(
−
1
4
)
+
{
4
÷
(
4
7
×
2
5
)
+
7
−
[
9
+
1
4
+
1
8
]
}

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**.

---

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.

{(-1/4) + {4 ÷ (4/7 × 2/5) + 7 - [9 + 1/4 + 1/8]}}

Step-by-step solution to the expression (-1/4) + {4 ÷ (4/7 × 2/5) + 7 - [9 + 1/4 + 1/8]}. Learn how to simplify complex expressions with fractions, use order of operations, and apply the reciprocal rule. Suitable for students in Grades 6-8.

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