Let's break down this problem step-by-step to evaluate the given expression based on the new meanings of the symbols.

The problem asks us to find the value of an expression where the standard mathematical operators (+, -, ×, ÷) have been assigned new meanings.

Understanding Symbol Substitution Rules

The rules for symbol substitution are as follows:

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

Applying Symbol Substitutions to the Expression

The original expression is:

$$ \frac{(24 – 8 \div 6 + 4 \times 7)}{(8 + 3 \times 19)} $$

Now, let's substitute the symbols according to the given rules:

• The '-' between 24 and 8 becomes '÷'.
• The '÷' between 8 and 6 becomes '+'.
• The '+' between 6 and 4 becomes '×'.
• The '×' between 4 and 7 becomes '-'.
• In the denominator, the '+' between 8 and 3 becomes '×'.
• The '×' between 3 and 19 becomes '-'.

After substitution, the new expression becomes:

$$ \frac{(24 \div 8 + 6 \times 4 - 7)}{(8 \times 3 - 19)} $$

Evaluating the Substituted Expression

We need to evaluate this new expression following the order of operations (BODMAS/PEMDAS: Brackets, Orders, Division/Multiplication, Addition/Subtraction).

Evaluating the Numerator

The numerator is \(24 \div 8 + 6 \times 4 - 7\).

Step 1: Perform division and multiplication from left to right.

• \(24 \div 8 = 3\)
• \(6 \times 4 = 24\)

The expression for the numerator now is \(3 + 24 - 7\).

Step 2: Perform addition and subtraction from left to right.

• \(3 + 24 = 27\)
• \(27 - 7 = 20\)

So, the value of the numerator is 20.

Evaluating the Denominator

The denominator is \(8 \times 3 - 19\).

Step 1: Perform multiplication.

• \(8 \times 3 = 24\)

The expression for the denominator now is \(24 - 19\).

Step 2: Perform subtraction.

• \(24 - 19 = 5\)

So, the value of the denominator is 5.

Final Calculation

Now we have the value of the numerator and the denominator. The expression is:

$$ \frac{\text{Numerator}}{\text{Denominator}} = \frac{20}{5} $$

Performing the division:

$$ 20 \div 5 = 4 $$

The final value of the expression after applying the symbol substitutions is 4.

Revision Table: Symbol Substitutions

Additional Information: Order of Operations

When evaluating mathematical expressions, it's crucial to follow the correct order of operations to ensure consistency and accuracy. A commonly used acronym for this order is BODMAS or PEMDAS.

• BODMAS: Brackets, Orders (powers, square roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).
• PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

In this problem, after substituting the symbols, we applied this order to solve the expression within the numerator and denominator before performing the final division.