Solving Mathematical Expressions Using Order of Operations

The question asks us to find the value of the given mathematical expression: $40 \div 5 \text{ of } 2 \times [18 \div 6 \times (12 - 9) \text{ of } 5 - (3 - 8)] \div 25$.

To solve this expression correctly, we must follow the order of operations, often remembered using acronyms like BODMAS or PEDMAS.

• B/P: Brackets or Parentheses
• O/E: Order (powers, square roots, etc.) or Exponents
• D/M: Division and Multiplication (from left to right)
• A/S: Addition and Subtraction (from left to right)

In this expression, 'of' acts like multiplication but is typically evaluated after brackets/parentheses and before standard multiplication or division.

Let's break down the calculation step by step:

Given Expression: $40 \div 5 \text{ of } 2 \times [18 \div 6 \times (12 - 9) \text{ of } 5 - (3 - 8)] \div 25$

Step 1: Solve the operations inside the innermost Brackets (Parentheses).

• $(12 - 9) = 3$
• $(3 - 8) = -5$

The expression becomes: $40 \div 5 \text{ of } 2 \times [18 \div 6 \times 3 \text{ of } 5 - (-5)] \div 25$

Step 2: Evaluate the 'of' operations.

• $5 \text{ of } 2 = 5 \times 2 = 10$
• $3 \text{ of } 5 = 3 \times 5 = 15$

The expression is now: $40 \div 10 \times [18 \div 6 \times 15 - (-5)] \div 25$

Step 3: Solve the operations inside the Square Brackets. Follow BODMAS/PEDMAS within the brackets.

• Inside brackets: $[18 \div 6 \times 15 - (-5)]$
• Perform Division and Multiplication from left to right:
  • $18 \div 6 = 3$
  • The expression inside becomes: $[3 \times 15 - (-5)]$
  • $3 \times 15 = 45$
  • The expression inside becomes: $[45 - (-5)]$
• Perform Subtraction:
  • $45 - (-5) = 45 + 5 = 50$

The value inside the square brackets is 50. The expression becomes: $40 \div 10 \times 50 \div 25$

Step 4: Solve the remaining operations outside the brackets. Follow BODMAS/PEDMAS from left to right.

• We have Division and Multiplication. Perform them from left to right.
• $40 \div 10 = 4$
• The expression becomes: $4 \times 50 \div 25$
• $4 \times 50 = 200$
• The expression becomes: $200 \div 25$
• $200 \div 25 = 8$

The final value of the expression is 8.

Let's summarize the steps in a table:

The value of the expression is 8.

Revision Table: Order of Mathematical Operations

Additional Information: The Role of 'of' in Expressions

The term 'of' in mathematical expressions is crucial for understanding the correct order of operations. It represents multiplication but is typically performed at the 'Orders' or 'Exponents' stage of the BODMAS/PEDMAS rule, meaning it takes precedence over standard multiplication and division.

For example, in the expression $10 \div 2 \text{ of } 5$:

• If we treated 'of' like regular multiplication and followed left-to-right for $\div$ and $\times$: $(10 \div 2) \times 5 = 5 \times 5 = 25$.
• However, following the correct order where 'of' comes before $\div$ and $\times$: $10 \div (2 \text{ of } 5) = 10 \div (2 \times 5) = 10 \div 10 = 1$.

As shown in our main problem, we calculated $5 \text{ of } 2$ first (as 10) and $3 \text{ of } 5$ first (as 15) before performing the divisions or other multiplications at the same level.

Understanding the correct hierarchy of operations, including the specific placement of 'of', is essential for accurately solving complex mathematical expressions.