When (1/2 - 1/4 + 1/5 - 1/6) is divided by(2/5 - 5/9 + 3/5 - 7/18) the result is:

We are given the expression:

\[
\frac{\left( \frac{1}{2} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} \right)}{\left( \frac{2}{5} - \frac{5}{9} + \frac{3}{5} - \frac{7}{18} \right)}
\]

### Step 1: Simplify the numerator
The numerator is:

\[
\frac{1}{2} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6}
\]

To simplify this, we need a common denominator. The least common denominator (LCD) for \(2, 4, 5, 6\) is 60. So we rewrite each fraction:

\[
\frac{1}{2} = \frac{30}{60}, \quad \frac{1}{4} = \frac{15}{60}, \quad \frac{1}{5} = \frac{12}{60}, \quad \frac{1}{6} = \frac{10}{60}
\]

Now substitute these values:

\[
\frac{30}{60} - \frac{15}{60} + \frac{12}{60} - \frac{10}{60} = \frac{30 - 15 + 12 - 10}{60} = \frac{17}{60}
\]

### Step 2: Simplify the denominator
The denominator is:

\[
\frac{2}{5} - \frac{5}{9} + \frac{3}{5} - \frac{7}{18}
\]

The least common denominator for \(5, 9, 18\) is 90. So we rewrite each fraction:

\[
\frac{2}{5} = \frac{36}{90}, \quad \frac{5}{9} = \frac{50}{90}, \quad \frac{3}{5} = \frac{54}{90}, \quad \frac{7}{18} = \frac{35}{90}
\]

Now substitute these values:

\[
\frac{36}{90} - \frac{50}{90} + \frac{54}{90} - \frac{35}{90} = \frac{36 - 50 + 54 - 35}{90} = \frac{5}{90} = \frac{1}{18}
\]

### Step 3: Divide the numerator by the denominator

We now divide the simplified numerator \(\frac{17}{60}\) by the simplified denominator \(\frac{1}{18}\):

\[
\frac{\frac{17}{60}}{\frac{1}{18}} = \frac{17}{60} \times \frac{18}{1} = \frac{17 \times 18}{60 \times 1} = \frac{306}{60}
\]

Simplify \(\frac{306}{60}\):

\[
\frac{306}{60} = \frac{51}{10} = 5.1
\]

### Final Answer:

The result of the expression is \( \boxed{5.1} \).

Would you like more details or further clarification? Here are five related questions to deepen understanding:

1. How do you find the least common denominator of fractions?
2. Why is it important to convert fractions to have a common denominator when adding or subtracting?
3. What are other methods to simplify complex fraction expressions?
4. How can you check if the result of a division of fractions is correct?
5. What is the process of multiplying and dividing fractions?

**Tip:** Always double-check your arithmetic steps, especially when working with fractions, to avoid mistakes in the final result!

When (1/2 - 1/4 + 1/5 - 1/6) is divided by (2/5 - 5/9 + 3/5 - 7/18) the result is:

Learn how to simplify the complex fraction expression (1/2 - 1/4 + 1/5 - 1/6) divided by (2/5 - 5/9 + 3/5 - 7/18) with a step-by-step solution. This problem covers key fraction simplification techniques, including finding the least common denominator and dividing fractions. Ideal for students in Grades 6-8.

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