Let's solve the given expression step by step:

The expression is:

$\left(-\frac{1}{4}\right) + \left\{ \frac{4}{\left(\frac{4}{7} \times \frac{2}{5}\right)} + 7 - \left[9 + \frac{1}{4} + \frac{1}{8}\right] \right\}$

Step 1: Simplify inside the brackets [...]

The expression inside the brackets is: $9 + \frac{1}{4} + \frac{1}{8}$

To add these, we need a common denominator:

• $\frac{1}{4} = \frac{2}{8}$

So: $9 + \frac{2}{8} + \frac{1}{8} = 9 + \frac{3}{8}$

So the expression becomes: $\left(-\frac{1}{4}\right) + \left\{ \frac{4}{\left(\frac{4}{7} \times \frac{2}{5}\right)} + 7 - \left(9 + \frac{3}{8}\right) \right\}$

Step 2: Simplify inside the braces {...}

Step 2.1: Simplify $\frac{4}{7} \times \frac{2}{5}$

$\frac{4}{7} \times \frac{2}{5} = \frac{8}{35}$

So the expression becomes: $\left(-\frac{1}{4}\right) + \left\{ \frac{4}{\left(\frac{8}{35}\right)} + 7 - \left(9 + \frac{3}{8}\right) \right\}$

Step 2.2: Simplify $\frac{4}{\frac{8}{35}}$

Dividing by a fraction is the same as multiplying by its reciprocal: $\frac{4}{\frac{8}{35}} = 4 \times \frac{35}{8} = \frac{140}{8} = 17.5$

Now, the expression is: $\left(-\frac{1}{4}\right) + \left\{ 17.5 + 7 - \left(9 + \frac{3}{8}\right) \right\}$

Step 2.3: Simplify $17.5 + 7$

$17.5 + 7 = 24.5$

So we have: $\left(-\frac{1}{4}\right) + \left\{ 24.5 - \left(9 + \frac{3}{8}\right) \right\}$

Step 2.4: Simplify $9 + \frac{3}{8}$

Convert $9$ into a fraction with denominator 8: $9 = \frac{72}{8}$ $\frac{72}{8} + \frac{3}{8} = \frac{75}{8} = 9.375$

So: $\left(-\frac{1}{4}\right) + \left\{ 24.5 - 9.375 \right\}$

Step 3: Simplify the expression inside the braces {...}

$24.5 - 9.375 = 15.125$

So the expression becomes: $\left(-\frac{1}{4}\right) + 15.125$

Step 4: Final Calculation

Convert $15.125$ to a fraction: $15.125 = \frac{121}{8}$ Now: $-\frac{1}{4} + \frac{121}{8}$

Convert $-\frac{1}{4}$ to have the same denominator: $-\frac{1}{4} = -\frac{2}{8}$

$-\frac{2}{8} + \frac{121}{8} = \frac{119}{8} = 14.875$

Thus, the final result is 14.875.

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Would you like more details or have any questions?

Here are 5 related questions to further explore:

1. How would you simplify an expression with multiple fractions?
2. What is the process of finding a common denominator?
3. How do you handle operations within different levels of brackets and braces?
4. Can you explain why dividing by a fraction is equivalent to multiplying by its reciprocal?
5. How can you convert a mixed number into an improper fraction?

Tip: Always solve expressions by carefully simplifying each part step by step, starting from the innermost brackets.