To solve the given mathematical expression, we need to follow the order of operations, often remembered by acronyms like BODMAS or PEMDAS.

The expression is:

$\qquad 45 – [30 – \{60 \div 3 – (6 – 9 \div 3) – 3\}]$

Understanding the Order of Operations (BODMAS/PEMDAS)

BODMAS stands for:

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

We solve the operations starting from the innermost brackets and work our way outwards.

Step-by-Step Solution Process

Let's solve the expression step by step following the BODMAS rule:

Step 1: Innermost Brackets (Parentheses)

First, we evaluate the expression inside the innermost parentheses: $(6 – 9 \div 3)$.

Inside these parentheses, we perform division before subtraction.

• Calculate $9 \div 3$:
• $9 \div 3 = 3$

Now substitute this back into the parentheses:

• $6 – 3 = 3$

The expression now becomes:

$\qquad 45 – [30 – \{60 \div 3 – 3 – 3\}]$

Step 2: Curly Braces

Next, we evaluate the expression inside the curly braces: $\{60 \div 3 – 3 – 3\}$.

Inside these curly braces, we perform division first, then subtractions from left to right.

• Calculate $60 \div 3$:
• $60 \div 3 = 20$

Substitute this back into the curly braces:

• $\{20 – 3 – 3\}$

Now perform the subtractions from left to right:

• $20 – 3 = 17$
• $17 – 3 = 14$

The expression now becomes:

$\qquad 45 – [30 – 14]$

Step 3: Square Brackets

Now, we evaluate the expression inside the square brackets: $[30 – 14]$.

• Calculate $30 – 14$:
• $30 – 14 = 16$

The expression now becomes:

$\qquad 45 – 16$

Step 4: Final Subtraction

Finally, we perform the remaining subtraction:

• $45 – 16 = 29$

Wait, let me recheck the calculation in the curly braces. $\{60 \div 3 – (6 – 9 \div 3) \div 3\}$ We found $(6 – 9 \div 3) = 3$. So the curly braces content is $\{60 \div 3 – 3 \div 3\}$. Division first: $60 \div 3 = 20$ $3 \div 3 = 1$ So the curly braces content is $\{20 - 1\}$. $\{20 - 1\} = 19$. The expression is $45 – [30 – \{60 \div 3 – (6 – 9 \div 3) \div 3\}]$. Substituting the inner part: $45 – [30 – \{60 \div 3 – (3) \div 3\}]$. Inside curly braces: $\{60 \div 3 – 3 \div 3\}$. $60 \div 3 = 20$. $3 \div 3 = 1$. So curly braces are $\{20 – 1\} = 19$. The expression is $45 – [30 – 19]$. Inside square brackets: $[30 – 19] = 11$. The expression is $45 – 11$. $45 - 11 = 34$.

My previous calculation for the curly braces was incorrect. Let's correct it.

Step 2 (Corrected): Curly Braces

Evaluate $\{60 \div 3 – (6 – 9 \div 3) \div 3\}$. We already found $(6 – 9 \div 3) = 3$.

Substitute this back:

$\{60 \div 3 – 3 \div 3\}$

Perform divisions from left to right:

• $60 \div 3 = 20$
• $3 \div 3 = 1$

Now perform the subtraction:

• $\{20 – 1\} = 19$

The expression now becomes:

$\qquad 45 – [30 – 19]$

Step 3 (Corrected): Square Brackets

Evaluate $[30 – 19]$.

• $30 – 19 = 11$

The expression now becomes:

$\qquad 45 – 11$

Step 4 (Corrected): Final Subtraction

Perform the final subtraction:

• $45 – 11 = 34$

Final Result

The value of the expression $45 – [30 – \{60 \div 3 – (6 – 9 \div 3) – 3\}]$ is 34.

Revision Table: Summary of BODMAS Steps

Additional Information on Order of Operations

The order of operations is a standard convention used in mathematics to ensure that expressions are evaluated consistently, leading to a unique result. Without this order, the same expression could be interpreted in multiple ways, yielding different answers. BODMAS (or PEMDAS) provides a clear hierarchy of operations to follow. Remember that division and multiplication have the same priority and are performed from left to right as they appear. Similarly, addition and subtraction have the same priority and are performed from left to right.