Evaluating Mathematical Expressions with BODMAS

To evaluate a mathematical expression like the one given, we need to follow a specific order of operations. This order is often remembered using the acronyms BODMAS or PEMDAS.

• B/P: Brackets (Parentheses) - Evaluate expressions inside brackets first, starting from the innermost ones.
• O/E: Orders (Exponents) - Evaluate powers and square roots.
• D/M: Division and Multiplication - Perform division and multiplication from left to right.
• A/S: Addition and Subtraction - Perform addition and subtraction from left to right.

The 'of' operator is sometimes included in BODMAS, and it usually means multiplication, performed after brackets but sometimes before division/multiplication in the D/M step, specifically when it appears in a structure like 'A of B' where A is the result of evaluating a bracket or a number immediately preceding 'of'. In the given expression, we have '[...]' followed by 'of 12'. This means the result of the expression inside the square brackets will be multiplied by 12, and this multiplication happens before the division outside the square brackets.

Step-by-Step Evaluation of the Expression

Let's evaluate the expression step-by-step:

The expression is: \(18 \div [26 - \{25 - (15 - 5) \div 2\}] \text{ of } 12 + 2 - 2 \div 4 \times 16\)

Step 1: Evaluate the innermost brackets.

\((15 - 5) = 10\)

The expression becomes: \(18 \div [26 - \{25 - 10 \div 2\}] \text{ of } 12 + 2 - 2 \div 4 \times 16\)

Step 2: Evaluate the expression inside the curly braces.

Inside the curly braces, we have \(\{25 - 10 \div 2\}\). Following BODMAS, perform the division first:

\(10 \div 2 = 5\)

Now, perform the subtraction inside the curly braces:

\(\{25 - 5\} = 20\)

The expression becomes: \(18 \div [26 - 20] \text{ of } 12 + 2 - 2 \div 4 \times 16\)

Step 3: Evaluate the expression inside the square brackets.

Inside the square brackets, we have \([26 - 20]\):

\([26 - 20] = 6\)

The expression becomes: \(18 \div 6 \text{ of } 12 + 2 - 2 \div 4 \times 16\)

Step 4: Evaluate the 'of' operation.

We have \(6 \text{ of } 12\), which means \(6 \times 12\). This multiplication takes precedence over the division outside the brackets.

\(6 \text{ of } 12 = 6 \times 12 = 72\)

The expression becomes: \(18 \div 72 + 2 - 2 \div 4 \times 16\)

Step 5: Perform Division and Multiplication from left to right.

First division: \(18 \div 72 = \frac{18}{72} = \frac{1}{4}\)

The expression is now: \(\frac{1}{4} + 2 - 2 \div 4 \times 16\)

Next division: \(2 \div 4 = \frac{2}{4} = \frac{1}{2}\)

The expression is now: \(\frac{1}{4} + 2 - \frac{1}{2} \times 16\)

Next multiplication: \(\frac{1}{2} \times 16 = \frac{16}{2} = 8\)

The expression is now: \(\frac{1}{4} + 2 - 8\)

Step 6: Perform Addition and Subtraction from left to right.

First addition: \(\frac{1}{4} + 2 = \frac{1}{4} + \frac{2 \times 4}{4} = \frac{1}{4} + \frac{8}{4} = \frac{1 + 8}{4} = \frac{9}{4}\)

The expression is now: \(\frac{9}{4} - 8\)

Finally, subtraction: \(\frac{9}{4} - 8 = \frac{9}{4} - \frac{8 \times 4}{4} = \frac{9}{4} - \frac{32}{4} = \frac{9 - 32}{4} = \frac{-23}{4}\)

The final value of the expression is \(-\frac{23}{4}\).

Revision Table: Key BODMAS Concepts

Additional Information on Order of Operations

Understanding the correct order of operations is crucial for solving mathematical expressions accurately. Without a standard order, the same expression could yield different results depending on which operation is performed first. BODMAS (or PEMDAS) provides this standard.

It's important to remember that division and multiplication have equal priority and should be performed from left to right as they appear in the expression. Similarly, addition and subtraction have equal priority and are performed from left to right.

The 'of' operator is a specific case, often encountered in problems from certain curricula. While it signifies multiplication, its place in the order is generally considered after brackets are resolved but before the general division and multiplication sequence begins. Always handle expressions within brackets completely before moving outside.