Understanding the Mathematical Operation Rules

The question provides a mathematical expression and a set of rules that redefine the meaning of standard mathematical operators. We need to carefully substitute the standard operators in the given expression with their new meanings according to the rules and then evaluate the expression using the correct order of operations.

Step-by-Step Evaluation with Changed Operators

Let's break down the problem:

1. Identify the given expression: $23 \minus 31 + 135 \times 15 \divide 16$
2. Identify the new meanings of the operators:
   • '+' means 'subtraction' ($\minus$)
   • '$\minus$' means 'multiplication' ($\times$)
   • '$\divide$' means 'addition' ($+$)
   • '$\times$' means 'division' ($\divide$)
3. Substitute the operators in the original expression with their new meanings:
   Original: $23 \minus 31 + 135 \times 15 \divide 16$
   Substituting the rules:
   • The first operator is '$\minus$', which means 'multiplication' ($\times$). So, $23 \minus 31$ becomes $23 \times 31$.
   • The second operator is '+', which means 'subtraction' ($\minus$). So, $... 31 + 135 ...$ becomes $... 31 \minus 135 ...$.
   • The third operator is '$\times$', which means 'division' ($\divide$). So, $... 135 \times 15 ...$ becomes $... 135 \divide 15 ...$.
   • The fourth operator is '$\divide$', which means 'addition' ($+$). So, $... 15 \divide 16$ becomes $... 15 + 16$.
   The new expression is: $23 \times 31 \minus 135 \divide 15 + 16$
4. Evaluate the new expression using the BODMAS/PEMDAS rule (Order of Operations):
   • Brackets / Parentheses
   • Orders (powers, square roots) / Exponents
   • Division and Multiplication (from left to right)
   • Addition and Subtraction (from left to right)
5. Apply BODMAS/PEMDAS to $23 \times 31 \minus 135 \divide 15 + 16$:
   • First, perform the division: $135 \divide 15 = 9$.
     The expression becomes: $23 \times 31 \minus 9 + 16$
   • Next, perform the multiplication: $23 \times 31$.
     $23 \times 31 = 23 \times (30 + 1) = 23 \times 30 + 23 \times 1 = 690 + 23 = 713$.
     The expression becomes: $713 \minus 9 + 16$
   • Now, perform addition and subtraction from left to right:
     $713 \minus 9 = 704$
     The expression becomes: $704 + 16$
   • Finally, perform the addition: $704 + 16 = 720$.

The value of the expression after substituting the new meanings of operators and evaluating is $720$.

Understanding Operator Precedence (BODMAS/PEMDAS)

The order of operations is crucial when evaluating mathematical expressions to ensure a consistent result. BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) are mnemonics used to remember this order.

Division and Multiplication have the same priority and are performed from left to right as they appear in the expression. Similarly, Addition and Subtraction have the same priority and are performed from left to right.

Revision Table: Operator Substitution

Additional Information: Solving Mathematical Expressions

Solving mathematical expressions with redefined operators is a common type of logical reasoning problem. The key steps are always:

• Carefully read the rules for operator substitution.
• Rewrite the entire expression substituting each original operator with its new meaning.
• Evaluate the new expression following the standard order of operations (BODMAS/PEMDAS).
• Double-check each step of the calculation to avoid errors.