1/2-(-1/2-1/3)-(1/3+1/4)+(1/4-1/5)-(-1/5-1/6)-(-1/6)

Let's solve the expression step by step:

The expression is:

\[
\frac{1}{2} - \left( -\frac{1}{2} - \frac{1}{3} \right) - \left( \frac{1}{3} + \frac{1}{4} \right) + \left( \frac{1}{4} - \frac{1}{5} \right) - \left( -\frac{1}{5} - \frac{1}{6} \right) - \left( -\frac{1}{6} \right)
\]

### Step 1: Simplify inside each parenthesis.

1. \(-\left( -\frac{1}{2} - \frac{1}{3} \right) = -\left( -\frac{1}{2} + \frac{1}{3} \right)\)
   - \(-\frac{1}{2} + \frac{1}{3} = -\frac{3}{6} + \frac{2}{6} = -\frac{1}{6}\)
   - So, this becomes \(\frac{1}{6}\).

2. \(-\left( \frac{1}{3} + \frac{1}{4} \right) = -\left( \frac{4}{12} + \frac{3}{12} \right) = -\frac{7}{12}\)

3. \(\frac{1}{4} - \frac{1}{5} = \frac{5}{20} - \frac{4}{20} = \frac{1}{20}\)

4. \(-\left( -\frac{1}{5} - \frac{1}{6} \right) = -\left( -\frac{6}{30} - \frac{5}{30} \right) = -\left( -\frac{11}{30} \right) = \frac{11}{30}\)

5. \(-\left( -\frac{1}{6} \right) = \frac{1}{6}\)

### Step 2: Substitute these simplified expressions back into the original equation.

\[
\frac{1}{2} + \frac{1}{6} - \frac{7}{12} + \frac{1}{20} + \frac{11}{30} + \frac{1}{6}
\]

### Step 3: Find a common denominator.

The least common denominator (LCD) of 2, 6, 12, 20, and 30 is 60.

Now, convert each fraction to have a denominator of 60:

- \(\frac{1}{2} = \frac{30}{60}\)
- \(\frac{1}{6} = \frac{10}{60}\)
- \(\frac{7}{12} = \frac{35}{60}\)
- \(\frac{1}{20} = \frac{3}{60}\)
- \(\frac{11}{30} = \frac{22}{60}\)
- \(\frac{1}{6} = \frac{10}{60}\)

### Step 4: Add and subtract the numerators.

\[
\frac{30}{60} + \frac{10}{60} - \frac{35}{60} + \frac{3}{60} + \frac{22}{60} + \frac{10}{60}
\]

Combine the numerators:

\[
30 + 10 - 35 + 3 + 22 + 10 = 40
\]

So the result is:

\[
\frac{40}{60} = \frac{2}{3}
\]

### Final Answer: 

\[
\boxed{\frac{2}{3}}
\]

Would you like further details on any step? 

Here are 5 related questions:
1. How do you find the least common denominator (LCD) in a fraction problem?
2. How do you simplify fractions with both addition and subtraction?
3. Why do we change the signs when subtracting negative numbers?
4. What are common tricks for simplifying complex fractions?
5. How do you solve expressions with multiple parentheses?

**Tip:** Always simplify within parentheses first before moving to other parts of the expression.

Step-by-step solution to solving the fractional expression 1/2-(-1/2-1/3)-(1/3+1/4)+(1/4-1/5)-(-1/5-1/6)-(-1/6). This problem involves simplifying fractions, using common denominators, and applying the order of operations. Ideal for students in Grades 7-9.

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