Understanding the Problem: Evaluating a Symbolically Transformed Expression

The question asks us to find the value of a given mathematical expression after replacing the standard mathematical operators with new meanings based on the provided rules. We are given the following transformation rules:

• ‘<’ means ‘multiplication’ ($\times$)
• ‘×’ means ‘subtraction’ ($-$)
• ‘÷’ means ‘addition’ ($+$)
• ‘+’ means ‘division’ ($\div$)

The expression to evaluate is:

5 ÷ 3 < 2 + (9 + 3) × 2

Applying the Symbol Transformations

Now, let's replace the operators in the given expression according to the rules provided. It's important to apply the rules consistently throughout the expression, including inside parentheses.

Original expression: $5 \div 3 < 2 + (9 + 3) \times 2$

Applying the transformations:

• ‘÷’ becomes ‘+’
• ‘<’ becomes ‘×’
• ‘+’ becomes ‘÷’
• ‘×’ becomes ‘-’

The transformed expression becomes:

$5 + 3 \times 2 \div (9 \div 3) - 2$

Evaluating the Transformed Expression using BODMAS

To find the value of the transformed expression, we must follow the order of operations, commonly known as BODMAS or PEMDAS:

1. Brackets (or Parentheses)
2. Orders (or Exponents)
3. Division and Multiplication (from left to right)
4. Addition and Subtraction (from left to right)

Let's evaluate the expression step by step:

Transformed expression: $5 + 3 \times 2 \div (9 \div 3) - 2$

Step 1: Solve the expression inside the Brackets (Parentheses)

The expression inside the bracket is $(9 \div 3)$.

$(9 \div 3) = 3$

Now, substitute this value back into the expression:

$5 + 3 \times 2 \div 3 - 2$

Step 2: Perform Division and Multiplication from left to right

We have $3 \times 2 \div 3$. We perform multiplication first as it appears first from the left.

$3 \times 2 = 6$

Substitute this value:

$5 + 6 \div 3 - 2$

Now, perform the division:

$6 \div 3 = 2$

Substitute this value:

$5 + 2 - 2$

Step 3: Perform Addition and Subtraction from left to right

We have $5 + 2 - 2$. Perform addition first.

$5 + 2 = 7$

Substitute this value:

$7 - 2$

Finally, perform the subtraction:

$7 - 2 = 5$

So, the value of the given expression after applying the symbol transformations is 5.

Final Answer

The value of the given expression after applying the symbol transformation rules is 5.

Revision Table: Symbol Substitution and Evaluation

Here's a summary of the process:

• Identify the symbol transformation rules.
• Rewrite the original expression by replacing each symbol with its new meaning.
• Evaluate the new expression step-by-step following the BODMAS/PEMDAS order of operations: Brackets, Orders, Division/Multiplication (left to right), Addition/Subtraction (left to right).

Additional Information: Order of Operations Explained

The order of operations is crucial in evaluating mathematical expressions to ensure a unique and correct result. BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction) and PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) are mnemonics used to remember this order. Division and Multiplication have equal priority and are performed from left to right as they appear. Similarly, Addition and Subtraction have equal priority and are performed from left to right as they appear.

In this specific problem, applying the transformations correctly and then strictly following the BODMAS rules for the transformed expression was key to finding the value.