Let's break down and solve the given mathematical expression step by step. The expression looks like this:

[ \frac{\left[9 - \frac{43}{8} \right] \times \left[ 4 \frac{5}{12} - 4 + 2 \frac{2}{3} + \left( 0.3 + 0.5 \div 4 \right) \times \frac{4}{7} \right]}{\frac{1}{25} + 0.25 \div 13 \frac{1}{3}} ]

Step 1: Simplify the numerator

Part 1: Simplifying $9 - \frac{43}{8}$

$9 - \frac{43}{8} = \frac{72}{8} - \frac{43}{8} = \frac{29}{8}$

Part 2: Simplifying $4 \frac{5}{12} - 4 + 2 \frac{2}{3} + (0.3 + 0.5 \div 4) \times \frac{4}{7}$

• $4 \frac{5}{12} = 4 + \frac{5}{12} = \frac{48}{12} + \frac{5}{12} = \frac{53}{12}$
• $2 \frac{2}{3} = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3}$
• $0.5 \div 4 = 0.125$
• $0.3 + 0.125 = 0.425$
• $0.425 \times \frac{4}{7} = \frac{1.7}{7}$

Now, combine all terms inside the second bracket: $\frac{53}{12} - 4 + \frac{8}{3} + \frac{1.7}{7}$

We can convert everything to a common denominator to simplify further, but first, let's simplify the terms inside.

Step 2: Simplify the denominator

The denominator is: $\frac{1}{25} + 0.25 \div 13 \frac{1}{3}$

We will now solve both the numerator and denominator and obtain the final result.The simplified result of the given mathematical expression is approximately $205.23$.

Would you like further details on any specific part of the solution or breakdown? Here are five related questions to expand your understanding:

1. How can we handle operations involving both fractions and decimals more effectively?
2. What strategies help simplify complex algebraic expressions step by step?
3. How do common denominators help in simplifying fraction operations?
4. What is the importance of parentheses and the order of operations in large expressions?
5. Can you explain how to convert mixed fractions to improper fractions for ease of calculation?

Tip: When working with complex expressions, it's often helpful to simplify one part at a time, combining like terms before performing division or multiplication.