Solving Equations by Interchanging Numbers

The problem asks us to determine which of the given equations becomes mathematically correct when the numbers 2 and 8 are interchanged within that equation. We need to substitute 8 for every 2 and 2 for every 8 in each option and then evaluate the resulting expression using the standard order of operations (like BODMAS or PEMDAS).

Evaluating Option 1 after Number Interchange

The original equation is: \(8 \div 1 + 4 \times 2 - 3 = 18\)

After interchanging 2 and 8, the equation becomes:

\(2 \div 1 + 4 \times 8 - 3\)

Now, let's evaluate this expression:

• Division: \(2 \div 1 = 2\)
• Multiplication: \(4 \times 8 = 32\)
• The expression is now: \(2 + 32 - 3\)
• Addition: \(2 + 32 = 34\)
• Subtraction: \(34 - 3 = 31\)

So, after interchanging 2 and 8, the equation becomes \(31 = 18\), which is incorrect.

Evaluating Option 2 after Number Interchange

The original equation is: \(7 \times 2 + 6 \div 3 - 8 = 60\)

After interchanging 2 and 8, the equation becomes:

\(7 \times 8 + 6 \div 3 - 2\)

Now, let's evaluate this expression:

• Multiplication: \(7 \times 8 = 56\)
• Division: \(6 \div 3 = 2\)
• The expression is now: \(56 + 2 - 2\)
• Addition: \(56 + 2 = 58\)
• Subtraction: \(58 - 2 = 56\)

So, after interchanging 2 and 8, the equation becomes \(56 = 60\), which is incorrect.

Evaluating Option 3 after Number Interchange

The original equation is: \(2 \times 4 - 8 + 6 \div 3 = 33\)

After interchanging 2 and 8, the equation becomes:

\(8 \times 4 - 2 + 6 \div 3\)

Now, let's evaluate this expression:

• Multiplication: \(8 \times 4 = 32\)
• Division: \(6 \div 3 = 2\)
• The expression is now: \(32 - 2 + 2\)
• Subtraction: \(32 - 2 = 30\)
• Addition: \(30 + 2 = 32\)

So, after interchanging 2 and 8, the equation becomes \(32 = 33\), which is incorrect.

Evaluating Option 4 after Number Interchange

The original equation is: \(8 \times 4 + 6 - 2 \div 1 = 6\)

After interchanging 2 and 8, the equation becomes:

\(2 \times 4 + 6 - 8 \div 1\)

Now, let's evaluate this expression using the order of operations:

• Multiplication: \(2 \times 4 = 8\)
• Division: \(8 \div 1 = 8\)
• The expression is now: \(8 + 6 - 8\)
• Addition: \(8 + 6 = 14\)
• Subtraction: \(14 - 8 = 6\)

So, after interchanging 2 and 8, the equation becomes \(6 = 6\), which is correct.

Therefore, interchanging the numbers 2 and 8 makes the fourth equation correct.

Revision Table: Order of Mathematical Operations (BODMAS/PEMDAS)

Additional Information: Understanding Mathematical Equations

A mathematical equation is a statement that shows that two expressions are equal. It uses an equals sign (\(=\)). To solve or verify an equation, we perform the operations on one or both sides until we can confirm if the equality holds true. When evaluating an expression with multiple operations, it's crucial to follow a specific order to get the correct result. This order is commonly remembered using acronyms like BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) or PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

In this problem, interchanging numbers means swapping their positions. If an equation originally has a '2', it becomes an '8', and if it has an '8', it becomes a '2'. This changes the entire value of the expression, and we must re-evaluate it completely to see if the equality still holds.