Solving the Given Mathematical Equation

The question asks us to find the value that replaces the '?' in the given equation:

\(4 + \frac{1}{6} \times \left[ {\left\{ { - 12 \times \left( {24 - 12 - 4} \right)} \right\} \div \left( {20 - 4} \right)} \right] = \;?\)

To solve this equation, we need to follow the order of operations, often remembered by the acronyms BODMAS or PEMDAS.

Understanding the Order of Operations (BODMAS/PEMDAS)

This rule dictates the sequence in which mathematical operations should be performed:

• Brackets (or Parentheses)
• Orders (or Exponents)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Let's simplify the expression step-by-step according to this rule.

Step-by-Step Solution to the Equation

We start by simplifying the innermost brackets.

Step 1: Simplify the innermost Parentheses

The equation contains two sets of innermost parentheses:

• \((24 - 12 - 4)\)
• \((20 - 4)\)

Let's calculate their values:

• \((24 - 12 - 4) = (12 - 4) = 8\)
• \((20 - 4) = 16\)

Substitute these values back into the equation:

\(4 + \frac{1}{6} \times \left[ {\left\{ { - 12 \times 8} \right\} \div 16} \right]\)

Step 2: Simplify the Curly Braces

Next, we simplify the expression inside the curly braces, which is a multiplication:

• \({ - 12 \times 8 } = -96\)

Substitute this value back into the equation:

\(4 + \frac{1}{6} \times \left[ {-96 \div 16} \right]\)

Step 3: Simplify the Square Brackets

Now, simplify the expression inside the square brackets, which is a division:

• \({-96 \div 16} = -6\)

Substitute this value back into the equation:

\(4 + \frac{1}{6} \times [-6]\)

Step 4: Perform the Multiplication

According to BODMAS, multiplication comes before addition. Perform the multiplication:

• \(\frac{1}{6} \times -6 = -\frac{6}{6} = -1\)

Substitute this value back into the equation:

\(4 + (-1)\)

Step 5: Perform the Addition

Finally, perform the addition:

• \(4 + (-1) = 4 - 1 = 3\)

Final Result

The value that comes in place of '?' is 3.

The simplified equation is:

\(4 + \frac{1}{6} \times \left[ {\left\{ { - 12 \times \left( {24 - 12 - 4} \right)} \right\} \div \left( {20 - 4} \right)} \right] = 3\)

Revision Table: Solving Equations with BODMAS

Additional Information: Importance of Order of Operations

The order of operations is fundamental in mathematics. Without a standard order, a single expression could be interpreted in multiple ways, leading to different results. For example, \(2 + 3 \times 4\). If you perform addition first (\(2+3=5\), then \(5 \times 4 = 20\)), you get 20. If you perform multiplication first (\(3 \times 4 = 12\), then \(2 + 12 = 14\)), you get 14. BODMAS/PEMDAS tells us multiplication comes before addition, so 14 is the correct answer. Following this order is essential for solving equations and expressions accurately, whether in basic arithmetic, algebra, or more advanced mathematics.

Mastering the application of BODMAS/PEMDAS to expressions involving fractions, negative numbers, and multiple layers of brackets is key to success in solving such problems.