Understanding the Mathematical Expression

We are asked to evaluate the mathematical expression: $18 \div \{(6 \text{ of } 2 - 4)\} \times 5(6 - 3)$.

This expression involves various operations like division, multiplication, subtraction, and includes brackets (both parentheses and curly braces). To correctly solve this, we must follow the order of operations.

Applying the Order of Operations (BODMAS/PEMDAS)

The order of operations dictates the sequence in which we perform calculations in an expression. A common acronym to remember this order is BODMAS or PEMDAS:

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

Also, the term 'of' in mathematics usually means multiplication.

Step-by-Step Calculation of the Expression

Let's break down the expression and solve it step by step according to the order of operations:

The expression is: $18 \div \{(6 \text{ of } 2 - 4)\} \times 5(6 - 3)$

1. Solve operations inside the innermost brackets/parentheses:
   $(6 - 3) = 3$
   The expression becomes: $18 \div \{(6 \text{ of } 2 - 4)\} \times 5(3)$
2. Solve 'of' operation inside the curly braces:
   $6 \text{ of } 2 = 6 \times 2 = 12$
   The expression becomes: $18 \div \{(12 - 4)\} \times 5(3)$
3. Solve operations inside the curly braces:
   $(12 - 4) = 8$
   The expression becomes: $18 \div \{8\} \times 5(3)$
   This can be written as: $18 \div 8 \times 5 \times 3$
4. Perform Division and Multiplication from left to right:
   First, perform the division:
   $18 \div 8 = \frac{18}{8}$
   The expression becomes: $\frac{18}{8} \times 5 \times 3$
5. Perform the remaining multiplications:
   $\frac{18}{8} \times 5 \times 3 = \frac{18 \times 5 \times 3}{8}$
   Calculate the numerator: $18 \times 5 = 90$, and $90 \times 3 = 270$.
   The expression becomes: $\frac{270}{8}$
6. Simplify the resulting fraction:
   The fraction is $\frac{270}{8}$. Both the numerator and the denominator are divisible by 2.
   Divide both by 2:
   $270 \div 2 = 135$
   $8 \div 2 = 4$
   The simplified fraction is $\frac{135}{4}$.

Thus, the value of the expression is $\frac{135}{4}$.

Final Answer Calculation

The step-by-step evaluation results in the value $\frac{135}{4}$.

Revision Table: Key Steps in Solving

Additional Information on Order of Operations

The order of operations is fundamental in mathematics to ensure consistent results when evaluating expressions. Without a standard order, expressions involving multiple operations could yield different answers depending on which operation is performed first.

• Parentheses/Brackets group parts of an expression, indicating that operations within them should be performed first.
• Exponents/Orders indicate powers or roots.
• Multiplication and Division have equal priority and are performed from left to right as they appear in the expression.
• Addition and Subtraction have equal priority and are performed from left to right as they appear in the expression.
• The term 'of' is often used in the context of fractions or percentages (e.g., 'half of 10' means $\frac{1}{2} \times 10$) and is treated as multiplication, usually performed before standard multiplication/division when encountered within brackets.

Mastering the order of operations is crucial for solving various mathematical problems correctly.