Solving Equations with Symbol Interchange

This problem requires us to evaluate a mathematical expression after interchanging certain operators according to given rules. We are given an equation and rules for swapping the '+' with '×' and '-' with '÷'. Let's break down the steps to solve this.

Understanding the Symbol Interchange Rules

The problem states the following interchange rules:

• The '+' symbol is interchanged with the '×' symbol. This means wherever we see '+', we should use '×', and wherever we see '×', we should use '+'.
• The '-' symbol is interchanged with the '÷' symbol. This means wherever we see '-', we should use '÷', and wherever we see '÷', we should use '-'.

We can represent these interchanges in a simple table:

Applying the Interchange to the Equation

The original equation given is:

$$2 + 6 \div 8 - 4 \times 14 = ?$$

Now, let's apply the symbol interchange rules to this equation:

• The '+' between 2 and 6 becomes '×'.
• The '÷' between 6 and 8 becomes '-'.
• The '-' between 8 and 4 becomes '÷'.
• The '×' between 4 and 14 becomes '+'.

So, the new equation after the symbol interchange is:

$$2 \times 6 - 8 \div 4 + 14 = ?$$

Solving the Transformed Equation

Now we need to solve the new equation: $2 \times 6 - 8 \div 4 + 14$. We must follow the order of operations (BODMAS/PEMDAS) which dictates the sequence of calculations: Brackets, Orders (powers and square roots), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Let's solve step-by-step:

1. Division: Perform the division first. $8 \div 4 = 2$. The equation becomes: $$2 \times 6 - 2 + 14$$
2. Multiplication: Next, perform the multiplication. $2 \times 6 = 12$. The equation becomes: $$12 - 2 + 14$$
3. Subtraction and Addition: Finally, perform the subtraction and addition from left to right. $$12 - 2 = 10$$ $$10 + 14 = 24$$

So, the value of the expression after the symbol interchange is 24.

Final Result

After interchanging the operators and solving the new equation according to the order of operations, we find that the value of the expression is 24.

Revision Table - Symbol Interchange

Here's a quick look at how the operations changed:

Putting it all together: $2 \times 6 - 8 \div 4 + 14$

Additional Information - Order of Operations (BODMAS/PEMDAS)

Understanding the order of operations is crucial for solving mathematical expressions correctly. The widely used mnemonics BODMAS and PEMDAS help remember the sequence:

• BODMAS:
  • Brackets
  • Orders (Powers, Roots)
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)
• PEMDAS:
  • Parentheses
  • Exponents (Powers, Roots)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

In our problem, we first performed division ($8 \div 4$), then multiplication ($2 \times 6$), and finally subtraction and addition from left to right ($12 - 2 + 14$).