \(\frac{1}{110} = \frac{1}{10} - \frac{1}{11}\)
8. \(\frac{1}{132} = \frac{1}{11} - \frac{1}{12}\)

### Step 2: Combine the rewritten terms
Now, we can combine all these terms:

\[ \left( \frac{1}{4} - \frac{1}{5} \right) + \left( \frac{1}{5} - \frac{1}{6} \right) + \left( \frac{1}{6} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{8} \right) + \left( \frac{1}{8} - \frac{1}{9} \right) + \left( \frac{1}{9} - \frac{1}{10} \right) + \left( \frac{1}{10} - \frac{1}{11} \right) + \left( \frac{1}{11} - \frac{1}{12} \right) \]

### Step 3: Observe cancellation
Notice that all intermediate terms cancel out:

\[ \frac{1}{4} - \frac{1}{12} \]

### Step 4: Calculate the remaining terms
Now we need to calculate:

\[ \frac{1}{4} - \frac{1}{12} \]

To do this, we need a common denominator, which is 12:

\[ \frac{1}{4} = \frac{3}{12} \]

Thus,

\[ \frac{3}{12} - \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \]

### Final Answer
The value of the original expression is:

\[ \frac{1}{6} \]