`1/30+1/42+1/56+1/72+1/90+1/110=?`

Options

A. `2/27`
B. `7/9`
C. `5/27`
D. `6/55`

Answer

To solve the problem \( \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90} + \frac{1}{110} \), we can rewrite each fraction in a form that makes it easier to combine them.

### Step-by-Step Solution:

1. **Identify the Denominators**:
The denominators are:
- \( 30 = 5 \times 6 \)
- \( 42 = 6 \times 7 \)
- \( 56 = 7 \times 8 \)
- \( 72 = 8 \times 9 \)
- \( 90 = 9 \times 10 \)
- \( 110 = 10 \times 11 \)

2. **Rewrite Each Fraction**:
We can express each fraction as a difference of two fractions:
\[ \frac{1}{30} = \frac{1}{5 \times 6} = \frac{1}{5} - \frac{1}{6} \]
\[ \frac{1}{42} = \frac{1}{6 \times 7} = \frac{1}{6} - \frac{1}{7} \]
\[ \frac{1}{56} = \frac{1}{7 \times 8} = \frac{1}{7} - \frac{1}{8} \]
\[ \frac{1}{72} = \frac{1}{8 \times 9} = \frac{1}{8} - \frac{1}{9} \]
\[ \frac{1}{90} = \frac{1}{9 \times 10} = \frac{1}{9} - \frac{1}{10} \]
\[ \frac{1}{110} = \frac{1}{10 \times 11} = \frac{1}{10} - \frac{1}{11} \]

3. **Combine All Fractions**:
Now, we can combine all these rewritten fractions:
\[ \left( \frac{1}{5} - \frac{1}{6} \right) + \left( \frac{1}{6} - \frac{1}{7} \right) + \left( \frac{1}{7} - \frac{1}{8} \right) + \left( \frac{1}{8} - \frac{1}{9} \right) + \left( \frac{1}{9} - \frac{1}{10} \right) + \left( \frac{1}{10} - \frac{1}{11} \right) \]

Notice that all intermediate terms cancel out:
\[ = \frac{1}{5} - \frac{1}{11} \]

4. **Calculate the Result**:
Now we need to find a common denominator to subtract these fractions. The least common multiple (LCM) of 5 and 11 is 55.
\[ \frac{1}{5} = \frac{11}{55}, \quad \frac{1}{11} = \frac{5}{55} \]
Therefore,
\[ \frac{1}{5} - \frac{1}{11} = \frac{11}{55} - \frac{5}{55} = \frac{6}{55} \]

5. **Final Answer**:
Thus, the final answer is:
\[ \frac{6}{55} \]