Evaluating Mathematical Expressions with Symbol Substitution

This question requires us to first understand the given rules for substituting mathematical operators and then evaluate the expression using the correct order of operations.

Understanding the Symbol Substitution Rules

The question provides specific rules for changing the standard mathematical symbols. Let's list them clearly:

• The symbol '–' stands for '×' (Multiplication).
• The symbol '×' stands for '+' (Addition).
• The symbol '+' stands for '÷' (Division).
• The symbol '÷' stands for '–' (Subtraction).

Applying the Substitution Rules to the Expression

The given expression is:

$516 \ndash 2558 + 172 \times 12 \div 7086$

Now, we replace each original symbol with its new meaning according to the rules:

• '–' is replaced by '×'
• '+' is replaced by '÷'
• '×' is replaced by '+'
• '÷' is replaced by '–'

The expression after applying the substitution rules becomes:

$516 \times 2558 \div 172 + 12 - 7086$

Evaluating the Modified Expression Using Order of Operations

We need to evaluate the new expression $516 \times 2558 \div 172 + 12 - 7086$ following the BODMAS or PEMDAS rule (Brackets/Parentheses, Orders/Exponents, Division and Multiplication from left to right, Addition and Subtraction from left to right).

The operations in our expression are Multiplication ($\times$), Division ($\div$), Addition (+), and Subtraction (-).

According to BODMAS, we first perform Division and Multiplication from left to right. In the expression $516 \times 2558 \div 172$, we have multiplication followed by division. While strict left-to-right would be $(516 \times 2558) \div 172$, noticing that 516 is divisible by 172 ($516 \div 172 = 3$), we can simplify the calculation by grouping $(516 \div 172) \times 2558$, which is mathematically equivalent for multiplication and division.

Let's perform the calculation step-by-step:

1. Step 1: Perform Division and Multiplication.

   Evaluate $(516 \div 172) \times 2558$

   $516 \div 172 = 3$

   Now, multiply the result by 2558:

   $3 \times 2558 = 7674$

   The expression now becomes:

   $7674 + 12 - 7086$

2. Step 2: Perform Addition and Subtraction from left to right.

   Next, perform the addition:

   $7674 + 12 = 7686$

   The expression is now:

   $7686 - 7086$

3. Step 3: Perform the remaining Subtraction.

   $7686 - 7086 = 600$

The final value of the expression after applying the symbol substitutions and evaluating is 600.

Comparing with Options

Let's check the given options:

• 500
• 400
• 600
• 700

Our calculated value, 600, matches one of the options.

Therefore, the correct value is 600.

Revision Table: Key Concepts Reviewed

Additional Information: Operator Precedence and Grouping

In mathematical expressions, the order of operations (BODMAS/PEMDAS) ensures that everyone gets the same result when evaluating the same expression. When you have multiplication and division together, they are performed from left to right.

For example, $A \times B \div C$ is evaluated as $(A \times B) \div C$. However, if $A$ is divisible by $C$, it is also mathematically correct to evaluate it as $(A \div C) \times B$. This reordering can sometimes simplify calculations, especially when aiming for integer results from friendly numbers, as seen in this problem where $516$ is divisible by $172$.

Similarly, for addition and subtraction, they are performed from left to right.

$A - B + C$ is evaluated as $(A - B) + C$.