Understanding the Question: Simplifying Expressions

The question asks us to simplify a mathematical expression involving several operations: division ($\div$), 'of', multiplication ($\times$), addition ($+$), and subtraction ($-$)

Applying the Order of Operations (BODMAS/PEMDAS)

The order of operations dictates the sequence in which calculations should be performed:

1. Brackets (Parentheses)
2. Of (Orders/Exponents/Indices) - 'of' is essentially multiplication but is done before standard division and multiplication.
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

Let's break down the given expression step-by-step:

The expression is: \(15 \div 3 \text{ of } 2 \times 4 + 9 \div 18 \text{ of } 2 \times 3 - 4 \div 8 \times 2\)

Step 1: Handle the 'of' operations

According to BODMAS, 'of' comes before division and multiplication. We have two 'of' terms:

• \(3 \text{ of } 2 = 3 \times 2 = 6\)
• \(18 \text{ of } 2 = 18 \times 2 = 36\)

Substitute these values back into the expression:

\(15 \div 6 \times 4 + 9 \div 36 \times 3 - 4 \div 8 \times 2\)

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

Now, we work through the expression from left to right, performing all division and multiplication operations.

Let's evaluate each term separated by addition or subtraction:

• First term: \(15 \div 6 \times 4\)
  Perform division first: \(15 \div 6 = \frac{15}{6} = \frac{5}{2}\)
  Now multiply: \(\frac{5}{2} \times 4 = \frac{5 \times 4}{2} = \frac{20}{2} = 10\)
  So, the first term simplifies to 10.
• Second term: \(9 \div 36 \times 3\)
  Perform division first: \(9 \div 36 = \frac{9}{36} = \frac{1}{4}\)
  Now multiply: \(\frac{1}{4} \times 3 = \frac{1 \times 3}{4} = \frac{3}{4}\)
  So, the second term simplifies to \(\frac{3}{4}\).
• Third term: \(4 \div 8 \times 2\)
  Perform division first: \(4 \div 8 = \frac{4}{8} = \frac{1}{2}\)
  Now multiply: \(\frac{1}{2} \times 2 = \frac{1 \times 2}{2} = \frac{2}{2} = 1\)
  So, the third term simplifies to 1.

Substitute these simplified terms back into the expression:

\(10 + \frac{3}{4} - 1\)

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

Finally, perform the addition and subtraction from left to right.

• Add 10 and \(\frac{3}{4}\): \(10 + \frac{3}{4} = 10\frac{3}{4}\)
• Subtract 1 from \(10\frac{3}{4}\): \(10\frac{3}{4} - 1 = 9\frac{3}{4}\)

Final Simplified Value

The simplified value of the expression is \(9\frac{3}{4}\).

Let's summarise the calculation process:

The final result obtained is \(9\frac{3}{4}\).

Revision Table: Key Concepts in Simplifying Expressions

Additional Information: Common Mistakes in Simplifying Expressions

When simplifying expressions like this, students often make mistakes by not strictly following the order of operations. Here are some common pitfalls:

• Ignoring 'of': Treating 'of' the same as regular multiplication and performing it after division instead of before.
• Incorrect Left-to-Right: Not performing division and multiplication (or addition and subtraction) strictly from left to right when they appear consecutively. For example, calculating \(4 \times 2 \div 8\) instead of \(4 \div 8 \times 2\) if multiplication appeared before division.
• Mixing Operations: Performing addition or subtraction before completing all division and multiplication operations.
• Fraction Errors: Making arithmetic errors when dealing with fractions resulting from division.

Always writing down each step clearly helps in avoiding these errors and ensures accuracy in simplifying complex expressions.