Evaluating Expressions with Symbol Substitutions

This problem requires us to first understand the new meanings assigned to the mathematical operators and then apply these rules to the given expression to find its numerical value. We will use the standard order of operations (BODMAS/PEMDAS) after the symbols have been substituted.

Understanding the Symbol Rules

The question provides a specific set of rules for substituting standard mathematical operators:

• The symbol '+' represents '\(\times\)' (multiplication).
• The symbol '-' represents '+' (addition).
• The symbol '\(\times\)' represents '\(\div\)' (division).
• The symbol '\(\div\)' represents '-' (subtraction).

Applying Substitutions to the Expression

The original expression given is:

\(\qquad 9 \times 3 + 6 \div 2\)

Now, we substitute the symbols in this expression according to the rules provided:

• The '\(\times\)' between 9 and 3 becomes '\(\div\)'.
• The '+' between 3 and 6 becomes '\(\times\)'.
• The '\(\div\)' between 6 and 2 becomes '-'.

So, the new expression after substitution is:

\(\qquad 9 \div 3 \times 6 - 2\)

Evaluating the Modified Expression using BODMAS/PEMDAS

We need to evaluate the expression \(9 \div 3 \times 6 - 2\) following the order of operations:

BODMAS/PEMDAS Order:

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

Let's evaluate step-by-step:

1. Division: Evaluate \(9 \div 3\).

   \(\qquad 9 \div 3 = 3\)

   The expression becomes: \(3 \times 6 - 2\).
2. Multiplication: Evaluate \(3 \times 6\).

   \(\qquad 3 \times 6 = 18\)

   The expression becomes: \(18 - 2\).
3. Subtraction: Evaluate \(18 - 2\).

   \(\qquad 18 - 2 = 16\)

The final value of the expression after applying the symbol substitutions and evaluating is 16.

Summary of Steps

Final Answer Evaluation

After performing the required symbol substitutions and following the order of operations, the value of the expression \(9 \times 3 + 6 \div 2\) (under the given rules) is 16.

Revision Table: Symbol Substitution and Expression Evaluation

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is a crucial rule in mathematics to ensure calculations are performed in a consistent sequence, leading to a unique correct answer. The acronyms BODMAS or PEMDAS help remember this order:

• Brackets / Parentheses: Operations inside brackets or parentheses are performed first.
• Orders / Exponents: Powers, square roots, and other orders are calculated next.
• Division and Multiplication: These are performed from left to right as they appear in the expression.
• Addition and Subtraction: These are performed last, also from left to right as they appear.

In the expression \(9 \div 3 \times 6 - 2\), division (\(9 \div 3\)) comes before multiplication (\(3 \times 6\)) because division appears first when reading from left to right among the division and multiplication operations. Then subtraction (\(18 - 2\)) is performed.