Understanding the BODMAS Rule for Calculations

To solve the given arithmetic expression, we must follow the BODMAS (or PEMDAS) rule. This rule dictates the correct order of operations to ensure a unique and accurate result for any given mathematical expression.

The BODMAS rule stands for:

• Brackets (Parentheses)
• Of (Orders, Exponents, roots)
• 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: \(16 \div 4 \text{ of } 4 \times [3 \div 4 \text{ of } \{4 \times 3 \div (3 + 3)\}] \div (2 \div 4 \text{ of } 8)\)

Step-by-Step Calculation using BODMAS

Step 1: Solve the innermost Brackets (Parentheses)

First, we evaluate the expression inside the innermost parentheses: \((3 + 3)\)

\((3 + 3) = 6\)

The expression now becomes: \(16 \div 4 \text{ of } 4 \times [3 \div 4 \text{ of } \{4 \times 3 \div 6\}] \div (2 \div 4 \text{ of } 8)\)

Step 2: Solve the innermost Brackets (Braces)

Next, evaluate the expression inside the braces: \(\{4 \times 3 \div 6\}\)

Within the braces, we have multiplication and division. According to BODMAS, we perform these from left to right.

• First, multiplication: \(4 \times 3 = 12\)
• Then, division: \(12 \div 6 = 2\)

So, \(\{4 \times 3 \div 6\} = 2\)

The expression now becomes: \(16 \div 4 \text{ of } 4 \times [3 \div 4 \text{ of } 2] \div (2 \div 4 \text{ of } 8)\)

Step 3: Solve the next set of Brackets (Square Brackets) and perform 'of' operations inside remaining brackets

Let's evaluate the expression inside the square brackets: \([3 \div 4 \text{ of } 2]\)

'Of' operation comes before division. \(4 \text{ of } 2\) means \(4 \times 2\).

\(4 \text{ of } 2 = 4 \times 2 = 8\)

So, \([3 \div 4 \text{ of } 2]\) becomes \([3 \div 8]\).

Now, let's look at the other set of parentheses: \((2 \div 4 \text{ of } 8)\)

Again, perform 'of' operation first. \(4 \text{ of } 8\) means \(4 \times 8\).

\(4 \text{ of } 8 = 4 \times 8 = 32\)

So, \((2 \div 4 \text{ of } 8)\) becomes \((2 \div 32)\).

The expression now becomes: \(16 \div 4 \text{ of } 4 \times [3 \div 8] \div (2 \div 32)\)

Step 4: Perform 'of' operation outside brackets

Now, perform the 'of' operation outside the brackets: \(4 \text{ of } 4\)

\(4 \text{ of } 4 = 4 \times 4 = 16\)

The expression is now simplified to: \(16 \div 16 \times [3 \div 8] \div (2 \div 32)\)

Step 5: Perform Division and Multiplication from left to right

The expression is: \(16 \div 16 \times (3/8) \div (2/32)\)

Remember that \(2 \div 32\) can be simplified: \(2 \div 32 = 2/32 = 1/16\)

The expression is now: \(16 \div 16 \times (3/8) \div (1/16)\)

Perform operations from left to right:

• First, division: \(16 \div 16 = 1\)
• Next, multiplication: \(1 \times (3/8) = 3/8\)
• Finally, division: \((3/8) \div (1/16)\)

Dividing by a fraction is the same as multiplying by its reciprocal:

\((3/8) \div (1/16) = (3/8) \times (16/1)\)

Multiply the fractions:

\((3/8) \times (16/1) = (3 \times 16) / (8 \times 1) = 48 / 8\)

Perform the final division:

\(48 \div 8 = 6\)

Thus, the value of the expression is 6.

Let's summarize the steps and results in a table:

The final value of the expression is 6.

Revision Table: Key Concepts

Additional Information: BODMAS vs. PEMDAS

The BODMAS rule is also known as PEMDAS in some regions. The acronyms are slightly different but represent the same order of operations:

• Parentheses (equivalent to Brackets)
• Exponents (equivalent to Orders/Of, typically covering powers and roots)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

The key difference is that 'Of' explicitly includes operations like percentages or fractions of a number, which often translate to multiplication, and it is performed before standard multiplication and division in BODMAS. PEMDAS groups 'Exponents' which covers powers and roots, and then standard multiplication/division. In practice, for expressions like the one solved, 'of' is treated as multiplication but is performed before division or multiplication that is explicitly written as \(\times\) or \(\div\) at the same level outside brackets. The left-to-right rule for division/multiplication and addition/subtraction is crucial in both systems.