Understanding Symbol Substitution in Mathematical Expressions

This question requires us to evaluate a mathematical expression after changing the meaning of the standard mathematical operators according to a given set of rules. This type of problem tests our ability to carefully follow instructions and apply the correct order of operations.

Decoding the Symbol Rules

The question provides the following rules for substituting the standard symbols:

• '+' means 'divided by' (\(\div\))
• '-' means 'add' (\(+\))
• '×' means 'minus' (\(-\))
• '÷' means 'multiplied by' (\(\times\))

Rewriting the Expression with New Meanings

The original expression is:

[{(18 × 16) - (5 ÷ 2)} + (5 - 1)] ÷ 1

Now, let's replace the symbols based on the given rules:

• '×' becomes '-'
• '÷' becomes '×'
• '-' becomes '+'
• '+' becomes '÷'

Substituting these into the expression, we get:

[{(18 - 16) + (5 × 2)} ÷ (5 + 1)] × 1

Evaluating the Substituted Expression

Now we need to evaluate the new expression following the standard order of operations (often remembered by acronyms like BODMAS or PEMDAS):

• Brackets (Parentheses)
• Orders (Exponents, Roots)
• Division and Multiplication (from left to right)
• Addition and Subtraction (from left to right)

Let's break down the calculation step by step:

The expression is: \( [\{(18 - 16) + (5 \times 2)\} \div (5 + 1)] \times 1 \)

Step 1: Evaluate the innermost parentheses.

• \( (18 - 16) = 2 \)
• \( (5 \times 2) = 10 \)
• \( (5 + 1) = 6 \)

Substitute these values back into the expression:

\( [\{2 + 10\} \div 6] \times 1 \)

Step 2: Evaluate the curly braces.

• \( \{2 + 10\} = 12 \)

Substitute this value back into the expression:

\( [12 \div 6] \times 1 \)

Step 3: Evaluate the square brackets.

• \( [12 \div 6] = 2 \)

Substitute this value back into the expression:

\( 2 \times 1 \)

Step 4: Perform the final multiplication.

• \( 2 \times 1 = 2 \)

The final value of the expression after applying the symbol substitutions and evaluating is 2.

Conclusion

By carefully substituting the symbols according to the given rules and then following the correct order of operations, we found the value of the expression to be 2.

Revision Table: Symbol Substitution Rules

Additional Information: Order of Operations (BODMAS/PEMDAS)

The order of operations is crucial in evaluating mathematical expressions correctly. It ensures that everyone gets the same answer when solving the same problem. The standard order is:

• B/P: Brackets or Parentheses - Evaluate expressions inside grouping symbols first.
• O/E: Orders or Exponents - Evaluate powers and roots next.
• DM: Division and Multiplication - Perform these operations from left to right.
• AS: Addition and Subtraction - Perform these operations from left to right.

Ignoring the order of operations can lead to incorrect results, especially in expressions involving multiple types of operators and grouping symbols.