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Let's break down the expression step by step to verify the result.

The problem is:

\[
\frac{\frac{7}{8} + \frac{1}{4} - \frac{3}{2} \cdot \frac{4}{9}}{\frac{5}{2} - \frac{11}{10} + \frac{1}{14} \cdot \frac{7}{5}} = ?
\]

### Numerator:
The numerator involves three terms:

\[
\frac{7}{8} + \frac{1}{4} - \left( \frac{3}{2} \cdot \frac{4}{9} \right)
\]

- First, multiply \(\frac{3}{2} \cdot \frac{4}{9} = \frac{12}{18} = \frac{2}{3}\).
- Now, add and subtract the fractions. To do this, convert all fractions to have a common denominator:

\[
\frac{7}{8} + \frac{1}{4} - \frac{2}{3}
\]

- The least common denominator (LCD) of 8, 4, and 3 is 24. Convert each fraction:

\[
\frac{7}{8} = \frac{21}{24}, \quad \frac{1}{4} = \frac{6}{24}, \quad \frac{2}{3} = \frac{16}{24}
\]

Now, compute:

\[
\frac{21}{24} + \frac{6}{24} - \frac{16}{24} = \frac{21 + 6 - 16}{24} = \frac{11}{24}
\]

### Denominator:
The denominator involves three terms:

\[
\frac{5}{2} - \frac{11}{10} + \left( \frac{1}{14} \cdot \frac{7}{5} \right)
\]

- First, multiply \(\frac{1}{14} \cdot \frac{7}{5} = \frac{7}{70} = \frac{1}{10}\).
- Now, subtract and add the fractions. To do this, convert all fractions to have a common denominator:

\[
\frac{5}{2} - \frac{11}{10} + \frac{1}{10}
\]

- The least common denominator (LCD) of 2 and 10 is 10. Convert each fraction:

\[
\frac{5}{2} = \frac{25}{10}, \quad \frac{11}{10} = \frac{11}{10}, \quad \frac{1}{10} = \frac{1}{10}
\]

Now, compute:

\[
\frac{25}{10} - \frac{11}{10} + \frac{1}{10} = \frac{25 - 11 + 1}{10} = \frac{15}{10} = \frac{3}{2}
\]

### Final Division:
Now, divide the numerator by the denominator:

\[
\frac{\frac{11}{24}}{\frac{3}{2}} = \frac{11}{24} \times \frac{2}{3} = \frac{11 \cdot 2}{24 \cdot 3} = \frac{22}{72} = \frac{11}{36}
\]

So, the answer is indeed \( \frac{11}{36} \), which matches the provided result.

---

Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How do you find the least common denominator of three or more fractions?
2. What is the process for simplifying complex fractions involving multiplication and division?
3. How can you multiply two fractions and simplify the result?
4. What are some strategies to check if a final fraction is in its simplest form?
5. Can you provide examples of dividing fractions using different methods?

**Tip:** When working with fractions, always simplify your results at each step to avoid overly complex numbers later in the calculation.

7/8 + 1/4 - (3/2 * 4/9) divided by (5/2 - 11/10 + 1/14 * 7/5)

This problem involves adding, subtracting, multiplying, and dividing fractions. The solution demonstrates step-by-step how to work with complex fractions by finding the least common denominator and simplifying results. Suitable for Grades 7-9.

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