Understanding Symbol Substitution in Mathematical Expressions

This problem requires us to evaluate a given mathematical expression after changing the meaning of the arithmetic symbols according to specific rules. We need to carefully substitute the symbols and then calculate the value of the new expression following the standard order of operations.

Given Symbol Substitution Rules

The rules for changing the meaning of symbols are as follows:

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

Original Expression

The original expression provided is:

\(13 + 6 \div 2 \times 32 - 2\)

Applying the Substitution Rules

Now, let's replace each symbol in the original expression with its new meaning as per the given rules:

• The '+' between 13 and 6 becomes '-'.
• The '÷' between 6 and 2 becomes '×'.
• The '×' between 2 and 32 becomes '+'.
• The '-' between 32 and 2 becomes '÷'.

So, the expression transforms from \(13 + 6 \div 2 \times 32 - 2\) to:

\(13 - 6 \times 2 + 32 \div 2\)

Evaluating the Modified Expression using Order of Operations (BODMAS/PEDMAS)

Now we evaluate the new expression \(13 - 6 \times 2 + 32 \div 2\). We must follow the order of operations, commonly remembered by acronyms like BODMAS or PEDMAS.

• B/P: Brackets/Parentheses
• O/E: Orders/Exponents
• D/M: Division/Multiplication (from left to right)
• A/S: Addition/Subtraction (from left to right)

In our expression \(13 - 6 \times 2 + 32 \div 2\), we have Subtraction, Multiplication, Addition, and Division. Following BODMAS/PEDMAS, we perform Division and Multiplication first, from left to right, and then Addition and Subtraction from left to right.

Step-by-Step Calculation:

1. First, perform the Division: \(32 \div 2 = 16\). The expression becomes \(13 - 6 \times 2 + 16\).
2. Next, perform the Multiplication: \(6 \times 2 = 12\). The expression becomes \(13 - 12 + 16\).
3. Now, perform Addition and Subtraction from left to right. First, Subtraction: \(13 - 12 = 1\). The expression becomes \(1 + 16\).
4. Finally, perform the Addition: \(1 + 16 = 17\).

Final Value of the Expression

After applying the symbol substitutions and evaluating the expression according to the order of operations, the final value is 17.

Revision Table: Symbol Substitution Problems

Here’s a quick table summarizing the key steps when tackling symbol substitution problems:

Additional Information: Importance of Order of Operations (BODMAS/PEDMAS)

The order of operations is crucial in mathematics to ensure that everyone gets the same answer when evaluating an expression. Without a standard order, different people might perform calculations in a different sequence and arrive at different results. BODMAS (or PEDMAS) provides this standard sequence:

• Parentheses/Brackets first: Simplify anything inside grouping symbols.
• Exponents/Orders next: Calculate powers and roots.
• Multiplication and Division: Perform these from left to right.
• Addition and Subtraction: Perform these from left to right.

In symbol substitution problems, once the symbols are changed, applying this standard order is key to finding the correct value of the resulting expression.