Understanding the Problem: Simplifying Arithmetic Expressions

The question asks us to find the value of a complex arithmetic expression involving division, multiplication, addition, and subtraction. To correctly solve this, we need to follow the standard order of operations, commonly known as BODMAS or PEMDAS.

Applying the BODMAS Rule

BODMAS is an acronym that helps remember the order of operations:

• B: Brackets (Parentheses)
• O: Of (Orders, Exponents, Roots)
• D: Division
• M: Multiplication
• A: Addition
• S: Subtraction

Operations of the same level (like Division and Multiplication, or Addition and Subtraction) are performed from left to right.

The expression is: $3 \div 18 \text{ of } 3 \times 6 + 21 \times 6 \div 18 - 3 \div 2 + 3 - 3 \div 9 \text{ of } 3 \times 9$

Step-by-Step Calculation using BODMAS

Step 1: Evaluate 'of' operations

The term 'of' signifies multiplication and has higher priority than regular multiplication and division in the BODMAS order.

• First 'of' term: $18 \text{ of } 3 = 18 \times 3 = 54$
• Second 'of' term: $9 \text{ of } 3 = 9 \times 3 = 27$

Substitute these values back into the expression:

Expression becomes: $3 \div 54 \times 6 + 21 \times 6 \div 18 - 3 \div 2 + 3 - 3 \div 27 \times 9$

Step 2: Perform Division and Multiplication (from left to right)

Let's evaluate each segment separated by addition/subtraction:

• Segment 1: $3 \div 54 \times 6$
  • $3 \div 54 = \frac{3}{54} = \frac{1}{18}$
  • $\frac{1}{18} \times 6 = \frac{6}{18} = \frac{1}{3}$
• Segment 2: $21 \times 6 \div 18$
  • $21 \times 6 = 126$
  • $126 \div 18 = 7$
• Segment 3: $3 \div 2 = \frac{3}{2}$
• Segment 4: $3 \div 27 \times 9$
  • $3 \div 27 = \frac{3}{27} = \frac{1}{9}$
  • $\frac{1}{9} \times 9 = 1$

Substitute these calculated values back into the expression:

Expression becomes: $\frac{1}{3} + 7 - \frac{3}{2} + 3 - 1$

Step 3: Perform Addition and Subtraction (from left to right)

Now, we combine the terms by performing addition and subtraction from left to right. It's helpful to work with a common denominator for the fractions.

Common denominator for $3$ and $2$ is $6$.

• $\frac{1}{3} + 7 = \frac{1}{3} + \frac{7 \times 3}{3} = \frac{1}{3} + \frac{21}{3} = \frac{1 + 21}{3} = \frac{22}{3}$
• $\frac{22}{3} - \frac{3}{2} = \frac{22 \times 2}{3 \times 2} - \frac{3 \times 3}{2 \times 3} = \frac{44}{6} - \frac{9}{6} = \frac{44 - 9}{6} = \frac{35}{6}$
• $\frac{35}{6} + 3 = \frac{35}{6} + \frac{3 \times 6}{6} = \frac{35}{6} + \frac{18}{6} = \frac{35 + 18}{6} = \frac{53}{6}$
• $\frac{53}{6} - 1 = \frac{53}{6} - \frac{1 \times 6}{6} = \frac{53}{6} - \frac{6}{6} = \frac{53 - 6}{6} = \frac{47}{6}$

Final Result

The value of the expression is $\frac{47}{6}$.

Revision Table: BODMAS Order

Additional Information: Common Pitfalls

When solving expressions like this, be careful with:

• The 'of' operation: Remember it's multiplication but done before division/multiplication.
• Left-to-right rule: For operations at the same level (like $\div$ and $\times$, or $+$ and $-$), always work from the left side of the expression to the right.
• Fraction arithmetic: Ensure you find a common denominator when adding or subtracting fractions.

Following BODMAS systematically ensures you arrive at the correct unique answer for any given arithmetic expression.