Solving Operator Substitution Problems

Let's break down the given expression where the mathematical operators have been redefined. We need to substitute the original operators with the new ones and then solve the expression following the order of operations (BODMAS/PEMDAS).

Understanding the Operator Changes

The question defines the following changes to the standard mathematical operators:

• '+' means subtraction ('-')
• '÷' means addition ('+')
• '<' means multiplication ('*')
• '>' means division ('/')

Rewriting the Expression with New Operators

The original expression given is: 66 &divide; 11 < 9 > (12 > 4) + 5

Substituting the operators based on the rules provided in the question, we get the new expression:

\(66 + 11 * 9 / (12 / 4) - 5\)

Step-by-Step Calculation using BODMAS/PEMDAS

We will now solve this new expression step-by-step following the BODMAS/PEMDAS rule, which dictates the order of operations:

• B/P: Brackets / Parentheses
• O/E: Orders / Exponents
• DM: Division and Multiplication (from left to right)
• AS: Addition and Subtraction (from left to right)

1. Step 1: Solve the expression inside the brackets.
   The innermost part of the expression is within the parentheses: \((12 / 4)\).
   Calculating this gives: \(12 / 4 = 3\)
   The expression now becomes: \(66 + 11 * 9 / 3 - 5\)
2. Step 2: Perform Division and Multiplication from left to right.
   Moving from left to right, the first operation is multiplication: \(11 * 9\).
   Calculating this gives: \(11 * 9 = 99\)
   The expression is now: \(66 + 99 / 3 - 5\)
   Next, we encounter division: \(99 / 3\).
   Calculating this gives: \(99 / 3 = 33\)
   The expression is now: \(66 + 33 - 5\)
3. Step 3: Perform Addition and Subtraction from left to right.
   Moving from left to right, the first operation is addition: \(66 + 33\).
   Calculating this gives: \(66 + 33 = 99\)
   The expression is now: \(99 - 5\)
   Finally, perform the subtraction: \(99 - 5\).
   Calculating this gives: \(99 - 5 = 94\)

Final Value of the Expression

After substituting the operators and solving the expression using the correct order of operations, the value of the expression \(66 &divide; 11 < 9 > (12 > 4) + 5\) is 94.

Revision Table: Operator Substitution Summary

Additional Information: Applying Order of Operations

Problems involving operator substitution require two main steps:

1. First, correctly replace all the given operators in the expression with their new meanings.
2. Second, evaluate the resulting expression by strictly following the standard order of operations (BODMAS/PEMDAS). It is crucial not to apply the operations in the order they appear in the written expression but according to the BODMAS/PEMDAS hierarchy. Always solve brackets first, then powers, then division and multiplication (from left to right), and finally addition and subtraction (from left to right).