To evaluate the given mathematical expression, we need to follow the order of operations, often remembered by the acronym BODMAS or PEMDAS.

• Brackets (Parentheses)
• Orders (Exponents, Square Roots, etc.) or Of
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

The expression is: \(4 \div 12 \text{ of } [3 \div 4 \text{ of } \{(4 - 2) \times 6 \div 2\}] - 2 \times 6 \div 8 + 3\)

Step-by-Step Evaluation of the Expression

We will evaluate the expression by working from the innermost brackets outwards and following the BODMAS rule.

Evaluating the Innermost Brackets (Parentheses)

First, evaluate the expression inside the parentheses \((4 - 2)\):

\((4 - 2) = 2\)

The expression becomes: \(4 \div 12 \text{ of } [3 \div 4 \text{ of } \{2 \times 6 \div 2\}] - 2 \times 6 \div 8 + 3\)

Evaluating the Curly Brackets

Next, evaluate the expression inside the curly brackets \(\{2 \times 6 \div 2\}\). Inside these brackets, we have multiplication and division. We perform them from left to right:

• \(2 \times 6 = 12\)
• \(12 \div 2 = 6\)

So, \(\{2 \times 6 \div 2\} = 6\). The expression becomes: \(4 \div 12 \text{ of } [3 \div 4 \text{ of } 6] - 2 \times 6 \div 8 + 3\)

Evaluating the Square Brackets

Now, evaluate the expression inside the square brackets \([3 \div 4 \text{ of } 6]\). Inside these brackets, we have 'of' and division. The 'of' operator is evaluated before division:

• Evaluate \(4 \text{ of } 6\): \(4 \text{ of } 6 = 4 \times 6 = 24\)

The expression inside the brackets becomes \([3 \div 24]\). Now, perform the division:

• \(3 \div 24 = \frac{3}{24} = \frac{1}{8}\)

So, \([3 \div 4 \text{ of } 6] = \frac{1}{8}\). The expression becomes: \(4 \div 12 \text{ of } \frac{1}{8} - 2 \times 6 \div 8 + 3\)

Evaluating the Remaining 'of' Operation

Next, evaluate the 'of' operation outside the brackets: \(12 \text{ of } \frac{1}{8}\). This is multiplication:

\(12 \text{ of } \frac{1}{8} = 12 \times \frac{1}{8} = \frac{12}{8} = \frac{3}{2}\)

The expression is now: \(4 \div \frac{3}{2} - 2 \times 6 \div 8 + 3\)

Performing Division and Multiplication

Now, we perform division and multiplication from left to right.

• First term: \(4 \div \frac{3}{2}\). Dividing by a fraction is the same as multiplying by its reciprocal: \(4 \times \frac{2}{3} = \frac{8}{3}\).
• Second term: \(2 \times 6 \div 8\). Perform multiplication first, then division: \(2 \times 6 = 12\), then \(12 \div 8 = \frac{12}{8} = \frac{3}{2}\).

The expression is now: \(\frac{8}{3} - \frac{3}{2} + 3\)

Performing Addition and Subtraction

Finally, we perform addition and subtraction from left to right. To do this, we need a common denominator for the fractions \(\frac{8}{3}\) and \(\frac{3}{2}\), and the integer \(3\). The least common multiple of 3 and 2 is 6.

• Convert \(\frac{8}{3}\): \(\frac{8 \times 2}{3 \times 2} = \frac{16}{6}\)
• Convert \(\frac{3}{2}\): \(\frac{3 \times 3}{2 \times 3} = \frac{9}{6}\)
• Convert \(3\): \(3 = \frac{3}{1} = \frac{3 \times 6}{1 \times 6} = \frac{18}{6}\)

The expression becomes: \(\frac{16}{6} - \frac{9}{6} + \frac{18}{6}\)

Perform subtraction:

\(\frac{16}{6} - \frac{9}{6} = \frac{16 - 9}{6} = \frac{7}{6}\)

Perform addition:

\(\frac{7}{6} + \frac{18}{6} = \frac{7 + 18}{6} = \frac{25}{6}\)

Converting to a Mixed Number

The result \(\frac{25}{6}\) is an improper fraction. To convert it to a mixed number, divide 25 by 6:

\(25 \div 6\)

6 goes into 25 four times with a remainder of 1. So, \(25 = 4 \times 6 + 1\).

Therefore, \(\frac{25}{6} = 4 \frac{1}{6}\).

The value of the expression is \(4\frac{1}{6}\).

Revision Table: Understanding Order of Operations (BODMAS/PEMDAS)

Additional Information: Working with Fractions and Mixed Numbers

When evaluating expressions, you often end up working with fractions. Here are some key concepts:

• Proper Fraction: Numerator is less than the denominator (e.g., \(\frac{1}{2}\)).
• Improper Fraction: Numerator is greater than or equal to the denominator (e.g., \(\frac{5}{3}\)).
• Mixed Number: A whole number combined with a proper fraction (e.g., \(1 \frac{2}{3}\)).
• Converting Mixed Number to Improper Fraction: Multiply the whole number by the denominator and add the numerator. Place the result over the original denominator. Example: \(1 \frac{2}{3} = \frac{(1 \times 3) + 2}{3} = \frac{5}{3}\).
• Converting Improper Fraction to Mixed Number: Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator, and the denominator stays the same. Example: \(\frac{25}{6}\). \(25 \div 6 = 4\) with remainder 1. So, \(4 \frac{1}{6}\).
• Adding/Subtracting Fractions: Find a common denominator. Convert the fractions, then add or subtract the numerators.

Understanding how to work with these forms is crucial for solving many mathematical problems.