To find the value of the given expression, we need to follow the order of operations. A commonly used rule for the order of operations is BODMAS or PEMDAS.

BODMAS stands for:

• Brackets
• Order (powers, roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Both rules give the same result as they represent the same hierarchy of operations.

Evaluating the Expression

The given expression is:

\(\frac{{3 \div 1 \times 2 + 5 - 2}}{{3 \times 3 - 2}}\)

We will evaluate the numerator and the denominator separately.

Calculating the Numerator

The numerator is \(3 \div 1 \times 2 + 5 - 2\).

Applying the BODMAS/PEMDAS rule:

1. Division and Multiplication (from left to right):
   • First, \(3 \div 1\). \(3 \div 1 = 3\). The expression becomes \(3 \times 2 + 5 - 2\).
   • Next, \(3 \times 2\). \(3 \times 2 = 6\). The expression becomes \(6 + 5 - 2\).
2. Addition and Subtraction (from left to right):
   • First, \(6 + 5\). \(6 + 5 = 11\). The expression becomes \(11 - 2\).
   • Next, \(11 - 2\). \(11 - 2 = 9\).

So, the value of the numerator is 9.

Calculating the Denominator

The denominator is \(3 \times 3 - 2\).

Applying the BODMAS/PEMDAS rule:

1. Multiplication:
   • First, \(3 \times 3\). \(3 \times 3 = 9\). The expression becomes \(9 - 2\).
2. Subtraction:
   • Next, \(9 - 2\). \(9 - 2 = 7\).

So, the value of the denominator is 7.

Finding the Final Value

Now, we combine the values of the numerator and the denominator:

\(\frac{\text{Numerator}}{\text{Denominator}} = \frac{9}{7}\)

The value of the expression is \(\frac{9}{7}\).

Revision Table: Order of Operations

Additional Information: Applying BODMAS Correctly

It is crucial to apply the BODMAS or PEMDAS rule carefully. When you have a mix of division and multiplication or a mix of addition and subtraction, you must perform these operations from left to right as they appear in the expression. They have equal priority.

For example, in \(6 \div 2 \times 3\):

• Correct: \(6 \div 2 = 3\), then \(3 \times 3 = 9\).
• Incorrect: \(2 \times 3 = 6\), then \(6 \div 6 = 1\).

Similarly, in \(10 - 3 + 5\):

• Correct: \(10 - 3 = 7\), then \(7 + 5 = 12\).
• Incorrect: \(3 + 5 = 8\), then \(10 - 8 = 2\).

Always work from left to right for operations at the same priority level (Division/Multiplication or Addition/Subtraction).