Decoding and Evaluating the Mathematical Expression

The question asks us to find the value of a mathematical expression after replacing the standard operators with new ones based on a given set of rules. This type of problem tests our ability to follow instructions precisely and apply the correct order of operations.

Understanding the Operator Replacement Rules

Let's first list the given rules for replacing the operators:

• '+' means '÷'
• '-' means '+'
• '×' means '-'
• '÷' means ' × '

Rewriting the Expression with New Operators

The original expression is: ${[{(38 \times 23) - (4 \div 3)} + (6 - 3)] \div 2}$.

Now, we replace each operator according to the rules:

• The '×' inside the first parentheses `(38 × 23)` becomes '-'. So, $(38 \times 23)$ becomes $(38 - 23)$.
• The '÷' inside the second parentheses `(4 ÷ 3)` becomes '×'. So, $(4 \div 3)$ becomes $(4 \times 3)$.
• The '-' inside the third parentheses `(6 - 3)` becomes '+'. So, $(6 - 3)$ becomes $(6 + 3)$.
• The '-' operator between the first two parentheses `{... - ...}` becomes '+'. So, `{(38 × 23) - (4 ÷ 3)}` becomes `{(38 - 23) + (4 \times 3)}`.
• The '+' operator between the curly braces and the third parentheses `[...] + (...)` becomes '÷'. So, `{[{(38 × 23) - (4 ÷ 3)} + (6 - 3)]}` becomes `{[(38 - 23) + (4 \times 3)] \div (6 + 3)}\`. Note the original square brackets are still there.
• The '÷' operator at the very end `[...] ÷ 2` becomes '×'. So, the entire expression `{[{(38 × 23) - (4 ÷ 3)} + (6 - 3)] \div 2}` becomes ${[{(38 - 23) + (4 \times 3)} \div (6 + 3)] \times 2}$.

The rewritten expression is: ${[{(38 - 23) + (4 \times 3)} \div (6 + 3)] \times 2}$.

Evaluating the Rewritten Expression

We will now evaluate the expression following the order of operations (Parentheses/Brackets, Exponents, Multiplication and Division, Addition and Subtraction), working from the innermost parts outwards.

1. Evaluate the innermost parentheses:
   • $(38 - 23) = 15$
   • $(4 \times 3) = 12$
   • $(6 + 3) = 9$
2. Substitute these values back into the expression:

   ${[{(15) + (12)} \div (9)] \times 2}$

   ${[{15 + 12} \div 9] \times 2}$

3. Evaluate the expression inside the curly braces:

   ${15 + 12 = 27}$

4. Substitute this value back:

   ${[27 \div 9] \times 2}$

5. Evaluate the expression inside the square brackets (perform division first):

   ${27 \div 9 = 3}$

6. Substitute this value back:

   ${3 \times 2}$

7. Perform the final multiplication:

   ${3 \times 2 = 6}$

So, the value of the expression after replacing the operators is 6.

Step-by-Step Calculation Summary

Let's list the steps clearly:

1. Original Expression: ${[{(38 \times 23) - (4 \div 3)} + (6 - 3)] \div 2}$
2. After Operator Replacement: ${[{(38 - 23) + (4 \times 3)} \div (6 + 3)] \times 2}$
3. Innermost Parentheses: ${[{(15) + (12)} \div (9)] \times 2}$
4. Curly Braces: ${[27 \div 9] \times 2}$
5. Square Brackets: ${[3] \times 2}$
6. Final Calculation: ${3 \times 2 = 6}$

The final calculated value of the expression is 6.

Revision Table: Operator Swapping Rules

Here is a summary of how the operators were swapped:

Additional Information: Order of Operations

When evaluating mathematical expressions, it is crucial to follow the correct order of operations to ensure a unique and correct result. A common acronym used to remember this order is PEMDAS or BODMAS.

• PEMDAS:
  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)
• BODMAS:
  • Brackets
  • Orders (powers, square roots, etc.)
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)

In this problem, we primarily dealt with parentheses (and curly/square brackets), multiplication, division, addition, and subtraction, applying the operations within brackets first, then multiplication/division, and finally addition/subtraction, always working from left to right for operations at the same level.