The question asks us to find the value of the mathematical expression: \(30 \div 6 \times 5 \text{ of } (2 + 3) - 12(3 \times 2)\). To solve this, we need to follow the order of operations, commonly known as BODMAS or PEMDAS.

Applying the Order of Operations (BODMAS/PEMDAS)

The BODMAS rule tells us the sequence in which operations should be performed:

• Brackets (Parentheses)
• Orders (Exponents, Roots) / Of
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Let's break down the expression step by step:

The expression is: \(30 \div 6 \times 5 \text{ of } (2 + 3) - 12(3 \times 2)\)

Step-by-Step Evaluation

Step 1: Solve the expressions inside the Brackets (Parentheses).

• The first bracket is \((2 + 3)\). Calculating this gives \(2 + 3 = 5\).
• The second bracket is \((3 \times 2)\). Calculating this gives \(3 \times 2 = 6\).

The expression now becomes: \(30 \div 6 \times 5 \text{ of } 5 - 12(6)\)

Step 2: Evaluate 'Of'.

• 'Of' represents multiplication. So, \(5 \text{ of } 5\) means \(5 \times 5\).
• Calculating this gives \(5 \times 5 = 25\).

The term \(12(6)\) also implies multiplication, which is \(12 \times 6\). While usually done after 'Of', it can be treated as multiplication alongside Division and Multiplication in the next step, following the left-to-right rule, or simplified now.

Let's rewrite the expression after step 2: \(30 \div 6 \times 25 - 12 \times 6\)

Step 3: Perform Division and Multiplication from left to right.

• First, Division: \(30 \div 6 = 5\).

The expression is now: \(5 \times 25 - 12 \times 6\)

• Next, Multiplication from left to right:
• \(5 \times 25 = 125\).
• \(12 \times 6 = 72\).

The expression is now: \(125 - 72\)

Step 4: Perform Addition and Subtraction from left to right.

• We have a single subtraction operation: \(125 - 72\).
• Calculating this gives \(125 - 72 = 53\).

Final Value of the Expression

Following the order of operations, the value of the expression \(30 \div 6 \times 5 \text{ of } (2 + 3) - 12(3 \times 2)\) is 53.

Revision Table: Understanding Order of Operations

Additional Information on Mathematical Expressions

The order of operations is crucial to ensure that everyone gets the same answer when evaluating a mathematical expression. Without a standard order, different interpretations could lead to different results.

Implied multiplication, like \(12(6)\) or \(5(5)\), is treated the same as explicit multiplication (\(12 \times 6\), \(5 \times 5\)). In BODMAS, 'Of' often refers to fractions, percentages, or ratios applied to a number (e.g., "50% of 100"), but in simpler contexts like this problem, it behaves like multiplication \(5 \text{ of } 5 = 5 \times 5\).