Understanding the Mathematical Expression

The problem asks us to find the value of a given mathematical expression involving several operations like division, multiplication, 'of', and different types of brackets (parentheses, curly braces, and square brackets). To solve this correctly, we must follow the order of operations, often remembered using acronyms like BODMAS or PEMDAS.

Applying the Order of Operations (BODMAS/PEMDAS)

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

• Brackets (Parentheses, Curly braces, Square brackets) - Solve expressions inside brackets first, from innermost to outermost.
• Orders (Powers, Square roots, etc.) or Of (means multiplication, done before standard multiplication/division)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Let's break down the given expression: \(90 \div 20 \text{ of } 6 \times [11 \div 4 \text{ of } \{3 \times 2 - (3 - 8)\}] \div (9 \div 3 \times 2)\).

Step-by-Step Calculation of the Expression

Step 1: Solve the innermost bracket (Parentheses)

First, calculate the expression inside the parentheses: \( (3 - 8) \).

\( (3 - 8) = -5 \)

The expression inside the curly braces becomes \( \{3 \times 2 - (-5)\} \).

Step 2: Solve the expression inside Curly Braces

Next, solve the expression inside the curly braces: \( \{3 \times 2 - (-5)\} \).

• Perform multiplication first: \( 3 \times 2 = 6 \)
• The expression becomes \( \{6 - (-5)\} \)
• Subtraction with a negative number means addition: \( 6 - (-5) = 6 + 5 = 11 \)

The expression inside the square brackets becomes \( [11 \div 4 \text{ of } \{11\}] \), which simplifies to \( [11 \div 4 \text{ of } 11] \).

Step 3: Solve the 'of' operation inside Square Brackets

Next, calculate '4 of 11' inside the square brackets. 'Of' means multiplication and is done before division/multiplication outside brackets.

\( 4 \text{ of } 11 = 4 \times 11 = 44 \)

The expression inside the square brackets becomes \( [11 \div 44] \).

Step 4: Solve the expression inside Square Brackets

Now, perform the division inside the square brackets:

\( [11 \div 44] = [\frac{11}{44}] = [\frac{1}{4}] \)

Step 5: Solve the expression in the Denominator Part

Look at the last part of the main expression: \( (9 \div 3 \times 2) \). This is treated as a single unit divided by the rest of the expression.

• Perform division and multiplication from left to right.
• \( 9 \div 3 = 3 \)
• Then, \( 3 \times 2 = 6 \)

So, \( (9 \div 3 \times 2) = 6 \).

Step 6: Solve the 'of' operation in the main expression

Now, look at the first part of the main expression: \( 90 \div 20 \text{ of } 6 \). Calculate '20 of 6'.

\( 20 \text{ of } 6 = 20 \times 6 = 120 \)

The main expression is now simplified to: \( 90 \div 120 \times [\frac{1}{4}] \div 6 \).

Step 7: Perform Division and Multiplication from Left to Right

Finally, solve the remaining divisions and multiplications from left to right.

• First, \( 90 \div 120 \): \( \frac{90}{120} = \frac{9}{12} = \frac{3}{4} \)
• The expression is now \( \frac{3}{4} \times \frac{1}{4} \div 6 \)
• Next, \( \frac{3}{4} \times \frac{1}{4} \): \( \frac{3 \times 1}{4 \times 4} = \frac{3}{16} \)
• The expression is now \( \frac{3}{16} \div 6 \)
• Dividing by 6 is the same as multiplying by \( \frac{1}{6} \): \( \frac{3}{16} \div 6 = \frac{3}{16} \times \frac{1}{6} \)
• \( \frac{3}{16} \times \frac{1}{6} = \frac{3 \times 1}{16 \times 6} = \frac{3}{96} \)

Step 8: Simplify the final fraction

Simplify the fraction \( \frac{3}{96} \) by dividing the numerator and the denominator by their greatest common divisor, which is 3.

\( \frac{3 \div 3}{96 \div 3} = \frac{1}{32} \)

Final Answer

The value of the expression is \( \frac{1}{32} \).

Revision Table: Key Steps in BODMAS Calculation

Additional Information on Order of Operations

Understanding the order of operations is crucial for correctly evaluating mathematical expressions. Without a standard order, the same expression could yield different results. BODMAS (or PEMDAS) provides this standard. Remember that division and multiplication have the same priority and should be performed from left to right as they appear in the expression. Similarly, addition and subtraction have the same priority and are performed from left to right.

The term 'of' is often used in problems involving fractions or percentages (e.g., 'half of 10', '20% of 50'). In the context of BODMAS, 'of' signifies multiplication but is typically evaluated before any other multiplication or division operations present in the same level of the expression (after brackets and orders/exponents).