The problem asks us to find which of the given equations becomes correct after interchanging the numbers 8 and 2. We need to go through each option, swap the positions of 8 and 2, and then evaluate the modified equation to see if the left side equals the right side.

When evaluating mathematical expressions, we must follow the order of operations, often remembered by the acronym BODMAS or PEMDAS:

• B/P: Brackets or Parentheses
• O/E: Orders (powers, square roots, etc.) or Exponents
• DM: Division and Multiplication (from left to right)
• AS: Addition and Subtraction (from left to right)

Evaluating Each Equation After Interchanging 8 and 2

Let's apply the interchange rule (8 becomes 2 and 2 becomes 8) to each equation and evaluate it.

Analyzing Option 1: Original Equation

The original equation is $7 \div 2 - 8 \times 4 + 1 = 0$.

After interchanging 8 and 2, the equation becomes:

$7 \div 8 - 2 \times 4 + 1$

Now, we evaluate this expression using BODMAS:

1. Perform Division and Multiplication from left to right:
   • $7 \div 8 = 0.875$
   • $2 \times 4 = 8$
2. Substitute these values back into the expression:
   • $0.875 - 8 + 1$
3. Perform Addition and Subtraction from left to right:
   • $0.875 - 8 = -7.125$
   • $-7.125 + 1 = -6.125$

The result is $-6.125$. The equation claims the result should be 0. Since $-6.125 \ne 0$, this equation is incorrect after interchanging 8 and 2.

Analyzing Option 2: Original Equation

The original equation is $2 \times 4 - 8 + 5 \div 1 = 50$.

After interchanging 8 and 2, the equation becomes:

$8 \times 4 - 2 + 5 \div 1$

Now, we evaluate this expression using BODMAS:

1. Perform Division and Multiplication from left to right:
   • $8 \times 4 = 32$
   • $5 \div 1 = 5$
2. Substitute these values back into the expression:
   • $32 - 2 + 5$
3. Perform Addition and Subtraction from left to right:
   • $32 - 2 = 30$
   • $30 + 5 = 35$

The result is 35. The equation claims the result should be 50. Since $35 \ne 50$, this equation is incorrect after interchanging 8 and 2.

Analyzing Option 3: Original Equation

The original equation is $8 \times 4 + 6 - 2 \div 1 = 6$.

After interchanging 8 and 2, the equation becomes:

$2 \times 4 + 6 - 8 \div 1$

Now, we evaluate this expression using BODMAS:

1. Perform Division and Multiplication from left to right:
   • $2 \times 4 = 8$
   • $8 \div 1 = 8$
2. Substitute these values back into the expression:
   • $8 + 6 - 8$
3. Perform Addition and Subtraction from left to right:
   • $8 + 6 = 14$
   • $14 - 8 = 6$

The result is 6. The equation claims the result should be 6. Since $6 = 6$, this equation is correct after interchanging 8 and 2.

Analyzing Option 4: Original Equation

The original equation is $8 \times 4 - 9 \div 3 + 2 = 35$.

After interchanging 8 and 2, the equation becomes:

$2 \times 4 - 9 \div 3 + 8$

Now, we evaluate this expression using BODMAS:

1. Perform Division and Multiplication from left to right:
   • $2 \times 4 = 8$
   • $9 \div 3 = 3$
2. Substitute these values back into the expression:
   • $8 - 3 + 8$
3. Perform Addition and Subtraction from left to right:
   • $8 - 3 = 5$
   • $5 + 8 = 13$

The result is 13. The equation claims the result should be 35. Since $13 \ne 35$, this equation is incorrect after interchanging 8 and 2.

Based on the analysis, only Option 3 becomes a correct equation after interchanging the numbers 8 and 2.

Summary of Results

Therefore, interchanging 8 and 2 makes the equation in Option 3 correct.

Revision Table: Key Concepts

Additional Information: Applying Order of Operations

Understanding the order of operations is crucial for correctly solving mathematical expressions, especially those involving multiple operations like addition, subtraction, multiplication, and division.

• Always start with operations inside brackets or parentheses.
• Next, calculate powers or exponents.
• Then, perform multiplication and division. If both are present, work from left to right.
• Finally, perform addition and subtraction. If both are present, work from left to right.

Mistakes in applying this order can lead to incorrect results. Practice with different types of expressions helps in mastering this rule.