Understanding Symbol Substitution in Mathematical Operations

This problem requires us to evaluate a mathematical expression after changing the meaning of the operators. We are given a set of rules for symbol substitution and then asked to calculate the value of a specific expression using the new meanings.

Rules for Symbol Substitution

The rules provided are:

• The symbol '+' means '×'.
• The symbol '-' means '÷'.
• The symbol '×' means '+'.
• The symbol '÷' means '-'.

The Expression to Evaluate

The original expression is:

\(512 - 8 + 5 \div 73 \times 92\)

Applying Symbol Substitution

Now, let's rewrite the expression by replacing the original symbols with their new meanings according to the given rules:

• Replace '-' with '÷'.
• Replace '+' with '×'.
• Replace '÷' with '-'.
• Replace '×' with '+'.

Applying these rules to the expression \(512 - 8 + 5 \div 73 \times 92\), we get:

\(512 \div 8 \times 5 - 73 + 92\)

Evaluating the Modified Expression

We need to evaluate the new expression \(512 \div 8 \times 5 - 73 + 92\) following the standard order of operations (BODMAS/PEMDAS), which dictates that we perform Division and Multiplication before Addition and Subtraction, working from left to right.

Step 1: Perform Division

Evaluate \(512 \div 8\):

\(512 \div 8 = 64\)

The expression becomes:

\(64 \times 5 - 73 + 92\)

Step 2: Perform Multiplication

Evaluate \(64 \times 5\):

\(64 \times 5 = 320\)

The expression becomes:

\(320 - 73 + 92\)

Step 3: Perform Subtraction (from left to right)

Evaluate \(320 - 73\):

\(320 - 73 = 247\)

The expression becomes:

\(247 + 92\)

Step 4: Perform Addition (from left to right)

Evaluate \(247 + 92\):

\(247 + 92 = 339\)

The final value of the expression after symbol substitution and evaluation is 339.

Therefore, the value of the expression \(512 - 8 + 5 \div 73 \times 92\) under the given symbol substitution rules is 339.

Revision Table: Key Concepts

Additional Information: Order of Operations Explained

Understanding the order of operations is crucial for correctly evaluating mathematical expressions. Different acronyms are used in different regions, but they all represent the same principle:

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

The key is that Multiplication and Division have equal priority and are performed from left to right as they appear. Similarly, Addition and Subtraction have equal priority and are performed from left to right as they appear, after all multiplication and division are done.

In our problem, after substitution, the expression was \(512 \div 8 \times 5 - 73 + 92\). Following the left-to-right rule for division and multiplication, we did \(512 \div 8\) first, then multiplied the result by 5. Then, for addition and subtraction, we did \(320 - 73\) first, and finally added 92.