Understanding Operator Substitution Problems

This type of question involves changing the meaning of standard mathematical operators according to a given set of rules. After substituting the operators based on the rules, we need to evaluate the new expression using the standard order of operations (BODMAS/PEMDAS).

Step 1: Identify the Operator Substitution Rules

The question provides the following substitution rules:

• ‘+’ represents ‘×’
• ‘÷’ represents ‘+’
• ‘-’ represents ‘÷’
• ‘×’ represents ‘-’

Step 2: Write Down the Original Expression

The expression given is:

\(8 + 4 \div 15 - 3 \times 1\)

Step 3: Substitute the Operators According to the Rules

Now, we will replace each original operator in the expression with its new meaning as per the rules:

• Replace ‘+’ with ‘×’
• Replace ‘÷’ with ‘+’
• Replace ‘-’ with ‘÷’
• Replace ‘×’ with ‘-’

Applying these rules to the original expression \(8 + 4 \div 15 - 3 \times 1\):

• 8 (replace +) becomes 8 \(\times\)
• 4 (replace ÷) becomes 4 +
• 15 (replace -) becomes 15 \(\div\)
• 3 (replace ×) becomes 3 -
• 1 remains 1

The new expression after substitution is:

\(8 \times 4 + 15 \div 3 - 1\)

Step 4: Evaluate the Substituted Expression Using BODMAS/PEMDAS

We evaluate the new expression \(8 \times 4 + 15 \div 3 - 1\) following the order of operations:

BODMAS/PEMDAS Order:

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

Let's evaluate step-by-step:

1. Division: First, calculate \(15 \div 3\).

   \(15 \div 3 = 5\)

   The expression becomes: \(8 \times 4 + 5 - 1\)

2. Multiplication: Next, calculate \(8 \times 4\).

   \(8 \times 4 = 32\)

   The expression becomes: \(32 + 5 - 1\)

3. Addition and Subtraction: Finally, perform addition and subtraction from left to right.

   \(32 + 5 = 37\)

   Then, \(37 - 1 = 36\)

The final value of the expression is 36.

Conclusion on Operator Substitution

By correctly substituting the operators according to the given rules and then applying the standard order of operations (BODMAS/PEMDAS), we found the value of the expression \(8 + 4 \div 15 - 3 \times 1\) with the new rules to be 36.

Revision Table: Operator Substitution Rules

Additional Information: Understanding BODMAS/PEMDAS Order of Operations

When solving mathematical expressions, it's crucial to follow a specific order to ensure consistency and accuracy. This order is commonly known as BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers, square roots), Division, Multiplication, Addition, Subtraction.
• PEMDAS: Parentheses, Exponents, Multiplication, Division, Addition, Subtraction.

Both acronyms represent the same order of priority. Division and Multiplication have equal priority and are performed from left to right as they appear in the expression. Similarly, Addition and Subtraction have equal priority and are performed from left to right.

Failing to follow this order will likely result in an incorrect answer when evaluating expressions.