Understanding the Problem

The question asks us to find the value of a mathematical expression: 45 - 3 × (4 of 6 + 12 ÷ 3 × 6 - 4 × 5) + 6. This expression involves several different arithmetic operations, including subtraction, multiplication, 'of', division, and addition. To correctly solve this, we need to follow the order of operations.

Applying the Order of Operations (BODMAS/PEMDAS)

The order of operations, commonly remembered by acronyms like BODMAS or PEMDAS, dictates the sequence in which operations should be performed:

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

Let's break down the given expression step by step:

The expression is: $$45 - 3 \times (4 \text{ of } 6 + 12 \div 3 \times 6 - 4 \times 5) + 6$$

Step 1: Solve the expression inside the Brackets

The expression inside the brackets is: $$(4 \text{ of } 6 + 12 \div 3 \times 6 - 4 \times 5)$$

Inside the brackets, we again follow BODMAS.

• First, calculate 'of': $$4 \text{ of } 6 = 4 \times 6 = 24$$

  The expression inside the brackets becomes: $$(24 + 12 \div 3 \times 6 - 4 \times 5)$$

• Next, perform Division and Multiplication from left to right:
  • $$12 \div 3 = 4$$

    The expression inside the brackets becomes: $$(24 + 4 \times 6 - 4 \times 5)$$

  • $$4 \times 6 = 24$$

    The expression inside the brackets becomes: $$(24 + 24 - 4 \times 5)$$

  • $$4 \times 5 = 20$$

    The expression inside the brackets becomes: $$(24 + 24 - 20)$$

• Finally, perform Addition and Subtraction from left to right inside the brackets:
  • $$24 + 24 = 48$$

    The expression inside the brackets becomes: $$(48 - 20)$$

  • $$48 - 20 = 28$$

    So, the value inside the brackets is 28.

Step 2: Substitute the Bracket value back into the main expression

The original expression now becomes: $$45 - 3 \times 28 + 6$$

Step 3: Perform Multiplication

Next, according to BODMAS, we perform multiplication:

• $$3 \times 28 = 84$$

  The expression becomes: $$45 - 84 + 6$$

Step 4: Perform Addition and Subtraction

Finally, perform addition and subtraction from left to right:

• $$45 - 84 = -39$$

  The expression becomes: $$-39 + 6$$

• $$-39 + 6 = -33$$

Final Value

The final value of the expression 45 - 3 × (4 of 6 + 12 ÷ 3 × 6 - 4 × 5) + 6 is -33.

Summary of Calculation Steps

Revision Table: Order of Operations

Additional Information: Understanding 'Of' in BODMAS

The term 'of' in BODMAS often represents multiplication, particularly when dealing with fractions or percentages (e.g., "half of 10" means 0.5 × 10). In expressions like "4 of 6", it is interpreted as 4 × 6. It typically ranks higher than standard multiplication and division in the order of operations, often done immediately after brackets and before division/multiplication.

Applying BODMAS carefully, especially the left-to-right rule for division/multiplication and addition/subtraction at their respective levels, is key to solving such mathematical expressions correctly.